Galactic Library - User contributions [en]

Black Hole Engineering

2026-08-01T23:05:13Z

Lwcamp: /* Gravity Generation */

Ah, black holes. Flaws in the fabric of the universe. Empty voids from which nothing can return. The ultimate unknowable mystery.

But what are they good for?

== Basics ==

Lets start with a brief introduction to black holes.

Things like planets and stars and other massive bodies have gravitational fields around them that tend to draw things toward them and trap stuff on them. In order to get away from such a body, you need to shoot yourself off it with a speed higher than its escape velocity. If you don't have that much speed, you can't get away. When you pack enough mass into a small enough volume, its gravity gets so high that the escape velocity is higher than the speed of light. Because nothing can go faster than light, nothing can escape. This is a black hole.

[[File:Black_hole_Schwarzschold.png|thumb|A diagram of the features of the Schwarzschild geometry, showing the event horizon (white circle) and central singularity.]]
That's the description motivated by Newtonian gravity, anyway. But when gravity gets really strong Newtonian gravity breaks down and you need to use general relativity instead. Curiously, the size and mass where light (and everything else) is trapped is the same as the Newtonian case. But instead of light and other things flying out, looping around, and coming back space-time gets strange. At the critical distance where light would be trapped you get a surface called an event horizon. Nothing that passes into an event horizon can ever get back out again. The gravity at and inside the event horizon is so strong that it rotates space and time enough that the direction inwards toward the center becomes your inevitable future. You can no more resist going toward the middle of the hole that you can avoid seeing what fate awaits you.

An uncharged and non-rotating black hole at rest is described by the Schwarzschild geometry. The radius of its event horizon is the Schwarzschild radius
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
rS = 2 G M / c2
</div>
where M is the mass of the black hole, G is the gravitational constant, and c is the speed of light in vacuum. As an example, a black hole with a mass of 100 million metric tons would have a Schwarzschild radius of 1.48 × 10-16 meters. This is slightly under one-fifth the radius of a proton.

At the center of a black hole lies a point at which our description of physics breaks down, called the singularity. While of immense scientific interest, it is irrelevant for engineering because it is inside the event horizon so it cannot possibly affect us or our environment.

Energy is conserved, and mass is a manifestation of energy that is not moving. So when matter or radiation is swallowed by the hole, its energy is added to that of the hole and the mass of the hole increases by E = m c2 to reflect this.

Charged and/or rotating black holes get more complicated:

[[File:Black_hole_Reissner-Nordstrom.png|thumb|A diagram of the features of the Reissner–Nordström geometry, showing the inner and outer event horizons (white solid circle), the location of the Schwarzschild event horizon for a black hole of equal mass but no charge (outer dashed circle), the location of the extremal horizon at half the Schwarzschild radius (inner dashed circle), and the central singularity.]]
=== Charged black holes ===
Charge is conserved. If electrically charged matter falls into a black hole, the hole itself will acquire the charge. The charge produces an electric field radiating away from the hole, much as the mass of the hole also creates a gravitational field.

A charged black hole is not expected to last long in the real world. The charge will draw in particles of the same charge and repel particles of the opposite charge, tending to neutralize it in any environment where any matter exists (even tenuous space plasma)<ref name="Gibbons 1974)>G. W. Gibbons, "Vacuum Polarization and the Spontaneous Loss of Charge by Black Holes", Commun. math. Phys. 44, 245-264 (1975)</ref>. An engineer intending to work with charged black holes will need to ensure it exists in a high vacuum environment and perhaps add additional features to slow the rate of neutralization or methods to top off its charge by adding additional charged particles. As will be seen later, a charged black hole will also spontaneously shed particles to get rid of its charge<ref name="Carter 1974">B. Carter, "Charge and Particle Conservation in Black-Hole Decay", Physical Review Letters Vol. 33 No. 9, pg. 558-561 (1974)</ref>, making keeping it charged even harder.

A charged black hole is described by the Reissner–Nordström geometry. For the same mass, a net charge will cause the event horizon to shrink. A second horizon will form inside the first horizon that will grow with increasing charge, although for the purpose of black hole engineering this is not particularly relevant because anything going through the outer horizon is lost to our universe one way or the other.

As charge is added, the two horizons approach each other until they meet at a distance of half of the Schwarzschild radius calculated for an uncharged hole of the same mass, with a charge of
<div align=center>Q = M √[4 π ε0 G] = M 8.61722×10-11 C/kg.</div>
This forms one example of an extremal black hole. In this case the mass-energy of the charge, considered as a sphere of charge located in a thin shell at the event horizon, makes up the entirety of the mass of the black hole with no room left over for mass from any matter or other kinds of energy. It is thus easy to see that simply adding more and more charge to a black hole that is not yet extremal cannot actually form an extremal black hole. Likewise, adding charge to an already extremal black hole at most keeps it extremal as you add electrostatic mass-energy that keeps up with the increase in charge (and all physical charged particles also have their own mass, which would take it out of the extremal condition). Some theories suggest that it is impossible for extremal black holes to form by any physical process, although these theories have been disputed.

[[File:Black_hole_Kerr.png|thumb|A diagram of the features of the Kerr geometry, showing the inner and outer event horizons (white ovals), outer boundary of the ergosphere (red oval), and ring singularity(dotted oval).]]

=== Rotating black holes ===
You get a rotating black hole when the hole devours things which have angular momentum and that angular momentum becomes a property of the hole. Black holes have no surface features so you can't actually see things on the hole going around. But the angular momentum manifests in other physically observable ways.

Most astrophysical processes that lead to the formation of black holes involve the collapse or collisions of rotating bodies with non-zero angular momentum. Hence it is expected that all naturally occurring black holes are born rotating. As we will see later, they may not remain rotating but large rotating holes are likely to remain rotating for long periods of time.

Massive rotating bodies exhibit a process called frame dragging, and rotating black holes are no exception. Frame dragging is a gravitational analogue of magnetic induction from moving electric charges. It induces motion in space-time near the body co-rotating with the body and objects therein will be moved along with the space-time. Because space-time is dragged faster near the body than far from it, a stationary object in a free-fall orbit around the hole will appear to be rotating in the opposite direction to the hole to a distant observer even though it is in an inertial reference frame.

A rotating black hole is described by the Kerr geometry. This has some similar behavior to the Reissner–Nordström geometry of charged black holes. You get the formation of an inner horizon that grows with increased rotation, and the outer horizon shrinks. Different from charged holes is that the singularity at the center forms a ring rather than a point. None of this is of any interest to the engineer, as it is all hidden behind an event horizon and cannot affect our world.

Also similar to charged black holes, a hole that is spinning fast enough can become extremal such that the spin alone is providing the energy for its mass term when the angular momentum J is
<div align=center> J = M2 G / c = M2 2.22615×10-19 m2/kg/s.</div>

Of more interest however, is that you get a region outside of the event horizon where it is impossible to stop moving. Here, frame dragging is so extreme that space-time is moving around the black hole faster than the speed of light. This region is called the ergosphere. Similar to how once you go past the event horizon time rotates so that your future is toward the center of the hole, in the ergosphere time rotates so that your future is in the direction of the hole's spin. You can no more come to a stop or go the other direction than you can go back in time.

=== Charged and rotating black holes ===
A black hole with both charge and angular momentum behaves much like you would expect from the solutions for charged black holes and rotating black holes. You get an ergosphere, frame dragging, electric field, and the possibility of extremal black holes. Extremal holes occur when
<div align=center> M2 - (J c / (G M))2 - (Q / √ [4 π ε0 G])2 = 0.</div>
The new feature is the presence of a magnetic field whose magnetic axis is aligned with the spin axis. For a black hole with charge Q, angular momentum J, and mass M, the magnetic moment m (as measured in the far-field) is
<div align=center> m = Q J / M</div>
This black hole is described by the Kerr-Newman geometry. The mathematics of this geometry allow for the event horizon to disappear and the ring singularity to be displayed to the world. However, to obtain this condition you need to go past the extremal case, which is generally thought to be physically impossible.

=== Caveats ===
All the above descriptions of black holes assumes a distribution of mass and charge that does not change with time. That is, it is static. It may be moving, as with the case of a rotating black hole, but the distribution of rotating stuff doesn't change. It may also be moving if you shift to a frame of reference where the hole is not at rest, but you can always find a frame of reference where the hole is at rest in the sense that it has no net linear momentum (and, in a more practical sense, isn't going anywhere. This also means that the occasionally encountered idea of "accelerate an object to such a high speed that it turns into a black hole" simply doesn't work and is not consistent with physics). If you have a static hole, it's properties are entirely defined by just the three quantities of its mass, charge, and angular momentum. Any two static black holes with these three quantities the same will be identical in every respect. To describe this, physicists use the somewhat odd terminology that "the black hole has no hair"; hair being things that do not directly derive from mass, spin, or charge.

Not all black holes need be static. At the moment of creation by the collision of two supermassive objects, for example, a black hole will momentarily have an event horizon that is elongated and wobbly. That is, it has "hair." However, it rapidly radiates gravitational waves until all its hair is shed and it settles down to a static state.

All of the above descriptions of different kinds of black holes assume that if you go far enough away from the black hole, space-time settles down into the ordinary mostly flat space-time where Newtonian gravity works and planets and satellites have regular orbits and geometry works like you would expect and things behave like we would otherwise naively expect them to. This is called asymptotic flatness, defined by the idea that if you go far enough away from the hole in any direction space-time will get as arbitrarily close to flat with increasing distance. Asymptotic flatness is a good approximation of our universe on scales up to and beyond galactic clusters. If you are only dealing with engineering projects within a single galactic cluster, you can generally assume that asymptotic flatness holds. There has been some work on black holes in universes that are not asymptotically flat, but we will not concern ourselves with that here as it is unlikely to be of relevance to engineering tasks.

The initial justification for nothing getting past the event horizon was that it would have to move faster than the speed of light, and nothing can move faster than light. But many science fiction works feature methods whereby information or objects (usually spacecraft) can go faster than light (FTL). Could a faster than light starship escape from inside the event horizon of a black hole? Possibly. It depends in the implementation, but under relativity FTL motion automatically implies time travel. And all of the results of relativity that inside a black hole the future is towards the center of the hole rather than forward in time would similarly be un-done by time traveling FTL. Likewise, your FTL spacecraft could likely go backwards around the ergosphere, if that's your thing. The article on [[Wormholes#Dropping_a_wormhole_into_a_black_hole|wormholes]] covers some of the details for wormholes interacting with black holes, illustrating one way to get information out of a black hole's event horizon and the difficulty of implementing it. This could, in principle, allow access to the interior of black holes that we formerly ignored. Such as using rotating black holes as a time machine (but we can already do that if we can get there and out in the first place) or as wormholes to other universes.

== Acquiring a black hole ==

If you want to do things with a black hole, first you need to get one. Here, we discuss various ways you might get your grubby little mitts on one of these monstrosities of physics.

=== Supermassive black holes ===

At the center of each galaxy resides a gigantic black hole with a mass ranging from tens of thousands to billions of times more massive than our sun. To acquire a supermassive black hole, you'll need to travel to the center of a galaxy. The mass of these black holes means that they can be difficult to take with you and you might need to do your work where you originally found the hole.

=== Stellar mass black holes ===

Stars do not readily form black holes, despite their immense gravity trying to pull them together. When you try to squish a star down to make a black hole, that squishing makes its temperature rise. A rising temperature makes the star hot, which increases its pressure, which pushes back against your squishing. This can be very annoying when trying to make a black hole. You need to wait for that thermal energy to radiate away. But even worse the hot, dense interior of the stuff you are squishing makes a great environment for thermonuclear fusion to occur. This fusion creates heat and you have to wait for that heat to radiate away, too, before you can get the stuff to contract down further.

But even after everything has fused, there can be limits to your squishing. As the stuff in the stars gets denser and denser, you get to a point where all the low energy places to park the electrons are all taken up. To make the star denser, you need to put the electrons in higher energy states. This takes energy to get the electrons there, which means even more pressure pushing back. This is a state of matter called electron degenerate matter, and the resulting object is called a white dwarf star. For stars with a mass of about 1.44 times the mass of our sun or less, the electron degeneracy pressure keeps the star from getting small enough to form a black hole. This threshold mass is called the [https://en.wikipedia.org/wiki/Chandrasekhar_limit|Chandrasekhar limit].

Okay, so you get together a star with more mass than the Chandrasekhar limit. Now you're good to go, right? You have enough mass to just push past that annoying electron degeneracy pressure. Not so fast, buckaroo! Once the energy of the electrons gets high enough it becomes energetically favorable for them to combine with protons to form neutrons (this happens for energies of about 0.78 MeV for free protons). Now you get a dense ball of neutrons and have the same issue that you previously had with electrons, but worse. This mass of degenerate neutrons is called a neutron star. It takes a mass of a bit more than twice the mass of the sun to overcome the pressure of degenerate neutron matter (the [https://en.wikipedia.org/wiki/Tolman%E2%80%93Oppenheimer%E2%80%93Volkoff_limit|Tolman–Oppenheimer–Volkoff limit]). But once you do that, there is nothing preventing the remains of the star from squishing down into a black hole under its gravity.

All of this is to show that it can be hard to make a black hole from stars. And that's not even considering other complications, like how stars tend to shed a lot of their mass as they collapse so you need considerably more mass than the Tolman–Oppenheimer–Volkoff limit to make your black hole.

But do not fret! The universe has been kind enough to make black holes out of stars for you. There has been enough time for many of the more massive stars to burn through their fusion fuel and collapse to make black holes. Even those that remain as neutron stars sometimes run in to other neutron stars and form black holes.

Needless to say, a stellar mass black hole is going to be very heavy. If your civilization cannot move stars around, this will be a location you go to rather than a piece of equipment you carry around with you.

Black holes may not be uncommon in the universe, but they can be dark (it's in their name, after all). So stellar mass black holes can be hard to find. But there are ways. If the black hole has a stellar companion, it can siphon gas from the companion to produce a bright x-ray source. If a dark black hole passes in front of another star, it can make that star temporarily brighter through gravitational lensing. So you may be able to locate a stellar mass black hole – we have already located a great many of them. The problem of getting to said stellar mass black hole is still an unsolved problem, however.

=== Primordial black holes ===

There are no known natural processes to make black holes in our universe with a mass less than the Tolman–Oppenheimer–Volkoff limit. However, it is possible that our universe might have been born with small black holes already in place. These primordial black holes could potentially be significantly smaller than stellar mass black holes. Primordial black holes with initial masses of less than five hundred million (5×108) tons will have evaporated by now<ref>MacGibbon, Jane H.; Carr, B. J.; Page, Don N. (2008). "Do Evaporating Black Holes Form Photospheres?". Physical Review D. 78 (6) 064043. arXiv:[https://arxiv.org/abs/0709.2380 0709.2380]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2008PhRvD..78f4043M abs/2003PhTea..41..299L 2008PhRvD..78f4043M]. doi:[https://doi.org/10.1103%2FPhysRevD.78.064043 10.1103/PhysRevD.78.064043]. S2CID [https://api.semanticscholar.org/CorpusID:119230843 119230843]</ref> (see below for why black holes evaporate). Some primordial black holes with masses slightly above this limit will survive to the present day with their masses since reduced to below this limit by the intervening evaporation. However, it does mean that black holes with mass smaller than this are going to be quite rare the wild.

It is not necessary for primordial black holes to be small<ref>Andi Hektor, Gert Hütsi and Martti Raidal, "Constraints on primordial black hole dark matter from Galactic center X-ray observations", Astronomy & Astrophysics Vol. 618, article no. A139 (2018) https://doi.org/10.1051/0004-6361/201833483</ref>. They could have initially formed at any size. Indeed, there has been discussion among the scientific community that the seeds of supermassive black holes were primordial black holes which would necessarily have been of large size.

Surviving primordial black holes that are not supermassive black holes would contribute to the dark matter of the universe<ref>Bernard Carr, Kazunori Kohri, Yuuiti Sendouda, and Jun'ichi Yokoyama, "Constraints on Primordial Black Holes", arXiv:2002.12778 [astro-ph.CO] https://arxiv.org/abs/2002.12778</ref>. Indeed, it is possible that most of the universe's dark matter consists of these primordial black holes. Ocasionally, a small primordial black hole might pass through a solar system and be detected by its minute gravitational effects on planetary orbits<ref>Valentin Thoss and Andreas Burkert, "Primordial Black Holes in the Solar System", arXiv:2409.04518 [astro-ph.EP] https://arxiv.org/abs/2409.04518</ref>.

=== Artificial black holes ===

If you can't find a hole, maybe you can make one. If your culture is capable of assembling massive stars and you're willing to wait a few tens or hundreds of millions of years, this is something that can be done. However, if you're looking to make holes of sub-stellar size, no one today has even the faintest idea of how it could be done.

For quite a while, one of the favorite ideas was a method called a kugelblitz<ref name="Crane_Westmoreland">L. Crane and S. Westmoreland, "Are Black Hole Starships Possible" https://arxiv.org/abs/0908.1803</ref>. Technically, this can be any arrangement of radiant energy or energy made of fields that surpasses the Schwarzschild critereon and forms a horizon, but since the development of the laser one of the favorite kugelblitzes has been to shine many enormously powerful laser pulses into a tiny spot. When the laser pulses simultaneously reach the focal spot, their combined energy is sufficient to form a black hole.

Unfortunately, it doesn't work<ref>Álvaro Álvarez-Domínguez, Luis J. Garay, Eduardo Martín-Martínez, and José Polo-Gómez, "No black holes from light", arXiv:2405.02389 [gr-qc]
https://doi.org/10.48550/arXiv.2405.02389; Physical Review Letters 133, 041401 (2024)
https://doi.org/10.1103/PhysRevLett.133.041401</ref><ref>Ball, Philip (July 26, 2024). "Black Holes Can't Be Created by Light". Physics. American Physical Society (APS). Retrieved June 22, 2025. https://physics.aps.org/articles/v17/119</ref>. Before the light can get concentrated enough to self-gravitate into a black hole, it gets intense enough for light to start interacting with light. This scatters the light out of the beam, preventing the light from focusing tightly enough to form a black hole.

So that's the current state of the art. If there are ways to make small black holes, we haven't thought of them yet.

== Energy ==

=== Hawking radiation ===

Famously, nothing that goes into a black hole can ever come back out again. But something comes out. For it turns out that black holes have a temperature and that, like everything with a temperature, they emit radiation. In fact, being perfectly black, they radiate as a perfect black body. This radiation is called Hawking radiation after its discoverer, physicist [https://en.wikipedia.org/wiki/Stephen_Hawking Stephen Hawking]. For normal sized black holes, those the size of stars or galaxies, this temperature is very small and the radiation power is absolutely minuscule. But the smaller the hole, the hotter it gets and the more power it radiates. For a Schwarzschild black hole with mass M, the Hawking temperature TH is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
TH = &hbar; c3 / (8 π G kb M)
</div>
where &hbar; is Planck's constant, π is the circle constant, and kb is Boltzmann's constant. Curiously, this means that the wavelengths around the peak emission of light in its spectrum is near the size of its event horizon. The power radiated by a hole of this temperature in the form of electromagnetic radiation is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
PH = &hbar; c6 / (15360 π (G M)3).
</div>
However, there are additional forms of radiation beyond electromagnetic energy which will add to this radiated power. If the black hole's temperature (in units of energy, so multiply the temperature by the Boltzmann constant to get the units right) is of the same order or higher than the rest mass-energy of a type of particle, that type of particle will also be emitted. The lowest mass particles known that are not electromagnetic radiation are neutrinos. Neutrinos are slippery elusive little fellows and we still don't know their rest masses, but an upper bound on the rest mass of the lightest neutrino species is approximately 0.1 eV. This corresponds to a temperature of 1160 K and a black hole mass of about a hundred thousand trillion (1017) tons. Temperatures higher than this and masses lower than this will need to take neutrino radiation into account. A black hole with a mass of less than twenty billion (2×1010) tons at a temperature of 6 billion kelvin will be radiating electrons and positrons. As the mass continues to decrease additional particle types such as muons and pions will start to contribute to the radiation; at even higher temperatures quarks and gluons will be produced that decay into particle jets creating various hadrons. Gravitational waves will also be radiated away at all temperatures similarly to electromagnetic radiation. The fraction of radiation coming off as various particle types is shown in the table below for black holes large enough to have insignificant muon, pion, and heavier particle radiation.
<table border=1> <tr><td align=center>
<table border=1>
<tr><td>Mass (tons) <td> >> 1 × 1017 <td> << 1 × 1017 & >> 2 × 1010 <td> << 2 × 1010 & >> 1 × 108
<tr><td>Temperature (K) <td> << 1200 <td> >> 1200 & << 6 × 109 <td> >> 6 × 109 & << 1.2 × 1012
<tr><td>Temperature (eV) <td> << 0.1 <td> >> 0.1 & << 500,000 <td> >> 500,000 & << 100,000,000
<tr><td>Electromagnetic fraction <td> 90% <td> 11.8% <td> 7.6%
<tr><td>Gravitational fraction <td> 10% <td> 1.4% <td> 0.9%
<tr><td>Neutrino fraction <td> 0 <td> 86.8% <td> 55.7%
<tr><td>Electron & Positron fraction <td> 0 <td> 0 <td> 35.8%
</table>
<tr><td>Fraction of power emitted as different kinds of radiation as a function of mass for larger mass black holes<ref>D. N. Page, "Particle emission rates from a black hole: Massless particles from an uncharged, nonrotating hole", Physical Review D Vol. 13, No. 2, pg. 198-206, (1976)</ref>. For black holes smaller than 1 × 108 tons, the radiation doesn't so neatly separate with many new kinds of radiation coming on-line without as obvious separations between them. Near the threshold masses, there is a gradual transition from one radiation scheme to another as the temperature gets high enough to occasionally excite the new particle type over the existence threshold.
</table>

The radiated energy comes from the black hole's mass-energy, so a black hole will shrink over time as its mass is radiated away. As the mass decreases, the temperature goes up and so does the power output. So you get a runaway process of the hole getting hotter and hotter and radiating more and more power until POOF! It's gone in a flash of light and radiation. If you only consider the radiated electromagnetic energy the lifetime remaining of any black hole, assuming more mass doesn't fall into it, is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
tH = 5120 π G2 M3 / (&hbar; c4).
</div>
As this does not take into account radiation of other particle types, it is an upper bound to the lifetime; the radiation of other kinds of particles will also carry away energy making the black hole lose mass faster. Details for including the emission of other kinds of particles can be found in reference <ref name="MacGibbon II">J. H. MacGibbon, "Quark- and gluon-jet emission from primordial black holes. II. The emission over the black-hole lifetime", Physical Review D Vol. 44, No. 2, pg. 376-392, (1991)</ref>. As an estimate, you can divide the electromagnetic lifetime by the ratio of the total radiated power to the electromagnetic power; although this does not take into account the variation in this ratio as the black hole changes mass you might expect most of its lifetime to be in a range where the types of particles emitted are not changing dramatically and in such a case this approximation applies.

This is a neat result. It allows perfect conversion of mass-energy into radiant energy (although the neutrino and gravitational radiation will be rather inconvenient to capture). However, the actual implementation can get a bit inconvenient.

Let's skip for the moment the details of how you get a black hole. We'll assume that you have a magic black hole making box that can pop out whatever size of hole you need. Now let's say you want a megawatt of usable power (so we ignore the gravitational waves and the neutrinos). What size of hole do you need? It turns out to be a cool 38 billion metric tons. A hole that size is rather hard to carry around with you. And its temperature will be 3.2 billion kelvin. At that temperature its usable radiation is primarily electrons and positrons, with a good dose of hard x-rays and gamma rays for good measure. On the plus side, it's about 2000 times smaller in radius than a typical atom. So you could slip it into your pocket; just don't expect it to stay there.

Here we see one of the issues on trying to utilize Hawking power from black holes. Usable amounts of power generally come with horrendous power to mass ratios with the energy released as highly penetrating ionizing radiation. And if you start getting to masses that are more practical to deal with, you've got more of a bomb than a reactor – a 1000 ton black hole will release all of its 20,000 gigatons TNT equivalent in under a second.

Let's take an example of a black hole with a mass of 100 million metric tons, for reasons that will become clear later. We have already found that this hole is only about a fifth the size of a proton. But that tiny speck of compact mass has a temperature of 1.23 × 1012 kelvin. It puts out a radiated power of 1.4 × 1012 watts (of which something like 7 × 1011 watts is usable), which is a rate of mass loss of 15.6 micrograms per second. Or in somewhat more descriptive terms, the interacting radiation has about the energy released by the detonation of 170 tons of TNT every second. Left to its own devices, it will slowly get brighter and brighter, losing mass faster and faster, until it eventually radiates itself away in about 67 million years.

The description of Hawking radiation so far has assumed a black hole without charge or angular momentum. These properties will change the amount of radiation emitted for a given amount of mass. In particular, an extremal black hole of any kind has a temperature of zero and emits no Hawking radiation. A rotating black hole preferentially emits particles with spin and orbital angular momentum aligned with its own; a charged black hole preferentially emits particles with a charge the same as its own. Consequently, Hawking radiation will tend to discharge charged black holes and spin down rotating black holes. As angular momentum is emitted at a higher rate than mass-energy, rotating black holes will spin down to black holes with negligible rotation over timescales where loss of mass is appreciable<ref>D. N. page, "Particle emission rates from a black hole. II. Massless particles from a rotating hole", Physical Review D Vol. 14, No. 12, pg. 3260-3273, (1976)</ref>. Similarly, charged black holes will rapidly discharge from hawking radiation on time scales far faster than their rate of mass loss<ref name="Carter 1974"></ref>.

=== Penrose process ===

In a rotating black hole, anything entering the ergosphere gets pulled around the black hole by the spinning space-time. If you dive into the ergosphere and then shoot something backward against the direction you're being swirled in, this is a rocket and you get pushed forward just like any other rocket. But if you do the math<ref> R. Penrose and R. M. Floyd, "Extraction of Rotational Energy from a Black Hole". Nature Physical Science. 229 (6): 177–179. (February 1971). Bibcode:[https://ui.adsabs.harvard.edu/abs/1971NPhS..229..177P 1971NPhS..229..177P]. [https://doi.org/10.1038%2Fphysci229177a0 doi:10.1038/physci229177a0]. [https://search.worldcat.org/issn/0300-8746 ISSN 0300-8746]</ref>, if you dive in deep enough (but still outside the event horizon!) when you come out of the ergosphere you can be going much faster than if you fired your rocket outside the black hole. What gives? How can you get more energy than you started with? Well, it turns out that the energy came from the black hole itself. You decreased both the black hole's mass-energy and its angular momentum when you did that, and got shot out with that extra energy and angular momentum.

This has obvious uses for getting energy. If you drop things into the black hole, and have them push stuff out backward to fall into the black hole, you can harvest the black hole's rotational energy by using the dropped things to do work when they come zipping back out.

For an uncharged extremal rotating black hole and a trajectory grazing the event horizon, up to 20.7% of the mass-energy of the ejected particle can be returned as kinetic energy by this process. However, for a charged rotating black hole there is no upper limit to the efficiency of the process<ref>M. Bhat, S. Dhurandhar, and N. Dadhich, "Energetics of the Kerr-Newman black hole by the penrose process". Journal of Astrophysics and Astronomy. 6 (2): 85–100. (1985). Bibcode:[https://ui.adsabs.harvard.edu/abs/1985JApA....6...85B 1985JApA....6...85B]. CiteSeerX [https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.512.1400 10.1.1.512.1400]. doi:[https://doi.org/10.1007%2FBF02715080 10.1007/BF02715080]. S2CID [https://api.semanticscholar.org/CorpusID:53513572 53513572]</ref>. In fact, you can gain more energy from the Penrose process with a charged black hole than was in the mass-energy of the particle you ejected!

==== Penrose batteries ====

For an uncharged extremal rotating black hole, nearly 30% of the mass-energy of the black hole can be extracted via the Penrose process<ref name="Rees 1984">M. J. Rees, "Black hole models for active galactic nuclei", Annual Review of Astronomy and Astrophysics Vol. 22 pp. 471-506 (1984)</ref>. This percentage can get even larger for a charged rotating black hole.

Of course, once you extract that energy, you can't use the black hole for the Penrose process any more. However, you could charge it up again by throwing matter into the hole with high angular momentum with respect to the hole. It is even better if the matter is highly charged. Assuming that the black hole is large enough that it can be fed efficiently (see below), you can re-use your black hole battery over and over again.

==== Superradiant scattering ====

An effect similar to the Penrose process with matter can be accomplished with radiation. Light is shone into the rotating black hole. A portion is absorbed by the black hole, but more energy than was lost is given to the light by the ergosphere, a process known as superradiant scattering<ref>Ya. B. Zel'dovich, "generation of waves by a rotating body", ZhETF Pisma Redaktsiiu Vol. 14 No. 4 pp. 270-272 (20 August 1971)</ref><ref>J. D. Bekenstein and M. Schiffer, "The many faces of superradiance", Physical Review D. Vol. 58 064014. [https://arxiv.org/abs/gr-qc/9803033 arXiv:gr-qc/9803033]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1998PhRvD..58f4014B 1998PhRvD..58f4014B]. doi:[https://doi.org/10.1103%2FPhysRevD.58.064014 10.1103/PhysRevD.58.064014]. S2CID [https://api.semanticscholar.org/CorpusID:14585592 14585592]</ref>. If this light is then reflected back into the black hole again and again, it can get amplified indefinitely – at least until the intensity of the light gets so high that it breaks your mirror. The idea of enclosing a rotating black hole with a mirrored shell is called a black hole bomb<ref>W. H. Press and S. A. Teukolsky, "Floating Orbits, Superradiant Scattering and the Black-hole Bomb", Nature Vol. 238 pp. 211–212 (July 28, 1972). Bibcode:[https://ui.adsabs.harvard.edu/abs/1972Natur.238..211P 1972Natur.238..211P]. doi:[https://doi.org/10.1038%2F238211a0 10.1038/238211a0]. ISSN [https://search.worldcat.org/issn/1476-4687 1476-4687]</ref>. All of this allows you to extract the energy of a rotating black hole using light and receiving energetic light in return. You no longer need worry about the energy coming out as extremely penetrating radiation of high energy particles.

=== Feeding a black hole ===

If you are extracting energy from a black hole, you might want to eventually put that energy back in to avoid using up your black hole too soon. You can do this by letting mass or other forms of energy fall into the hole, passing through its event horizon to get trapped forever. If the infalling matter is charged, the black hole will aquire that charge. If the infalling matter is off-center or spinning, the black hole will acquire the angular momentum of the system once the matter is absorbed.

==== Tidal disruption ====

If you have something smaller in size than a black hole's event horizon and you drop it straight in, it should enter the hole without any particular complications. But as the object approaches the hole, the hole's changing gravity will affect different parts of the object differently. Gravity drops off with distance, so the parts of the object nearest the hole will be getting pulled harder than those furthest away. This means that once you account for the average force on the object accelerating it toward the hole, you have an additional force acting on the body to tear it apart along the direction to the hole. Meanwhile the direction of gravity is toward the center of the hole, pointing radially inward. Again, after accounting for the average force on the object this means that the parts furthest to the left are experience a residual force pointing to the right and vice versa. So the net result is that tidal forces stretch an object along the direction towards the center of the hole and squish it together in the directions transverse to that direction. This is called "spaghettification".

Tidal forces fall off faster than the average force of gravity on an object. Whereas gravity falls off with the square of the distance, tides fall off with the cube of the distance. So far out from a black hole, you might be falling comfortably but as you get closer the tides get strong quickly. Very large black holes, like the supermassive black holes at the center of galaxies, might not generate any noticeable tides even as you fall though the event horizon. Smaller holes, on the scale of stellar mass black holes, do generate enough tides to spaghettify any astronaut unlucky enough to fall into them.

==== Accretion disks and astrophysical jets ====

If the thing you drop into a black hole isn't dropping straight in – maybe it has a bit of transverse velocity as it gets sucked down – it is likely to miss the event horizon and slingshot around on an orbit. However, even as it misses the all-devouring beast at the center tidal disruption is still pulling the object apart. A close enough approach will have the tides rip apart the object and smear it out into a smudge of debris. The inner parts of the debris cloud will be orbiting faster than the outer parts, leading to shear flow and friction and drag. This leads to heating of the debris, coming from the object's kinetic energy. After enough passes the former object will get spread out into a ring around the hole, called an accretion disk. The closer the debris is to the hole, the faster the difference in speed between adjacent streamlines and the more heating will occur. So you can get the inner parts of the ring glowing brightly with radiated heat.

Most physical process that can feed matter into a black hole start with the infalling matter having some angular momentum. Because the angular momentum is conserved it naturally results in accretion disks forming as the matter falls in.

As the inner part of the disk radiates heat, it loses kinetic energy and gets a little bit closer to the event horizon. As it gets closer it gains heat at a greater rate and its temperature increases. When it gets hot enough, the matter turns into a plasma. To a good approximation, plasmas cannot cross magnetic field lines. A strong field with a diffuse plasma will have the plasma move along the field line direction. A dense, fast plasma, on the other hand, can bully through weak field lines, stretching out the field so that it moves with the plasma. In a turbulent plasma, or, in this case, a circulating plasma, the field gets stretched out enough that it can come back and meet itself, getting stronger and stronger. This dynamo effect will amplify even very weak fields within the accretion disk, forming a strong magnetic field near the black hole.

And this is where things get a bit weird. Something happens – we're still not entirely sure what – and the interaction of the strong field with the energetic plasma right near the event horizon creates jets of fast moving plasma, high energy particles, and electromagnetic radiation shooting out along the axis of the accretion disk, usually in both directions.

In some cases, the circling debris may puff up into a shape more like a doughnut than a flat disk. These toruses are generally expected to be less efficient at radiating energy out of the infalling matter<ref name="Rees 1984"></ref>, with the radiation getting trapped in the torus and serving to puff it out rather than escaping.

The accretion disk process around a non-rotating, uncharged black hole can extract up to 5.7% of the mass energy of infalling matter into radiated energy and energy of the jets. The efficiency at radiation can increase to up to 42% for an extremal rotating black hole<ref name="Rees 1984"></ref>. If this radiated energy from the accretion disk can be collected, it can provide an additional source of energy beyond what you can get from Hawking radiation and its somewhat inconvenient limits. So now we must see what limits the rate of accretion to see how much energy we can get out of it and also how fast we can recharge our hole for the extraction of Hawking and Penrose energy.

==== Mass collection rates ====

Suppose you have a black hole inside of some material. This might be a rock, or a star-hot plasma, or the diffuse gas of interstellar space.

If you are at rest with respect to the surrounding material, you'll get that material falling toward you. It will pile up as it crams together trying to get to the hole, until you reach a point where the flow turns super-sonic and the material free-falls the rest of the way into the hole. Finding the feeding rate is thus a [https://en.wikipedia.org/wiki/Choked_flow choked flow] problem.

If the hole is moving through the material faster than the speed of sound, material passing close to the hole will get deflected by the hole's gravity to converge in a wake behind it. Where it collides with other gas coming in from all directions in the wake, the gas comes to a halt and from there it can freely fall into the hole from behind.

The analysis of these two limits may be combined to give the Bondi-Hoyle accrection rate<ref>Edgar, Richard (21 Jun 2004). "A Review of Bondi-Hoyle-Lyttleton Accretion" https://ned.ipac.caltech.edu/level5/March09/Edgar/Edgar2.html https://arxiv.org/abs/astro-ph/0406166</ref>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ṁBH = 4 π ρ G2 M2/ (cs2 + v2)3/2
</div>
where ρ is the density of the stuff the hole is in, cs is the speed of sound in the medium, and v is the speed of the hole through the medium. The distance at which the in-falling material goes from subsonic choked flow to supersonic free-fall is the Bondi radius
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
rB = 2 G M / cs3.
</div>
The speed of sound in a solid makes a useful approximation for where inertial effects overcome material strength effects. Thus, the Bondi radius can serve as a useful approximation of how big of a channel will be ripped out of something that has a black hole pass through it.

If the Bondi-Hoyle accretion rate is too low, the black hole will be losing mass faster to Hawking radiation than it will be gaining mass to accretion. This depends on the variables described above, but let's look at what happens if we drop it into solid rock. Assuming a typical density of rock of 2.7 grams per square centimeter and a sound speed in rock of about 5 kilometers per second, we find that holes that are larger than 105 million metric tons are able to absorb a net gain in mass while those below this limit lose more mass to Hawking radiation than they gain by eating the rock. If you want to feed your hole with rock, you'll need it to be bigger than 105 million metric tons. The Bondi radius for such a black hole will be about half a micrometer, or about 5000 atoms in radius, so the tunnel it will make falling through rock will be fairly small.

The best material for feeding your black hole, according to the Bondi-Hoyle accretion rate, is the heavy metal thallium. If you drop your hole into a blob of thallium, it can achieve a net mass gain at a mass of only 22 million metric tons. For black hole masses below this, you cannot feed a black hole on normal matter at room temperature and pressure (whether it can feed at the crazy high pressures at the cores of planets or stars is a subject not explored here).

==== Radiation pressure ====

Both the Hawking radiation and the radiation from the accretion disk will be shining out of an accreting black hole. This radiation will encounter material from the accretion disk. The radiated light can scatter off electrons in the disk material; on average, this will push them outward. The electrons will then drag any assorted atomic nuclei in the disk material with them. This puts a limit on how much material can flow into the black hole – if it is too bright, it will push everything away. If the hole gets brighter than this limit, it can no longer feed.

This is often referenced in terms of the Eddington luminosity
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
LE = 4 π G M (A/Z) mp c / σT
</div>
where A is the average atomic weight of the plasma, Z is the average atomic number, mp = 1.672622 × 10-27 kg is the mass of a proton, and σT = 6.65246 × 10-29 m2 is the Thompson cross section for scattering light off an electron. If something is shining with the Eddington luminosity, it will keep matter from falling in. Strictly speaking, this assumes hydrostatic equilibrium; for problems that are time varying or with steady-state flows the Eddington limit does not necessarily apply. However, it is often a good first guess to figure out when the radiation chokes off the inflow in accretion disks. There are some configurations of accretion disks that can support luminosity higher than the Eddington limit, but most are at or below this limit.

If we assume that our black hole's accretion disk is Eddington limited, we can find out how big it needs to be in order to accrete any matter at all, or to achieve net mass gain after its Hawking radiation losses are accounted for. In hydrogen gas, with A/Z = 1, we find that a hole must have a mass of at least about 104 million metric tons for any matter to fall in past the Hawking radiation pressure. The hole's mass has to be in the 109 to 125 million metric ton range to gain mass via accretion faster than it is lost to Hawking radiation, depending on the efficiency at which matter in the accretion disk is converted into radiation. If you drop the hole into rock or other light elements you'll have an A/Z ratio of 2 or very slightly higher. Setting A/Z = 2, we find that you can't get any accretion for masses under 85 million metric tons and, again depending on the radiative efficiency of the accretion disk, you need somewhere in the range of 90 to 103 million metric tons to reach breakeven in terms of mass loss versus mass gain. Even for very heavy elements like lead or uranium, with an A/Z ratio of approximately 2.5, you need at least 80 million metric tons to accrete matter at all and somewhere between 84 and 97 million metric tons to break even.

In other words, if you want to be able to add mass to your black hole by having it gobble up surrounding matter, you'll want it bigger than many tens of millions of metric tons.

Interestingly, the limit for net mass gain for the Eddington limit is very similar to that of the Bondi_Hoyle limit. In order to get a black hole that gains mass, you're pretty much going to need at least a mass somewhere near the 100 million metric ton range.

==== Reaction rates at sub-atomic sizes ====

We now know the rate at which matter can fall on to a black hole, getting past both the radiation coming from the hole and its inner accretion disk and for getting past the choked flow of the material getting in its own way. But what about when it reaches the hole? Obviously, if the hole is bigger than the size of an atom any atoms it touches will immediately get sucked in. But a lot of holes of engineering interest are much smaller than this. A black hole with a mass of 100 million tons would have a Schwarzschild radius of about 5.7 times smaller than that of a proton. If a hydrogen atom fell into the hole, it would end up sitting there with the black hole inside of the proton. How quickly could the hole slurp up that proton and its companion electron?

 Consuming protons and neutrons 

It is easy enough to get an estimate of how fast a proton or neutron will get eaten once a black hole is inside of it. Both protons and neutrons have a radius of about 8.4 × 10-16 meters. Both are made up of three quarks. This gives a quark density of about 1.21 × 1045 / m3 inside of the proton or neutron. Because the binding energy of the quarks is much larger than the mass-energies of the quarks, we can assume that they are highly relativistic and are moving at about light speed. Multiply the density by the speed to get the flux (particles passing through per area per time) of about 3.62 × 1053 quarks / m2 / s. Then multiply by the surface area of the hole to get the absorption rate of the quarks. Once one quark is eaten, color confinement ensures that the rest of the quarks cannot leave and the particle is stuck to the black hole until the rest of it is eaten, which time we can guestimate by the time needed to eat three quarks. For our 100 million ton black hole, this shakes out to about 3 × 10-23 seconds to eat a proton or neutron, or 3.3 × 1022 protons and neutrons eaten per second. If we multiply by the mass of a proton or neutron, we find that the 100 megaton black hole can eat protons and neutrons at a rate of about 5.6 × 10-5 kg/s if it has a constant supply of protons and neutrons ready to immediately fall into the hole once the previous one was eaten. Which is comfortably higher than the loss to Hawking radiation of 1.56 × 10-5 kg/s.

This is okay for neutrons (if you can somehow find a supply of free neutrons), but for protons there is a problem. For every proton the hole eats, it gains one unit of elementary charge (that is, the charge that the proton had gets added to the charge of the hole). If it eats enough protons, it will gain enough charge to repel away any other proton (or atomic nucleus) that comes near enough to it that the electrons around the atom can no longer screen the electric charge of the proton or nucleus. The potential energy of a proton or nucleus bound to the black hole by their mutual gravitational attraction is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
UG = -mp A M G / r
</div>
and the potential energy of the repulsion between the proton or nucleus and a charged hole that has absorbed Y other protons is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
UE = [Y Z q2 / (4 π ε0)] / r.
</div>
Here, Z is the number of protons in the nucleus under consideration (Z = 1 for a single proton), A is the number of protons + neutrons in the nucleus (A = 1 for a single proton), q = 1.602176487 × 10-19 C is one unit of elementary charge, ε0 = 8.854187817620 × 10-12 C2 / J / m is the permittivity of free space, mp = 1.67262192369 × 10-27 kg is the mass of a proton, and r is the distance between the black hole and the proton or nucleus.
If the sum UG + UE is negative, the hole still attracts the proton or nucleus and matter free-falling into the hole can collide with the hole without issue. If the sum is positive the force is repulsive and the proton or nucleus cannot approach the hole. We see that this happens when
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Y = 4 π ε0 mp (A/Z) M G / q2.
</div>
For our 100 megaton black hole eating hydrogen (which has only protons as a nucleus), the hole can charge up to a maximum of Y = 49. For heavier nuclei with a mass to charge (A/Z) ratio of 2, the hole can charge up to Y = 97. Whatever the case, if the hole cannot get rid of this charge fast enough, the hole will get too much charge to freely eat everything falling into it.

 Discharging via Hawking radiation 

There are many ways that the hole can shed its charge. It's gravitational field and positive electric charge pulls negatively charged electrons in to a high density, it can simply eat these electrons to reduce its charge. Alternately, the electrons densely packed around the protons might get captured by the protons to form neutrons that can fall into the hole and keep feeding it. For this case, however, the most efficient means of reducing the hole's charge is from its Hawking radiation.

The hole will have a chemical potential for electrons of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
μ = q2 Y / (4 π ε0 rs),
</div>
which is the potential energy to bring an electron from far away to the event horizon. If the chemical potential is significantly larger than the Hawking temperature (in energy units) and if the Hawking temperature (in energy units) is significantly larger than the mass energy of an electron
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
μ > kB TH > me c2
</div>
then the rate of positron emission from the hole is approximately μ/&hbar;<ref name="Carter 1974"></ref>. Our 100 million ton hole with Y > 10 meets both these criteria. For Y = 11 the rate of positron emission is 1.6 × 1023, a full order of magnitude larger than the rate at which protons can be absorbed, and only increases as the charge goes up. This discharges the hole faster than it is charged by gobbling up protons. We thus see that nothing prevents matter from falling into the hole at the macroscopic accretion rates.

== Propulsion ==

People often like to get from one place to another. A black hole gives you various options for moving things around.

=== Penrose launcher ===

If you have a large enough rapidly rotating black hole, you can drop an entire spacecraft in it. If you get deep enough into the ergosphere, you can use the Penrose process by firing your rockets at the point of closest approach. Now you can get yeeted out at ridiculous speeds. If you can survive the tidal forces that close to the event horizon, you can potentially get a machine for flinging you around the galaxy at relativistic speeds.

=== Black hole rockets ===

Taking a black hole with you has the advantage that you don't need to rely on any black hole based infrastructure at your destination. An obvious method of propelling yourself with a black hole is to use the energy emitted by a hole to energize your propellant, rather than using a chemical or nuclear reaction for your rocket thrust. Perhaps you can directly use the astrophysical jet as your rocket propellant. Or the radiant light or energy from Hawking radiation<ref>[https://www.researchgate.net/publication/293633217_Acceleration_of_a_Schwarzschild_Kugelblitz_Starship J. S. Lee, "Acceleration of a Schwarzschild Kugelblitz Starship", Journal of the British Interplanetary Society pp. 105-116 (2015) ]</ref><ref name="Crane_Westmoreland"></ref> or a black hole bomb as a photon drive. All of these methods will require careful engineering to avoid very low accelerations from the high mass of the black hole while avoiding getting a black hole so small that it immediately evaporates in an explosion far larger than your spacecraft can survive.

== Making Holes in Things ==
Sometimes, you need to put a hole in something. Not in the sense of putting a black hole inside of something, but drilling a cylindrical hole through something. Perhaps you are interested in machining part out of difficult to work materials. Perhaps you want to build a weapon that perforates your enemies. In either case, if you have a black hole available you could imagine sending the black hole through the target object and leaving a hole ... or at least a region of gravitationally disrupted material ... behind.

For its frontal surface area, a black hole has an enormous mass. It's sectional density and the pressures it exerts on the material it passes through will be so high that it will essentially ignore the material in its way. After passing through enough material, it will eventually be slowed down both by accumulating mass and through drag forces, but that will occur over distances well beyond what we are concerned with here. For practical purposes, the black hole will just punch through without being impeded in any way by the object in its path. Our goal is to figure out what happens to that object.

=== Direct absorption ===
Obviously, anything which directly encounters the event horizon will be lost forever. This gives us a lower bound on the size of the hole left as the black hole diameter of twice the Schwarzschild radius 2 rS.

=== Gravitational disruption ===
A more significant effect is how the black hole will gravitationally accrete the material it passes through and eventually consume it. We have already looked at [[Black_Hole_Engineering#Mass_collection_rates|Bondi-Hoyle accretion]]. The choked flow treatment takes as a cutoff where the infalling fluid transitions from subsonic to supersonic speeds at the speed of sound. But the speed of sound is also a reasonable estimate of where inertial effects overcome material strength effects. Motion due to gravity is fundamentally inertial, so we can take the Bondi radius rB as a rough estimate of the distance where the black hole's gravity is able to rip material apart. If the black hole is moving slowly compared to the speed of sound, this material will be consumed; if it is moving much faster than the speed of sound it merely leaves a gravitationally disrupted trail behind it. In either case we are left with a region of diameter 2 rB where the target object is torn apart.

=== Vapor explosions ===
The black hole will emit radiation into the target object as it passes, either from Hawking radiation or from the radiation coming from its accretion disk. In practice, much of the Hawking radiation from small black holes will be in the form of highly penetrating radiation. But if we make the assumption that the radiation is absorbed locally (a reasonable assumption for larger black holes where the temperature is on the order of 10 keV or less) we can find the energy deposited per distance traveled by a black hole moving with speed v as dE/dx = PH/v. Any neutrinos or gravitational waves emitted will be far too penetrating to affect this calculation; consider only the Hawking power from interacting particles (and even then, the muons, pions, hadronic showers, and weak vector bosons that you get from the smaller black holes all put a significant fraction of their decay energy into neutrinos, so only part of their energy can be used).

The radiation from the accretion disk is likely to be more amenable to local absorption. Find the rate of accretion, multiply by the square of the speed of light to find the mass-energy accretion rate, and then by the efficiency ε of turning accretion disk mass energy into radiation that was discussed earlier. Then divide by the speed to find the energy deposited per distance traveled to get dE/dx = ṁBH c2 ε / v. Add this to the Hawking energy deposition to get the total dE/dx. If the accretion is Eddington limited, the accretion rate cannot bring the energy deposition above LE/v.

Under the assumption that this energy is absorbed locally, it will heat a cylinder of material to a high pressure vapor. This vapor will then expand, pushing surrounding material violently away. The radius of the resulting cavity can be found if you know the cavity strength of the material Kc. This can be found from the compressive strength K and the shear modulus G, both of which can usually be looked up for many common materials:
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Kc = (2/3) K + (1 + ln(2 G/K))
</div>
The volume of a cavity blown out by an energetic event will be Kc times the energy release. This gives a radius of the cylinder exploded out of the target object of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
rv = √[(dE/dx) / (π Kc )]
</div>
The diameter of the exploded hole will be twice the radius.

Reference <ref>Robert J. Scherrer, "Gravitational Effects of a Small Primordial Black Hole Passing Through the Human Body", [https://arxiv.org/abs/2502.09734 arXiv:2502.09734 [astro-ph.CO]]</ref> gives one attempt to estimate the effects of a micro black hole passing through the human body. Here, they assume that the black hole has a speed on the order of the dark matter velocity dispersion of around 200 km/s, and find a minimum mass for serious injury or death to a human victim of 1.4×1014 kg. That work used different assumptions than are used here. If we take a black hole of that mass and speed passing through the human body (taking water as the primary constituent such that density 1 gram/cubic centimeter, A = 18, Z = 10, and a speed of sound of 1500 m/s) the Bondi accretion limit is 0.14 g/s (far less than the Eddington limit, so we are Bondi limited rather than Eddington limited). The Bondi radius is 8.3 mm, so we can assume that the gravitationally disrupted tissue alone is equivalent to the effect of a 16.6 mm bullet. If we assume a 5% efficiency at turning the mass-energy of the accretion disk into radiation, we get an accretion power of 616 GW, leading to a linear energy deposition of 3.08 MJ/m. The Hawking radiation is negligible compared to this, so we ignore it. The cavity strength can be crudely approximated as 1.2 MPa, which gives results roughly consistent with ballistics gelatin results. Crunching through the calculations, we find that the vapor explosion blows out a hole 90 cm in radius (180 cm in diameter), which is enough to explosively disassemble the entire person into splattered gibbets. We therefore see that the vapor explosion is the most significant factor and that the given 1.4×1014 kg is a significant overestimate of the minimum dangerous mass of a black hole.

== Gravity Generation ==

People are healthiest when living in gravity. If you want to go out in space, there is no gravity. Even on worlds, if the world is small enough there might not be enough gravity for good health.

There are many proposals to address this, and they mostly involve spinning things around in centrifuges. Which, to be perfectly honest, is probably always going to be a better approach to making gravity than black holes. But we're not here for practicality, so lets look at using black holes as a gravity source.

The source of gravity we are most familiar with here on Earth is gravity from mass. You need a lot of mass to generate just a little bit of gravity, so it seems rather inefficient. However, the closer you can get to your mass the more gravity you get, following Newton's law of universal gravitation
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
g = G M / r2
</div>
where lower case g is the acceleration due to gravity, upper case G = 6.67430×10−11 m3/kg/s2 is the gravitational constant, M is the mass making the gravity, and r is the distance between the center of the mass and the place where you are measuring the gravitational acceleration. Technically, this is only for point masses or spherically symmetric masses, but we will be dealing with planets and black holes which are generally pretty close to spherical in most cases so we're okay. Given this, we can get more gravity the closer we can get to the source of our mass without going inside of it which in turn argues for using the densest source of mass we can find. Which is black holes.

Gravity on Earth has a value of g&oplus; = 9.8 m/s2. If we know the mass of our black hole, we can plug this in to the law of universal gravitation to find how far away we need to be to get a comfortable gravity. However, there is another consideration. Your head and your feet will be at different distances from the center of the hole, so if you are standing up your feet will experience more gravity than your head. The average person is somewhere around 1.5 to 2 meters tall, so if you need to be 10 cm from the black hole for 1 g&oplus; at your feet your head will nearly be in freefall. So we also want the distance for 1 g&oplus; to be significantly larger than a human height.
;
Let's take, for example, a case where we have 1 g&oplus; at a distance of 10 meters. Plugging this in to the law of universal gravitation, we find that we need a mass of 14.7 billion tons. Given that we need to pack all of this into a sphere with a radius of 10 meters or less, we require a density of more than 3.5 million grams per cubic centimeter. The densest material known is osmium, which is 22.6 grams per cubic centimeter. As we need a density five orders of magnitude more than this, normal materials will not cut it. Electron degenerate matter can approach these densities, but electron degenerate matter cannot hold itself together and will spontaneously explode under environmental conditions suitable for human life (specifically, if the gravity is only 1 g&oplus;) so we can rule that out. Neutron degenerate matter has the same issue. Which leaves black holes as our only option.

Such a hole would be smaller than an atom, although substantially larger than an atomic nucleus. It will produce about 20 MW of hard radiation but most of that is neutrinos; only a bit over 8 MW is going to interact with normal matter – mainly several hundred keV gamma rays, positrons, and electrons which are all easy enough to shield against. The black hole will last much longer than the current age of the universe and if you need to feed it the Eddington limited rate is a few grams per second while the Bondi limit is about a quarter kg/s for rock, a few kg/s for water, or a couple hundred kg/s for thallium. As far as the gravity, if your feet are at 1 g&oplus;, then (assuming you are 1.7 m tall) your head will experience about 3/4 g&oplus;. This is probably both healthy and comfortable, the black hole is relatively benign, and so this presents one option for artificial gravity.

== Computation ==

A black hole's event horizon has a temperature. This implies, via thermodynamics, that it has an entropy. In information theory, the entropy of a system is a measure of its information content, and thus the Hawking radiation coming out of the black hole is the rate at which information is returned to the outside world. This brings up the idea of, what if you could input information via coded messages into the black hole, have the black hole process that information, and then return that information as patterns and correlations in its Hawking radiation?

If this all sounds very hand-wavy, that's because it is. You could apply the same argument to the glow coming off of a bar of hot iron. But one work<ref>G.R. Andrews III, "Black hole thermodynamics", Results in Physics,
Volume 13,
2019,
102188,
ISSN 2211-3797,
https://doi.org/10.1016/j.rinp.2019.102188.
(https://www.sciencedirect.com/science/article/pii/S2211379719304036)</ref> has looked into this concept and found ways, at least in principle, to make black holes Turing complete so that they can be used, again in principle, as a computer. This raises the possibility of arbitrarily advanced civilizations with near omniscient abilities to measure radiation using black holes as the ultimate computation device<ref>S. Lloyd and Y. J. Ng, "Black Hole Computers", Scientific American (April 1, 2007) https://www.scientificamerican.com/article/black-hole-computers-2007-04/</ref>.

== Containment ==
<blockquote>
There were a dozen other questions that Duncan was longing to ask. How were these tiny yet immensely massive objects handled? Now that Sirius was in free fall, the node would remain floating where it was--but what kept it from shooting out of the drive tube as soon as acceleration started? He assumed that some combination of powerful electric and magnetic fields held it in place, and transmitted its thrust to the ship.

Arthur C. Clarke, Imperial Earth
</blockquote>

So, you have a black hole. And let's say you want to use it for a mobile application. This means you need to move it around. As you are likely dealing with something that has a mass of millions of tons or more, it will take a lot of force to accelerate it just a little bit. If you are going to use it for thrust for your spacecraft, or even if you need to move it around somewhere using a spacecraft, you're going to want to make sure it doesn't get left behind when your spacecraft moves. As you can see from the quote above, even some of the foremost minds in science fiction simply hand-waved this detail away.

This can get particularly bothersome if you are on a planet. A basic 100 million ton black hole weighs, well, 100 million tons. Or about a trillion newtons of force. It's smaller than the nucleus of an atom. Any chemical bond will fail with a force of about 0.010 μN; the black hole will exert something like fourteen orders of magnitude more force than is needed to break any known force holding it to other atoms in matter. The pressure of all the force concentrated into such a tiny area means that nothing material could keep it from simply falling down. After which it will end up orbiting through the planet, mostly ignoring the matter in the way but gradually slowing down over geological time spans. If this happens and you wanted to do something other than geoengineering with your black hole, you're probably out of luck.

So how can you exert a force on a black hole?

By Newton's third law of motion, anything that gets gravitationally attracted to a black hole also exerts the same force back on a black hole. A black hole near something else massive will be tugged toward the massive thing as the massive thing pulls the black hole. So if that massive thing is made out of matter, you can pull the thing which can pull the black hole. Unfortunately, the resulting force is probably going to be really weak. If you had a 200 meter diameter ball of osmium (the densest material known) it would have a mass of 95 million tons. At the surface of the ball, it would attract a black hole with a gravitational acceleration of 0.63 mm/s2; about 1/15,500 that of Earth's gravity. The acceleration is pitiful, and you're going to have to be carrying around a lot of extra mass (whether it is a significant amount of extra mass compared to your black hole is another matter). But you can apply the acceleration continuously over long periods of time. If you use this to couple your black hole rocket to your spacecraft you can accelerate at 54 m/s per day; or a km/s every 20 days. Perhaps surprisingly, this is not entirely unworkable.

Note that this method does not provide overall propulsion. Conservation of momentum dictates that you still must use some kind of thruster than expels or exchanges momentum with the outside environment. Rather, this gives you the limits at which your black hole can be accelerated by whatever method you are using to move your spacecraft and the hole without the hole falling away.

You can also electrically charge the black hole. This will give it an electric field. If the black hole is also spinning, the combination of spin and charge will give it a magnetic field. You can then push or pull on the black hole with beefy capacitor plates or electromagnets. However, it can be challenging to give a black hole a large charge, or to have it keep its charge for long.

One problem is the electrical potential of the hole.
A black hole will have a capacitance of
<div align=center> C = 4 π ε0 rS</div>
where ε0 = 8.8541878188×10−12 F/m is the vacuum permittivity.
The potential &Vscr;, in volts, for a black hole with a charge Q in coulombs, is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"> &Vscr; = Q / C </div>
and the energy to charge the black hole up is
<div align=center> W = (1/2) C &Vscr;2.</div>
Generally, the charge you can achieve is limited by the voltage (or energy per particle, expressed in eV) you can get with your particle accelerator. For a given &Vscr;, this means the most charge you can put on your hole is
<div align=center> Q = C &Vscr;.</div>

With modern accelerators, we might get electrons up to an energy of 1 TeV (1×1012 eV), for a potential of &Vscr; = 1×1012 V.
For our example 100 million ton black hole, this gives a charge of Q = 1.65×10-14 C with a negligible charging energy. We can put this next to a highly charged capacitor plate to accelerate it. You can generate fields as high as the vacuum breakdown limit for the materials used to make your plate, which is typically about 𝓔 ~= 108 V/m. The force is F = Q 𝓔, or about (very roughly) 1 μN. Using F = M a, the acceleration a produced is a rather pathetic a ~= 10-17 m/s2, or about 10-18 g&oplus;. This is not going to get anyone anywhere in a reasonable time! But you can at least see the math needed to figure out how to move the hole so you can work other examples for yourself.

For electric containment, it is interesting to note that because rS</div> is proportional to the black hole mass, the capacitance is also proportional to the mass. So for a given attainable voltage the charge on the black hole is proportional to the mass. And consequently, for a given electric field the force on the black hole is proportional to the mass. With the final result that for a fixed voltage and electric field strength, the acceleration of the black hole you can get with electric methods is entirely independent of its mass.

If you have a charged rotating black hole, as described earlier it will have a magnetic moment. If you put a magnetic moment in a magnetic field gradient dB/dx the magnetic moment will experience a force F = m dB/dx. If we take our 100 million ton black hole charged up to a trillion volts from above, and give it enough spin that it becomes extremal, you will have an angular momentum of J = 2.2×10-8 kg m2/s. This gives it a magnetic dipole moment of m = 3.7×10-33 A m2. The highest magnetic field gradients we have managed to achieve have been about a GT/m<ref>[Zablotskii, V., Polyakova, T., Lunov, O. et al. How a High-Gradient Magnetic Field Could Affect Cell Life. Sci Rep 6, 37407 (2016). https://doi.org/10.1038/srep37407</ref>. Thus, we have a force of approximately 3.7×10-21 N and an acceleration of about 3.7×10-32 m/s2, which is many orders of magnitude worse than the already pathetic electric field case. But again, using these tools you can work out for yourself the best way to move your black hole if your black hole is not 100 million tons or is charged to a different potential. In particular, for a given voltage and magnetic field gradient, the acceleration should scale linearly with the black hole mass, thus favoring larger black holes.

But there is another issue to consider. If e &Vscr; / (TH kB), for e the fundamental charge, is not much less than 1, you will get significant discharging from the hawking radiation emitting unbalanced numbers of electrons and positrons. For e &Vscr; / (TH kB) much larger than 1 and for TH kB / (me c2) much larger than 1, the discharge rate is approximately e2 &Vscr; / &hbar;<ref name="Carter 1974"></ref>. In our previous example with a 100 million ton black hole, e &Vscr; / (TH kB) is about 10,000 and TH kB / (me c2) is about 200. Because these are much larger than 1 we can use our discharging estimate to find a discharge current of I = 24 million A. In a tiny fraction of a second, our charged black hole would be neutral again. Keeping it charged requires a power of P = I &Vscr; = 24 million terawatts from our particle accelerator.

But we have one more lever left to pull here. Momentum is conserved, so if we can get our black hole to consume matter moving at high speed the momentum of the matter the black hole eats will be transferred to the black hole. With a little bit of calculus you can find that for a Bondi-limited black hole, the optimum speed to shoot your mass stream at the black hole is v = √2 cs. The force on the black hole is v ṁBH.

Again for our example 100 million ton black hole, if we shoot it with a jet of thallium at 1157 m/s (the optimum for thallium's speed of sound) the black hole will experience a force of 2.7 N and an acceleration of 2.7×10-11 m/s2. This is still much less than the gravity tractor that was the first suggestion we floated for pulling a black hole; but at least it is much better than using electric or magnetic fields! Again, this is just one example. Black holes with different masses will get different results. In particular, because the Bondi accretion rate increases proportionally to the square of the mass, the acceleration you can get from shooting your black hole with a mass jet will increase linearly with its mass and thus favor larger black holes for more reasonable accelerations.

== Credit ==
Author: Luke Campbell

== References ==

[[Category:Engineering‏‎]][[Category:Physics & Engineering‏‎]][[Category:Physics & Math & Engineering]][[Category:Physics]][[Category:Astronomy & Cosmology]][[Category:Infrastructure]][[Category:Propulsion]]

Black Hole Engineering

2026-08-01T22:16:08Z

Lwcamp: /* Rotating black holes */

Ah, black holes. Flaws in the fabric of the universe. Empty voids from which nothing can return. The ultimate unknowable mystery.

But what are they good for?

== Basics ==

Lets start with a brief introduction to black holes.

Things like planets and stars and other massive bodies have gravitational fields around them that tend to draw things toward them and trap stuff on them. In order to get away from such a body, you need to shoot yourself off it with a speed higher than its escape velocity. If you don't have that much speed, you can't get away. When you pack enough mass into a small enough volume, its gravity gets so high that the escape velocity is higher than the speed of light. Because nothing can go faster than light, nothing can escape. This is a black hole.

[[File:Black_hole_Schwarzschold.png|thumb|A diagram of the features of the Schwarzschild geometry, showing the event horizon (white circle) and central singularity.]]
That's the description motivated by Newtonian gravity, anyway. But when gravity gets really strong Newtonian gravity breaks down and you need to use general relativity instead. Curiously, the size and mass where light (and everything else) is trapped is the same as the Newtonian case. But instead of light and other things flying out, looping around, and coming back space-time gets strange. At the critical distance where light would be trapped you get a surface called an event horizon. Nothing that passes into an event horizon can ever get back out again. The gravity at and inside the event horizon is so strong that it rotates space and time enough that the direction inwards toward the center becomes your inevitable future. You can no more resist going toward the middle of the hole that you can avoid seeing what fate awaits you.

An uncharged and non-rotating black hole at rest is described by the Schwarzschild geometry. The radius of its event horizon is the Schwarzschild radius
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
rS = 2 G M / c2
</div>
where M is the mass of the black hole, G is the gravitational constant, and c is the speed of light in vacuum. As an example, a black hole with a mass of 100 million metric tons would have a Schwarzschild radius of 1.48 × 10-16 meters. This is slightly under one-fifth the radius of a proton.

At the center of a black hole lies a point at which our description of physics breaks down, called the singularity. While of immense scientific interest, it is irrelevant for engineering because it is inside the event horizon so it cannot possibly affect us or our environment.

Energy is conserved, and mass is a manifestation of energy that is not moving. So when matter or radiation is swallowed by the hole, its energy is added to that of the hole and the mass of the hole increases by E = m c2 to reflect this.

Charged and/or rotating black holes get more complicated:

[[File:Black_hole_Reissner-Nordstrom.png|thumb|A diagram of the features of the Reissner–Nordström geometry, showing the inner and outer event horizons (white solid circle), the location of the Schwarzschild event horizon for a black hole of equal mass but no charge (outer dashed circle), the location of the extremal horizon at half the Schwarzschild radius (inner dashed circle), and the central singularity.]]
=== Charged black holes ===
Charge is conserved. If electrically charged matter falls into a black hole, the hole itself will acquire the charge. The charge produces an electric field radiating away from the hole, much as the mass of the hole also creates a gravitational field.

A charged black hole is not expected to last long in the real world. The charge will draw in particles of the same charge and repel particles of the opposite charge, tending to neutralize it in any environment where any matter exists (even tenuous space plasma)<ref name="Gibbons 1974)>G. W. Gibbons, "Vacuum Polarization and the Spontaneous Loss of Charge by Black Holes", Commun. math. Phys. 44, 245-264 (1975)</ref>. An engineer intending to work with charged black holes will need to ensure it exists in a high vacuum environment and perhaps add additional features to slow the rate of neutralization or methods to top off its charge by adding additional charged particles. As will be seen later, a charged black hole will also spontaneously shed particles to get rid of its charge<ref name="Carter 1974">B. Carter, "Charge and Particle Conservation in Black-Hole Decay", Physical Review Letters Vol. 33 No. 9, pg. 558-561 (1974)</ref>, making keeping it charged even harder.

A charged black hole is described by the Reissner–Nordström geometry. For the same mass, a net charge will cause the event horizon to shrink. A second horizon will form inside the first horizon that will grow with increasing charge, although for the purpose of black hole engineering this is not particularly relevant because anything going through the outer horizon is lost to our universe one way or the other.

As charge is added, the two horizons approach each other until they meet at a distance of half of the Schwarzschild radius calculated for an uncharged hole of the same mass, with a charge of
<div align=center>Q = M √[4 π ε0 G] = M 8.61722×10-11 C/kg.</div>
This forms one example of an extremal black hole. In this case the mass-energy of the charge, considered as a sphere of charge located in a thin shell at the event horizon, makes up the entirety of the mass of the black hole with no room left over for mass from any matter or other kinds of energy. It is thus easy to see that simply adding more and more charge to a black hole that is not yet extremal cannot actually form an extremal black hole. Likewise, adding charge to an already extremal black hole at most keeps it extremal as you add electrostatic mass-energy that keeps up with the increase in charge (and all physical charged particles also have their own mass, which would take it out of the extremal condition). Some theories suggest that it is impossible for extremal black holes to form by any physical process, although these theories have been disputed.

[[File:Black_hole_Kerr.png|thumb|A diagram of the features of the Kerr geometry, showing the inner and outer event horizons (white ovals), outer boundary of the ergosphere (red oval), and ring singularity(dotted oval).]]

=== Rotating black holes ===
You get a rotating black hole when the hole devours things which have angular momentum and that angular momentum becomes a property of the hole. Black holes have no surface features so you can't actually see things on the hole going around. But the angular momentum manifests in other physically observable ways.

Most astrophysical processes that lead to the formation of black holes involve the collapse or collisions of rotating bodies with non-zero angular momentum. Hence it is expected that all naturally occurring black holes are born rotating. As we will see later, they may not remain rotating but large rotating holes are likely to remain rotating for long periods of time.

Massive rotating bodies exhibit a process called frame dragging, and rotating black holes are no exception. Frame dragging is a gravitational analogue of magnetic induction from moving electric charges. It induces motion in space-time near the body co-rotating with the body and objects therein will be moved along with the space-time. Because space-time is dragged faster near the body than far from it, a stationary object in a free-fall orbit around the hole will appear to be rotating in the opposite direction to the hole to a distant observer even though it is in an inertial reference frame.

A rotating black hole is described by the Kerr geometry. This has some similar behavior to the Reissner–Nordström geometry of charged black holes. You get the formation of an inner horizon that grows with increased rotation, and the outer horizon shrinks. Different from charged holes is that the singularity at the center forms a ring rather than a point. None of this is of any interest to the engineer, as it is all hidden behind an event horizon and cannot affect our world.

Also similar to charged black holes, a hole that is spinning fast enough can become extremal such that the spin alone is providing the energy for its mass term when the angular momentum J is
<div align=center> J = M2 G / c = M2 2.22615×10-19 m2/kg/s.</div>

Of more interest however, is that you get a region outside of the event horizon where it is impossible to stop moving. Here, frame dragging is so extreme that space-time is moving around the black hole faster than the speed of light. This region is called the ergosphere. Similar to how once you go past the event horizon time rotates so that your future is toward the center of the hole, in the ergosphere time rotates so that your future is in the direction of the hole's spin. You can no more come to a stop or go the other direction than you can go back in time.

=== Charged and rotating black holes ===
A black hole with both charge and angular momentum behaves much like you would expect from the solutions for charged black holes and rotating black holes. You get an ergosphere, frame dragging, electric field, and the possibility of extremal black holes. Extremal holes occur when
<div align=center> M2 - (J c / (G M))2 - (Q / √ [4 π ε0 G])2 = 0.</div>
The new feature is the presence of a magnetic field whose magnetic axis is aligned with the spin axis. For a black hole with charge Q, angular momentum J, and mass M, the magnetic moment m (as measured in the far-field) is
<div align=center> m = Q J / M</div>
This black hole is described by the Kerr-Newman geometry. The mathematics of this geometry allow for the event horizon to disappear and the ring singularity to be displayed to the world. However, to obtain this condition you need to go past the extremal case, which is generally thought to be physically impossible.

=== Caveats ===
All the above descriptions of black holes assumes a distribution of mass and charge that does not change with time. That is, it is static. It may be moving, as with the case of a rotating black hole, but the distribution of rotating stuff doesn't change. It may also be moving if you shift to a frame of reference where the hole is not at rest, but you can always find a frame of reference where the hole is at rest in the sense that it has no net linear momentum (and, in a more practical sense, isn't going anywhere. This also means that the occasionally encountered idea of "accelerate an object to such a high speed that it turns into a black hole" simply doesn't work and is not consistent with physics). If you have a static hole, it's properties are entirely defined by just the three quantities of its mass, charge, and angular momentum. Any two static black holes with these three quantities the same will be identical in every respect. To describe this, physicists use the somewhat odd terminology that "the black hole has no hair"; hair being things that do not directly derive from mass, spin, or charge.

Not all black holes need be static. At the moment of creation by the collision of two supermassive objects, for example, a black hole will momentarily have an event horizon that is elongated and wobbly. That is, it has "hair." However, it rapidly radiates gravitational waves until all its hair is shed and it settles down to a static state.

All of the above descriptions of different kinds of black holes assume that if you go far enough away from the black hole, space-time settles down into the ordinary mostly flat space-time where Newtonian gravity works and planets and satellites have regular orbits and geometry works like you would expect and things behave like we would otherwise naively expect them to. This is called asymptotic flatness, defined by the idea that if you go far enough away from the hole in any direction space-time will get as arbitrarily close to flat with increasing distance. Asymptotic flatness is a good approximation of our universe on scales up to and beyond galactic clusters. If you are only dealing with engineering projects within a single galactic cluster, you can generally assume that asymptotic flatness holds. There has been some work on black holes in universes that are not asymptotically flat, but we will not concern ourselves with that here as it is unlikely to be of relevance to engineering tasks.

The initial justification for nothing getting past the event horizon was that it would have to move faster than the speed of light, and nothing can move faster than light. But many science fiction works feature methods whereby information or objects (usually spacecraft) can go faster than light (FTL). Could a faster than light starship escape from inside the event horizon of a black hole? Possibly. It depends in the implementation, but under relativity FTL motion automatically implies time travel. And all of the results of relativity that inside a black hole the future is towards the center of the hole rather than forward in time would similarly be un-done by time traveling FTL. Likewise, your FTL spacecraft could likely go backwards around the ergosphere, if that's your thing. The article on [[Wormholes#Dropping_a_wormhole_into_a_black_hole|wormholes]] covers some of the details for wormholes interacting with black holes, illustrating one way to get information out of a black hole's event horizon and the difficulty of implementing it. This could, in principle, allow access to the interior of black holes that we formerly ignored. Such as using rotating black holes as a time machine (but we can already do that if we can get there and out in the first place) or as wormholes to other universes.

== Acquiring a black hole ==

If you want to do things with a black hole, first you need to get one. Here, we discuss various ways you might get your grubby little mitts on one of these monstrosities of physics.

=== Supermassive black holes ===

At the center of each galaxy resides a gigantic black hole with a mass ranging from tens of thousands to billions of times more massive than our sun. To acquire a supermassive black hole, you'll need to travel to the center of a galaxy. The mass of these black holes means that they can be difficult to take with you and you might need to do your work where you originally found the hole.

=== Stellar mass black holes ===

Stars do not readily form black holes, despite their immense gravity trying to pull them together. When you try to squish a star down to make a black hole, that squishing makes its temperature rise. A rising temperature makes the star hot, which increases its pressure, which pushes back against your squishing. This can be very annoying when trying to make a black hole. You need to wait for that thermal energy to radiate away. But even worse the hot, dense interior of the stuff you are squishing makes a great environment for thermonuclear fusion to occur. This fusion creates heat and you have to wait for that heat to radiate away, too, before you can get the stuff to contract down further.

But even after everything has fused, there can be limits to your squishing. As the stuff in the stars gets denser and denser, you get to a point where all the low energy places to park the electrons are all taken up. To make the star denser, you need to put the electrons in higher energy states. This takes energy to get the electrons there, which means even more pressure pushing back. This is a state of matter called electron degenerate matter, and the resulting object is called a white dwarf star. For stars with a mass of about 1.44 times the mass of our sun or less, the electron degeneracy pressure keeps the star from getting small enough to form a black hole. This threshold mass is called the [https://en.wikipedia.org/wiki/Chandrasekhar_limit|Chandrasekhar limit].

Okay, so you get together a star with more mass than the Chandrasekhar limit. Now you're good to go, right? You have enough mass to just push past that annoying electron degeneracy pressure. Not so fast, buckaroo! Once the energy of the electrons gets high enough it becomes energetically favorable for them to combine with protons to form neutrons (this happens for energies of about 0.78 MeV for free protons). Now you get a dense ball of neutrons and have the same issue that you previously had with electrons, but worse. This mass of degenerate neutrons is called a neutron star. It takes a mass of a bit more than twice the mass of the sun to overcome the pressure of degenerate neutron matter (the [https://en.wikipedia.org/wiki/Tolman%E2%80%93Oppenheimer%E2%80%93Volkoff_limit|Tolman–Oppenheimer–Volkoff limit]). But once you do that, there is nothing preventing the remains of the star from squishing down into a black hole under its gravity.

All of this is to show that it can be hard to make a black hole from stars. And that's not even considering other complications, like how stars tend to shed a lot of their mass as they collapse so you need considerably more mass than the Tolman–Oppenheimer–Volkoff limit to make your black hole.

But do not fret! The universe has been kind enough to make black holes out of stars for you. There has been enough time for many of the more massive stars to burn through their fusion fuel and collapse to make black holes. Even those that remain as neutron stars sometimes run in to other neutron stars and form black holes.

Needless to say, a stellar mass black hole is going to be very heavy. If your civilization cannot move stars around, this will be a location you go to rather than a piece of equipment you carry around with you.

Black holes may not be uncommon in the universe, but they can be dark (it's in their name, after all). So stellar mass black holes can be hard to find. But there are ways. If the black hole has a stellar companion, it can siphon gas from the companion to produce a bright x-ray source. If a dark black hole passes in front of another star, it can make that star temporarily brighter through gravitational lensing. So you may be able to locate a stellar mass black hole – we have already located a great many of them. The problem of getting to said stellar mass black hole is still an unsolved problem, however.

=== Primordial black holes ===

There are no known natural processes to make black holes in our universe with a mass less than the Tolman–Oppenheimer–Volkoff limit. However, it is possible that our universe might have been born with small black holes already in place. These primordial black holes could potentially be significantly smaller than stellar mass black holes. Primordial black holes with initial masses of less than five hundred million (5×108) tons will have evaporated by now<ref>MacGibbon, Jane H.; Carr, B. J.; Page, Don N. (2008). "Do Evaporating Black Holes Form Photospheres?". Physical Review D. 78 (6) 064043. arXiv:[https://arxiv.org/abs/0709.2380 0709.2380]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2008PhRvD..78f4043M abs/2003PhTea..41..299L 2008PhRvD..78f4043M]. doi:[https://doi.org/10.1103%2FPhysRevD.78.064043 10.1103/PhysRevD.78.064043]. S2CID [https://api.semanticscholar.org/CorpusID:119230843 119230843]</ref> (see below for why black holes evaporate). Some primordial black holes with masses slightly above this limit will survive to the present day with their masses since reduced to below this limit by the intervening evaporation. However, it does mean that black holes with mass smaller than this are going to be quite rare the wild.

It is not necessary for primordial black holes to be small<ref>Andi Hektor, Gert Hütsi and Martti Raidal, "Constraints on primordial black hole dark matter from Galactic center X-ray observations", Astronomy & Astrophysics Vol. 618, article no. A139 (2018) https://doi.org/10.1051/0004-6361/201833483</ref>. They could have initially formed at any size. Indeed, there has been discussion among the scientific community that the seeds of supermassive black holes were primordial black holes which would necessarily have been of large size.

Surviving primordial black holes that are not supermassive black holes would contribute to the dark matter of the universe<ref>Bernard Carr, Kazunori Kohri, Yuuiti Sendouda, and Jun'ichi Yokoyama, "Constraints on Primordial Black Holes", arXiv:2002.12778 [astro-ph.CO] https://arxiv.org/abs/2002.12778</ref>. Indeed, it is possible that most of the universe's dark matter consists of these primordial black holes. Ocasionally, a small primordial black hole might pass through a solar system and be detected by its minute gravitational effects on planetary orbits<ref>Valentin Thoss and Andreas Burkert, "Primordial Black Holes in the Solar System", arXiv:2409.04518 [astro-ph.EP] https://arxiv.org/abs/2409.04518</ref>.

=== Artificial black holes ===

If you can't find a hole, maybe you can make one. If your culture is capable of assembling massive stars and you're willing to wait a few tens or hundreds of millions of years, this is something that can be done. However, if you're looking to make holes of sub-stellar size, no one today has even the faintest idea of how it could be done.

For quite a while, one of the favorite ideas was a method called a kugelblitz<ref name="Crane_Westmoreland">L. Crane and S. Westmoreland, "Are Black Hole Starships Possible" https://arxiv.org/abs/0908.1803</ref>. Technically, this can be any arrangement of radiant energy or energy made of fields that surpasses the Schwarzschild critereon and forms a horizon, but since the development of the laser one of the favorite kugelblitzes has been to shine many enormously powerful laser pulses into a tiny spot. When the laser pulses simultaneously reach the focal spot, their combined energy is sufficient to form a black hole.

Unfortunately, it doesn't work<ref>Álvaro Álvarez-Domínguez, Luis J. Garay, Eduardo Martín-Martínez, and José Polo-Gómez, "No black holes from light", arXiv:2405.02389 [gr-qc]
https://doi.org/10.48550/arXiv.2405.02389; Physical Review Letters 133, 041401 (2024)
https://doi.org/10.1103/PhysRevLett.133.041401</ref><ref>Ball, Philip (July 26, 2024). "Black Holes Can't Be Created by Light". Physics. American Physical Society (APS). Retrieved June 22, 2025. https://physics.aps.org/articles/v17/119</ref>. Before the light can get concentrated enough to self-gravitate into a black hole, it gets intense enough for light to start interacting with light. This scatters the light out of the beam, preventing the light from focusing tightly enough to form a black hole.

So that's the current state of the art. If there are ways to make small black holes, we haven't thought of them yet.

== Energy ==

=== Hawking radiation ===

Famously, nothing that goes into a black hole can ever come back out again. But something comes out. For it turns out that black holes have a temperature and that, like everything with a temperature, they emit radiation. In fact, being perfectly black, they radiate as a perfect black body. This radiation is called Hawking radiation after its discoverer, physicist [https://en.wikipedia.org/wiki/Stephen_Hawking Stephen Hawking]. For normal sized black holes, those the size of stars or galaxies, this temperature is very small and the radiation power is absolutely minuscule. But the smaller the hole, the hotter it gets and the more power it radiates. For a Schwarzschild black hole with mass M, the Hawking temperature TH is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
TH = &hbar; c3 / (8 π G kb M)
</div>
where &hbar; is Planck's constant, π is the circle constant, and kb is Boltzmann's constant. Curiously, this means that the wavelengths around the peak emission of light in its spectrum is near the size of its event horizon. The power radiated by a hole of this temperature in the form of electromagnetic radiation is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
PH = &hbar; c6 / (15360 π (G M)3).
</div>
However, there are additional forms of radiation beyond electromagnetic energy which will add to this radiated power. If the black hole's temperature (in units of energy, so multiply the temperature by the Boltzmann constant to get the units right) is of the same order or higher than the rest mass-energy of a type of particle, that type of particle will also be emitted. The lowest mass particles known that are not electromagnetic radiation are neutrinos. Neutrinos are slippery elusive little fellows and we still don't know their rest masses, but an upper bound on the rest mass of the lightest neutrino species is approximately 0.1 eV. This corresponds to a temperature of 1160 K and a black hole mass of about a hundred thousand trillion (1017) tons. Temperatures higher than this and masses lower than this will need to take neutrino radiation into account. A black hole with a mass of less than twenty billion (2×1010) tons at a temperature of 6 billion kelvin will be radiating electrons and positrons. As the mass continues to decrease additional particle types such as muons and pions will start to contribute to the radiation; at even higher temperatures quarks and gluons will be produced that decay into particle jets creating various hadrons. Gravitational waves will also be radiated away at all temperatures similarly to electromagnetic radiation. The fraction of radiation coming off as various particle types is shown in the table below for black holes large enough to have insignificant muon, pion, and heavier particle radiation.
<table border=1> <tr><td align=center>
<table border=1>
<tr><td>Mass (tons) <td> >> 1 × 1017 <td> << 1 × 1017 & >> 2 × 1010 <td> << 2 × 1010 & >> 1 × 108
<tr><td>Temperature (K) <td> << 1200 <td> >> 1200 & << 6 × 109 <td> >> 6 × 109 & << 1.2 × 1012
<tr><td>Temperature (eV) <td> << 0.1 <td> >> 0.1 & << 500,000 <td> >> 500,000 & << 100,000,000
<tr><td>Electromagnetic fraction <td> 90% <td> 11.8% <td> 7.6%
<tr><td>Gravitational fraction <td> 10% <td> 1.4% <td> 0.9%
<tr><td>Neutrino fraction <td> 0 <td> 86.8% <td> 55.7%
<tr><td>Electron & Positron fraction <td> 0 <td> 0 <td> 35.8%
</table>
<tr><td>Fraction of power emitted as different kinds of radiation as a function of mass for larger mass black holes<ref>D. N. Page, "Particle emission rates from a black hole: Massless particles from an uncharged, nonrotating hole", Physical Review D Vol. 13, No. 2, pg. 198-206, (1976)</ref>. For black holes smaller than 1 × 108 tons, the radiation doesn't so neatly separate with many new kinds of radiation coming on-line without as obvious separations between them. Near the threshold masses, there is a gradual transition from one radiation scheme to another as the temperature gets high enough to occasionally excite the new particle type over the existence threshold.
</table>

The radiated energy comes from the black hole's mass-energy, so a black hole will shrink over time as its mass is radiated away. As the mass decreases, the temperature goes up and so does the power output. So you get a runaway process of the hole getting hotter and hotter and radiating more and more power until POOF! It's gone in a flash of light and radiation. If you only consider the radiated electromagnetic energy the lifetime remaining of any black hole, assuming more mass doesn't fall into it, is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
tH = 5120 π G2 M3 / (&hbar; c4).
</div>
As this does not take into account radiation of other particle types, it is an upper bound to the lifetime; the radiation of other kinds of particles will also carry away energy making the black hole lose mass faster. Details for including the emission of other kinds of particles can be found in reference <ref name="MacGibbon II">J. H. MacGibbon, "Quark- and gluon-jet emission from primordial black holes. II. The emission over the black-hole lifetime", Physical Review D Vol. 44, No. 2, pg. 376-392, (1991)</ref>. As an estimate, you can divide the electromagnetic lifetime by the ratio of the total radiated power to the electromagnetic power; although this does not take into account the variation in this ratio as the black hole changes mass you might expect most of its lifetime to be in a range where the types of particles emitted are not changing dramatically and in such a case this approximation applies.

This is a neat result. It allows perfect conversion of mass-energy into radiant energy (although the neutrino and gravitational radiation will be rather inconvenient to capture). However, the actual implementation can get a bit inconvenient.

Let's skip for the moment the details of how you get a black hole. We'll assume that you have a magic black hole making box that can pop out whatever size of hole you need. Now let's say you want a megawatt of usable power (so we ignore the gravitational waves and the neutrinos). What size of hole do you need? It turns out to be a cool 38 billion metric tons. A hole that size is rather hard to carry around with you. And its temperature will be 3.2 billion kelvin. At that temperature its usable radiation is primarily electrons and positrons, with a good dose of hard x-rays and gamma rays for good measure. On the plus side, it's about 2000 times smaller in radius than a typical atom. So you could slip it into your pocket; just don't expect it to stay there.

Here we see one of the issues on trying to utilize Hawking power from black holes. Usable amounts of power generally come with horrendous power to mass ratios with the energy released as highly penetrating ionizing radiation. And if you start getting to masses that are more practical to deal with, you've got more of a bomb than a reactor – a 1000 ton black hole will release all of its 20,000 gigatons TNT equivalent in under a second.

Let's take an example of a black hole with a mass of 100 million metric tons, for reasons that will become clear later. We have already found that this hole is only about a fifth the size of a proton. But that tiny speck of compact mass has a temperature of 1.23 × 1012 kelvin. It puts out a radiated power of 1.4 × 1012 watts (of which something like 7 × 1011 watts is usable), which is a rate of mass loss of 15.6 micrograms per second. Or in somewhat more descriptive terms, the interacting radiation has about the energy released by the detonation of 170 tons of TNT every second. Left to its own devices, it will slowly get brighter and brighter, losing mass faster and faster, until it eventually radiates itself away in about 67 million years.

The description of Hawking radiation so far has assumed a black hole without charge or angular momentum. These properties will change the amount of radiation emitted for a given amount of mass. In particular, an extremal black hole of any kind has a temperature of zero and emits no Hawking radiation. A rotating black hole preferentially emits particles with spin and orbital angular momentum aligned with its own; a charged black hole preferentially emits particles with a charge the same as its own. Consequently, Hawking radiation will tend to discharge charged black holes and spin down rotating black holes. As angular momentum is emitted at a higher rate than mass-energy, rotating black holes will spin down to black holes with negligible rotation over timescales where loss of mass is appreciable<ref>D. N. page, "Particle emission rates from a black hole. II. Massless particles from a rotating hole", Physical Review D Vol. 14, No. 12, pg. 3260-3273, (1976)</ref>. Similarly, charged black holes will rapidly discharge from hawking radiation on time scales far faster than their rate of mass loss<ref name="Carter 1974"></ref>.

=== Penrose process ===

In a rotating black hole, anything entering the ergosphere gets pulled around the black hole by the spinning space-time. If you dive into the ergosphere and then shoot something backward against the direction you're being swirled in, this is a rocket and you get pushed forward just like any other rocket. But if you do the math<ref> R. Penrose and R. M. Floyd, "Extraction of Rotational Energy from a Black Hole". Nature Physical Science. 229 (6): 177–179. (February 1971). Bibcode:[https://ui.adsabs.harvard.edu/abs/1971NPhS..229..177P 1971NPhS..229..177P]. [https://doi.org/10.1038%2Fphysci229177a0 doi:10.1038/physci229177a0]. [https://search.worldcat.org/issn/0300-8746 ISSN 0300-8746]</ref>, if you dive in deep enough (but still outside the event horizon!) when you come out of the ergosphere you can be going much faster than if you fired your rocket outside the black hole. What gives? How can you get more energy than you started with? Well, it turns out that the energy came from the black hole itself. You decreased both the black hole's mass-energy and its angular momentum when you did that, and got shot out with that extra energy and angular momentum.

This has obvious uses for getting energy. If you drop things into the black hole, and have them push stuff out backward to fall into the black hole, you can harvest the black hole's rotational energy by using the dropped things to do work when they come zipping back out.

For an uncharged extremal rotating black hole and a trajectory grazing the event horizon, up to 20.7% of the mass-energy of the ejected particle can be returned as kinetic energy by this process. However, for a charged rotating black hole there is no upper limit to the efficiency of the process<ref>M. Bhat, S. Dhurandhar, and N. Dadhich, "Energetics of the Kerr-Newman black hole by the penrose process". Journal of Astrophysics and Astronomy. 6 (2): 85–100. (1985). Bibcode:[https://ui.adsabs.harvard.edu/abs/1985JApA....6...85B 1985JApA....6...85B]. CiteSeerX [https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.512.1400 10.1.1.512.1400]. doi:[https://doi.org/10.1007%2FBF02715080 10.1007/BF02715080]. S2CID [https://api.semanticscholar.org/CorpusID:53513572 53513572]</ref>. In fact, you can gain more energy from the Penrose process with a charged black hole than was in the mass-energy of the particle you ejected!

==== Penrose batteries ====

For an uncharged extremal rotating black hole, nearly 30% of the mass-energy of the black hole can be extracted via the Penrose process<ref name="Rees 1984">M. J. Rees, "Black hole models for active galactic nuclei", Annual Review of Astronomy and Astrophysics Vol. 22 pp. 471-506 (1984)</ref>. This percentage can get even larger for a charged rotating black hole.

Of course, once you extract that energy, you can't use the black hole for the Penrose process any more. However, you could charge it up again by throwing matter into the hole with high angular momentum with respect to the hole. It is even better if the matter is highly charged. Assuming that the black hole is large enough that it can be fed efficiently (see below), you can re-use your black hole battery over and over again.

==== Superradiant scattering ====

An effect similar to the Penrose process with matter can be accomplished with radiation. Light is shone into the rotating black hole. A portion is absorbed by the black hole, but more energy than was lost is given to the light by the ergosphere, a process known as superradiant scattering<ref>Ya. B. Zel'dovich, "generation of waves by a rotating body", ZhETF Pisma Redaktsiiu Vol. 14 No. 4 pp. 270-272 (20 August 1971)</ref><ref>J. D. Bekenstein and M. Schiffer, "The many faces of superradiance", Physical Review D. Vol. 58 064014. [https://arxiv.org/abs/gr-qc/9803033 arXiv:gr-qc/9803033]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1998PhRvD..58f4014B 1998PhRvD..58f4014B]. doi:[https://doi.org/10.1103%2FPhysRevD.58.064014 10.1103/PhysRevD.58.064014]. S2CID [https://api.semanticscholar.org/CorpusID:14585592 14585592]</ref>. If this light is then reflected back into the black hole again and again, it can get amplified indefinitely – at least until the intensity of the light gets so high that it breaks your mirror. The idea of enclosing a rotating black hole with a mirrored shell is called a black hole bomb<ref>W. H. Press and S. A. Teukolsky, "Floating Orbits, Superradiant Scattering and the Black-hole Bomb", Nature Vol. 238 pp. 211–212 (July 28, 1972). Bibcode:[https://ui.adsabs.harvard.edu/abs/1972Natur.238..211P 1972Natur.238..211P]. doi:[https://doi.org/10.1038%2F238211a0 10.1038/238211a0]. ISSN [https://search.worldcat.org/issn/1476-4687 1476-4687]</ref>. All of this allows you to extract the energy of a rotating black hole using light and receiving energetic light in return. You no longer need worry about the energy coming out as extremely penetrating radiation of high energy particles.

=== Feeding a black hole ===

If you are extracting energy from a black hole, you might want to eventually put that energy back in to avoid using up your black hole too soon. You can do this by letting mass or other forms of energy fall into the hole, passing through its event horizon to get trapped forever. If the infalling matter is charged, the black hole will aquire that charge. If the infalling matter is off-center or spinning, the black hole will acquire the angular momentum of the system once the matter is absorbed.

==== Tidal disruption ====

If you have something smaller in size than a black hole's event horizon and you drop it straight in, it should enter the hole without any particular complications. But as the object approaches the hole, the hole's changing gravity will affect different parts of the object differently. Gravity drops off with distance, so the parts of the object nearest the hole will be getting pulled harder than those furthest away. This means that once you account for the average force on the object accelerating it toward the hole, you have an additional force acting on the body to tear it apart along the direction to the hole. Meanwhile the direction of gravity is toward the center of the hole, pointing radially inward. Again, after accounting for the average force on the object this means that the parts furthest to the left are experience a residual force pointing to the right and vice versa. So the net result is that tidal forces stretch an object along the direction towards the center of the hole and squish it together in the directions transverse to that direction. This is called "spaghettification".

Tidal forces fall off faster than the average force of gravity on an object. Whereas gravity falls off with the square of the distance, tides fall off with the cube of the distance. So far out from a black hole, you might be falling comfortably but as you get closer the tides get strong quickly. Very large black holes, like the supermassive black holes at the center of galaxies, might not generate any noticeable tides even as you fall though the event horizon. Smaller holes, on the scale of stellar mass black holes, do generate enough tides to spaghettify any astronaut unlucky enough to fall into them.

==== Accretion disks and astrophysical jets ====

If the thing you drop into a black hole isn't dropping straight in – maybe it has a bit of transverse velocity as it gets sucked down – it is likely to miss the event horizon and slingshot around on an orbit. However, even as it misses the all-devouring beast at the center tidal disruption is still pulling the object apart. A close enough approach will have the tides rip apart the object and smear it out into a smudge of debris. The inner parts of the debris cloud will be orbiting faster than the outer parts, leading to shear flow and friction and drag. This leads to heating of the debris, coming from the object's kinetic energy. After enough passes the former object will get spread out into a ring around the hole, called an accretion disk. The closer the debris is to the hole, the faster the difference in speed between adjacent streamlines and the more heating will occur. So you can get the inner parts of the ring glowing brightly with radiated heat.

Most physical process that can feed matter into a black hole start with the infalling matter having some angular momentum. Because the angular momentum is conserved it naturally results in accretion disks forming as the matter falls in.

As the inner part of the disk radiates heat, it loses kinetic energy and gets a little bit closer to the event horizon. As it gets closer it gains heat at a greater rate and its temperature increases. When it gets hot enough, the matter turns into a plasma. To a good approximation, plasmas cannot cross magnetic field lines. A strong field with a diffuse plasma will have the plasma move along the field line direction. A dense, fast plasma, on the other hand, can bully through weak field lines, stretching out the field so that it moves with the plasma. In a turbulent plasma, or, in this case, a circulating plasma, the field gets stretched out enough that it can come back and meet itself, getting stronger and stronger. This dynamo effect will amplify even very weak fields within the accretion disk, forming a strong magnetic field near the black hole.

And this is where things get a bit weird. Something happens – we're still not entirely sure what – and the interaction of the strong field with the energetic plasma right near the event horizon creates jets of fast moving plasma, high energy particles, and electromagnetic radiation shooting out along the axis of the accretion disk, usually in both directions.

In some cases, the circling debris may puff up into a shape more like a doughnut than a flat disk. These toruses are generally expected to be less efficient at radiating energy out of the infalling matter<ref name="Rees 1984"></ref>, with the radiation getting trapped in the torus and serving to puff it out rather than escaping.

The accretion disk process around a non-rotating, uncharged black hole can extract up to 5.7% of the mass energy of infalling matter into radiated energy and energy of the jets. The efficiency at radiation can increase to up to 42% for an extremal rotating black hole<ref name="Rees 1984"></ref>. If this radiated energy from the accretion disk can be collected, it can provide an additional source of energy beyond what you can get from Hawking radiation and its somewhat inconvenient limits. So now we must see what limits the rate of accretion to see how much energy we can get out of it and also how fast we can recharge our hole for the extraction of Hawking and Penrose energy.

==== Mass collection rates ====

Suppose you have a black hole inside of some material. This might be a rock, or a star-hot plasma, or the diffuse gas of interstellar space.

If you are at rest with respect to the surrounding material, you'll get that material falling toward you. It will pile up as it crams together trying to get to the hole, until you reach a point where the flow turns super-sonic and the material free-falls the rest of the way into the hole. Finding the feeding rate is thus a [https://en.wikipedia.org/wiki/Choked_flow choked flow] problem.

If the hole is moving through the material faster than the speed of sound, material passing close to the hole will get deflected by the hole's gravity to converge in a wake behind it. Where it collides with other gas coming in from all directions in the wake, the gas comes to a halt and from there it can freely fall into the hole from behind.

The analysis of these two limits may be combined to give the Bondi-Hoyle accrection rate<ref>Edgar, Richard (21 Jun 2004). "A Review of Bondi-Hoyle-Lyttleton Accretion" https://ned.ipac.caltech.edu/level5/March09/Edgar/Edgar2.html https://arxiv.org/abs/astro-ph/0406166</ref>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ṁBH = 4 π ρ G2 M2/ (cs2 + v2)3/2
</div>
where ρ is the density of the stuff the hole is in, cs is the speed of sound in the medium, and v is the speed of the hole through the medium. The distance at which the in-falling material goes from subsonic choked flow to supersonic free-fall is the Bondi radius
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
rB = 2 G M / cs3.
</div>
The speed of sound in a solid makes a useful approximation for where inertial effects overcome material strength effects. Thus, the Bondi radius can serve as a useful approximation of how big of a channel will be ripped out of something that has a black hole pass through it.

If the Bondi-Hoyle accretion rate is too low, the black hole will be losing mass faster to Hawking radiation than it will be gaining mass to accretion. This depends on the variables described above, but let's look at what happens if we drop it into solid rock. Assuming a typical density of rock of 2.7 grams per square centimeter and a sound speed in rock of about 5 kilometers per second, we find that holes that are larger than 105 million metric tons are able to absorb a net gain in mass while those below this limit lose more mass to Hawking radiation than they gain by eating the rock. If you want to feed your hole with rock, you'll need it to be bigger than 105 million metric tons. The Bondi radius for such a black hole will be about half a micrometer, or about 5000 atoms in radius, so the tunnel it will make falling through rock will be fairly small.

The best material for feeding your black hole, according to the Bondi-Hoyle accretion rate, is the heavy metal thallium. If you drop your hole into a blob of thallium, it can achieve a net mass gain at a mass of only 22 million metric tons. For black hole masses below this, you cannot feed a black hole on normal matter at room temperature and pressure (whether it can feed at the crazy high pressures at the cores of planets or stars is a subject not explored here).

==== Radiation pressure ====

Both the Hawking radiation and the radiation from the accretion disk will be shining out of an accreting black hole. This radiation will encounter material from the accretion disk. The radiated light can scatter off electrons in the disk material; on average, this will push them outward. The electrons will then drag any assorted atomic nuclei in the disk material with them. This puts a limit on how much material can flow into the black hole – if it is too bright, it will push everything away. If the hole gets brighter than this limit, it can no longer feed.

This is often referenced in terms of the Eddington luminosity
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
LE = 4 π G M (A/Z) mp c / σT
</div>
where A is the average atomic weight of the plasma, Z is the average atomic number, mp = 1.672622 × 10-27 kg is the mass of a proton, and σT = 6.65246 × 10-29 m2 is the Thompson cross section for scattering light off an electron. If something is shining with the Eddington luminosity, it will keep matter from falling in. Strictly speaking, this assumes hydrostatic equilibrium; for problems that are time varying or with steady-state flows the Eddington limit does not necessarily apply. However, it is often a good first guess to figure out when the radiation chokes off the inflow in accretion disks. There are some configurations of accretion disks that can support luminosity higher than the Eddington limit, but most are at or below this limit.

If we assume that our black hole's accretion disk is Eddington limited, we can find out how big it needs to be in order to accrete any matter at all, or to achieve net mass gain after its Hawking radiation losses are accounted for. In hydrogen gas, with A/Z = 1, we find that a hole must have a mass of at least about 104 million metric tons for any matter to fall in past the Hawking radiation pressure. The hole's mass has to be in the 109 to 125 million metric ton range to gain mass via accretion faster than it is lost to Hawking radiation, depending on the efficiency at which matter in the accretion disk is converted into radiation. If you drop the hole into rock or other light elements you'll have an A/Z ratio of 2 or very slightly higher. Setting A/Z = 2, we find that you can't get any accretion for masses under 85 million metric tons and, again depending on the radiative efficiency of the accretion disk, you need somewhere in the range of 90 to 103 million metric tons to reach breakeven in terms of mass loss versus mass gain. Even for very heavy elements like lead or uranium, with an A/Z ratio of approximately 2.5, you need at least 80 million metric tons to accrete matter at all and somewhere between 84 and 97 million metric tons to break even.

In other words, if you want to be able to add mass to your black hole by having it gobble up surrounding matter, you'll want it bigger than many tens of millions of metric tons.

Interestingly, the limit for net mass gain for the Eddington limit is very similar to that of the Bondi_Hoyle limit. In order to get a black hole that gains mass, you're pretty much going to need at least a mass somewhere near the 100 million metric ton range.

==== Reaction rates at sub-atomic sizes ====

We now know the rate at which matter can fall on to a black hole, getting past both the radiation coming from the hole and its inner accretion disk and for getting past the choked flow of the material getting in its own way. But what about when it reaches the hole? Obviously, if the hole is bigger than the size of an atom any atoms it touches will immediately get sucked in. But a lot of holes of engineering interest are much smaller than this. A black hole with a mass of 100 million tons would have a Schwarzschild radius of about 5.7 times smaller than that of a proton. If a hydrogen atom fell into the hole, it would end up sitting there with the black hole inside of the proton. How quickly could the hole slurp up that proton and its companion electron?

 Consuming protons and neutrons 

It is easy enough to get an estimate of how fast a proton or neutron will get eaten once a black hole is inside of it. Both protons and neutrons have a radius of about 8.4 × 10-16 meters. Both are made up of three quarks. This gives a quark density of about 1.21 × 1045 / m3 inside of the proton or neutron. Because the binding energy of the quarks is much larger than the mass-energies of the quarks, we can assume that they are highly relativistic and are moving at about light speed. Multiply the density by the speed to get the flux (particles passing through per area per time) of about 3.62 × 1053 quarks / m2 / s. Then multiply by the surface area of the hole to get the absorption rate of the quarks. Once one quark is eaten, color confinement ensures that the rest of the quarks cannot leave and the particle is stuck to the black hole until the rest of it is eaten, which time we can guestimate by the time needed to eat three quarks. For our 100 million ton black hole, this shakes out to about 3 × 10-23 seconds to eat a proton or neutron, or 3.3 × 1022 protons and neutrons eaten per second. If we multiply by the mass of a proton or neutron, we find that the 100 megaton black hole can eat protons and neutrons at a rate of about 5.6 × 10-5 kg/s if it has a constant supply of protons and neutrons ready to immediately fall into the hole once the previous one was eaten. Which is comfortably higher than the loss to Hawking radiation of 1.56 × 10-5 kg/s.

This is okay for neutrons (if you can somehow find a supply of free neutrons), but for protons there is a problem. For every proton the hole eats, it gains one unit of elementary charge (that is, the charge that the proton had gets added to the charge of the hole). If it eats enough protons, it will gain enough charge to repel away any other proton (or atomic nucleus) that comes near enough to it that the electrons around the atom can no longer screen the electric charge of the proton or nucleus. The potential energy of a proton or nucleus bound to the black hole by their mutual gravitational attraction is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
UG = -mp A M G / r
</div>
and the potential energy of the repulsion between the proton or nucleus and a charged hole that has absorbed Y other protons is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
UE = [Y Z q2 / (4 π ε0)] / r.
</div>
Here, Z is the number of protons in the nucleus under consideration (Z = 1 for a single proton), A is the number of protons + neutrons in the nucleus (A = 1 for a single proton), q = 1.602176487 × 10-19 C is one unit of elementary charge, ε0 = 8.854187817620 × 10-12 C2 / J / m is the permittivity of free space, mp = 1.67262192369 × 10-27 kg is the mass of a proton, and r is the distance between the black hole and the proton or nucleus.
If the sum UG + UE is negative, the hole still attracts the proton or nucleus and matter free-falling into the hole can collide with the hole without issue. If the sum is positive the force is repulsive and the proton or nucleus cannot approach the hole. We see that this happens when
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Y = 4 π ε0 mp (A/Z) M G / q2.
</div>
For our 100 megaton black hole eating hydrogen (which has only protons as a nucleus), the hole can charge up to a maximum of Y = 49. For heavier nuclei with a mass to charge (A/Z) ratio of 2, the hole can charge up to Y = 97. Whatever the case, if the hole cannot get rid of this charge fast enough, the hole will get too much charge to freely eat everything falling into it.

 Discharging via Hawking radiation 

There are many ways that the hole can shed its charge. It's gravitational field and positive electric charge pulls negatively charged electrons in to a high density, it can simply eat these electrons to reduce its charge. Alternately, the electrons densely packed around the protons might get captured by the protons to form neutrons that can fall into the hole and keep feeding it. For this case, however, the most efficient means of reducing the hole's charge is from its Hawking radiation.

The hole will have a chemical potential for electrons of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
μ = q2 Y / (4 π ε0 rs),
</div>
which is the potential energy to bring an electron from far away to the event horizon. If the chemical potential is significantly larger than the Hawking temperature (in energy units) and if the Hawking temperature (in energy units) is significantly larger than the mass energy of an electron
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
μ > kB TH > me c2
</div>
then the rate of positron emission from the hole is approximately μ/&hbar;<ref name="Carter 1974"></ref>. Our 100 million ton hole with Y > 10 meets both these criteria. For Y = 11 the rate of positron emission is 1.6 × 1023, a full order of magnitude larger than the rate at which protons can be absorbed, and only increases as the charge goes up. This discharges the hole faster than it is charged by gobbling up protons. We thus see that nothing prevents matter from falling into the hole at the macroscopic accretion rates.

== Propulsion ==

People often like to get from one place to another. A black hole gives you various options for moving things around.

=== Penrose launcher ===

If you have a large enough rapidly rotating black hole, you can drop an entire spacecraft in it. If you get deep enough into the ergosphere, you can use the Penrose process by firing your rockets at the point of closest approach. Now you can get yeeted out at ridiculous speeds. If you can survive the tidal forces that close to the event horizon, you can potentially get a machine for flinging you around the galaxy at relativistic speeds.

=== Black hole rockets ===

Taking a black hole with you has the advantage that you don't need to rely on any black hole based infrastructure at your destination. An obvious method of propelling yourself with a black hole is to use the energy emitted by a hole to energize your propellant, rather than using a chemical or nuclear reaction for your rocket thrust. Perhaps you can directly use the astrophysical jet as your rocket propellant. Or the radiant light or energy from Hawking radiation<ref>[https://www.researchgate.net/publication/293633217_Acceleration_of_a_Schwarzschild_Kugelblitz_Starship J. S. Lee, "Acceleration of a Schwarzschild Kugelblitz Starship", Journal of the British Interplanetary Society pp. 105-116 (2015) ]</ref><ref name="Crane_Westmoreland"></ref> or a black hole bomb as a photon drive. All of these methods will require careful engineering to avoid very low accelerations from the high mass of the black hole while avoiding getting a black hole so small that it immediately evaporates in an explosion far larger than your spacecraft can survive.

== Making Holes in Things ==
Sometimes, you need to put a hole in something. Not in the sense of putting a black hole inside of something, but drilling a cylindrical hole through something. Perhaps you are interested in machining part out of difficult to work materials. Perhaps you want to build a weapon that perforates your enemies. In either case, if you have a black hole available you could imagine sending the black hole through the target object and leaving a hole ... or at least a region of gravitationally disrupted material ... behind.

For its frontal surface area, a black hole has an enormous mass. It's sectional density and the pressures it exerts on the material it passes through will be so high that it will essentially ignore the material in its way. After passing through enough material, it will eventually be slowed down both by accumulating mass and through drag forces, but that will occur over distances well beyond what we are concerned with here. For practical purposes, the black hole will just punch through without being impeded in any way by the object in its path. Our goal is to figure out what happens to that object.

=== Direct absorption ===
Obviously, anything which directly encounters the event horizon will be lost forever. This gives us a lower bound on the size of the hole left as the black hole diameter of twice the Schwarzschild radius 2 rS.

=== Gravitational disruption ===
A more significant effect is how the black hole will gravitationally accrete the material it passes through and eventually consume it. We have already looked at [[Black_Hole_Engineering#Mass_collection_rates|Bondi-Hoyle accretion]]. The choked flow treatment takes as a cutoff where the infalling fluid transitions from subsonic to supersonic speeds at the speed of sound. But the speed of sound is also a reasonable estimate of where inertial effects overcome material strength effects. Motion due to gravity is fundamentally inertial, so we can take the Bondi radius rB as a rough estimate of the distance where the black hole's gravity is able to rip material apart. If the black hole is moving slowly compared to the speed of sound, this material will be consumed; if it is moving much faster than the speed of sound it merely leaves a gravitationally disrupted trail behind it. In either case we are left with a region of diameter 2 rB where the target object is torn apart.

=== Vapor explosions ===
The black hole will emit radiation into the target object as it passes, either from Hawking radiation or from the radiation coming from its accretion disk. In practice, much of the Hawking radiation from small black holes will be in the form of highly penetrating radiation. But if we make the assumption that the radiation is absorbed locally (a reasonable assumption for larger black holes where the temperature is on the order of 10 keV or less) we can find the energy deposited per distance traveled by a black hole moving with speed v as dE/dx = PH/v. Any neutrinos or gravitational waves emitted will be far too penetrating to affect this calculation; consider only the Hawking power from interacting particles (and even then, the muons, pions, hadronic showers, and weak vector bosons that you get from the smaller black holes all put a significant fraction of their decay energy into neutrinos, so only part of their energy can be used).

The radiation from the accretion disk is likely to be more amenable to local absorption. Find the rate of accretion, multiply by the square of the speed of light to find the mass-energy accretion rate, and then by the efficiency ε of turning accretion disk mass energy into radiation that was discussed earlier. Then divide by the speed to find the energy deposited per distance traveled to get dE/dx = ṁBH c2 ε / v. Add this to the Hawking energy deposition to get the total dE/dx. If the accretion is Eddington limited, the accretion rate cannot bring the energy deposition above LE/v.

Under the assumption that this energy is absorbed locally, it will heat a cylinder of material to a high pressure vapor. This vapor will then expand, pushing surrounding material violently away. The radius of the resulting cavity can be found if you know the cavity strength of the material Kc. This can be found from the compressive strength K and the shear modulus G, both of which can usually be looked up for many common materials:
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Kc = (2/3) K + (1 + ln(2 G/K))
</div>
The volume of a cavity blown out by an energetic event will be Kc times the energy release. This gives a radius of the cylinder exploded out of the target object of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
rv = √[(dE/dx) / (π Kc )]
</div>
The diameter of the exploded hole will be twice the radius.

Reference <ref>Robert J. Scherrer, "Gravitational Effects of a Small Primordial Black Hole Passing Through the Human Body", [https://arxiv.org/abs/2502.09734 arXiv:2502.09734 [astro-ph.CO]]</ref> gives one attempt to estimate the effects of a micro black hole passing through the human body. Here, they assume that the black hole has a speed on the order of the dark matter velocity dispersion of around 200 km/s, and find a minimum mass for serious injury or death to a human victim of 1.4×1014 kg. That work used different assumptions than are used here. If we take a black hole of that mass and speed passing through the human body (taking water as the primary constituent such that density 1 gram/cubic centimeter, A = 18, Z = 10, and a speed of sound of 1500 m/s) the Bondi accretion limit is 0.14 g/s (far less than the Eddington limit, so we are Bondi limited rather than Eddington limited). The Bondi radius is 8.3 mm, so we can assume that the gravitationally disrupted tissue alone is equivalent to the effect of a 16.6 mm bullet. If we assume a 5% efficiency at turning the mass-energy of the accretion disk into radiation, we get an accretion power of 616 GW, leading to a linear energy deposition of 3.08 MJ/m. The Hawking radiation is negligible compared to this, so we ignore it. The cavity strength can be crudely approximated as 1.2 MPa, which gives results roughly consistent with ballistics gelatin results. Crunching through the calculations, we find that the vapor explosion blows out a hole 90 cm in radius (180 cm in diameter), which is enough to explosively disassemble the entire person into splattered gibbets. We therefore see that the vapor explosion is the most significant factor and that the given 1.4×1014 kg is a significant overestimate of the minimum dangerous mass of a black hole.

== Gravity Generation ==

People are healthiest when living in gravity. If you want to go out in space, there is no gravity. Even on worlds, if the world is small enough there might not be enough gravity for good health.

There are many proposals to address this, and they mostly involve spinning things around in centrifuges. Which, to be perfectly honest, is probably always going to be a better approach to making gravity than black holes. But we're not here for practicality, so lets look at using black holes as a gravity source.

The source of gravity we are most familiar with here on Earth is gravity from mass. You need a lot of mass to generate just a little bit of gravity, so it seems rather inefficient. However, the closer you can get to your mass the more gravity you get, following Newton's law of universal gravitation
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
g = G M / r2
</div>
where lower case g is the acceleration due to gravity, upper case G = 6.67430×10−11 m3/kg/s2 is the gravitational constant, M is the mass making the gravity, and r is the distance between the center of the mass and the place where you are measuring the gravitational acceleration. Technically, this is only for point masses or spherically symmetric masses, but we will be dealing with planets and black holes which are generally pretty close to spherical in most cases so we're okay. Given this, we can get the same gravity the closer we can get to the source of our mass without going inside of it which in turn argues for using the densest source of mass we can find. Which is black holes.

Gravity on Earth has a value of g&oplus; = 9.8 m/s2. If we know the mass of our black hole, we can plug this in to the law of universal gravitation to find how far away we need to be to get a comfortable gravity. However, there is another consideration. Your head and your feet will be at different distances from the center of the hole, so if you are standing up your feet will experience more gravity than your head. The average person is somewhere around 1.5 to 2 meters tall, so if you need to be 10 cm from the black hole for 1 g&oplus; at your feet your head will nearly be in freefall. So we also want the distance for 1 g&oplus; to be significantly larger than a human height.
;
Let's take, for example, a case where we have 1 g&oplus; at a distance of 10 meters. Plugging this in to the law of universal gravitation, we find that we need a mass of 14.7 billion tons. Given that we need to pack all of this into a sphere with a radius of 10 meters or less, we require a density of more than 3.5 million grams per cubic centimeter. The densest material known is osmium, which is 22.6 grams per cubic centimeter. As we need a density five orders of magnitude more than this, normal materials will not cut it. Electron degenerate matter can approach these densities, but electron degenerate matter cannot hold itself together and will spontaneously explode under environmental conditions suitable for human life (specifically, if the gravity is only 1 g&oplus;) so we can rule that out. Neutron degenerate matter has the same issue. Which leaves black holes as our only option.

Such a hole would be smaller than an atom, although substantially larger than an atomic nucleus. It will produce about 20 MW of hard radiation but most of that is neutrinos; only a bit over 8 MW is going to interact with normal matter – mainly several hundred keV gamma rays, positrons, and electrons which are all easy enough to shield against. The black hole will last much longer than the current age of the universe and if you need to feed it the Eddington limited rate is a few grams per second while the Bondi limit is about a quarter kg/s for rock, a few kg/s for water, or a couple hundred kg/s for thallium. As far as the gravity, if your feet are at 1 g&oplus;, then (assuming you are 1.7 m tall) your head will experience about 3/4 g&oplus;. This is probably both healthy and comfortable, the black hole is relatively benign, and so this presents one option for artificial gravity.

== Computation ==

A black hole's event horizon has a temperature. This implies, via thermodynamics, that it has an entropy. In information theory, the entropy of a system is a measure of its information content, and thus the Hawking radiation coming out of the black hole is the rate at which information is returned to the outside world. This brings up the idea of, what if you could input information via coded messages into the black hole, have the black hole process that information, and then return that information as patterns and correlations in its Hawking radiation?

If this all sounds very hand-wavy, that's because it is. You could apply the same argument to the glow coming off of a bar of hot iron. But one work<ref>G.R. Andrews III, "Black hole thermodynamics", Results in Physics,
Volume 13,
2019,
102188,
ISSN 2211-3797,
https://doi.org/10.1016/j.rinp.2019.102188.
(https://www.sciencedirect.com/science/article/pii/S2211379719304036)</ref> has looked into this concept and found ways, at least in principle, to make black holes Turing complete so that they can be used, again in principle, as a computer. This raises the possibility of arbitrarily advanced civilizations with near omniscient abilities to measure radiation using black holes as the ultimate computation device<ref>S. Lloyd and Y. J. Ng, "Black Hole Computers", Scientific American (April 1, 2007) https://www.scientificamerican.com/article/black-hole-computers-2007-04/</ref>.

== Containment ==
<blockquote>
There were a dozen other questions that Duncan was longing to ask. How were these tiny yet immensely massive objects handled? Now that Sirius was in free fall, the node would remain floating where it was--but what kept it from shooting out of the drive tube as soon as acceleration started? He assumed that some combination of powerful electric and magnetic fields held it in place, and transmitted its thrust to the ship.

Arthur C. Clarke, Imperial Earth
</blockquote>

So, you have a black hole. And let's say you want to use it for a mobile application. This means you need to move it around. As you are likely dealing with something that has a mass of millions of tons or more, it will take a lot of force to accelerate it just a little bit. If you are going to use it for thrust for your spacecraft, or even if you need to move it around somewhere using a spacecraft, you're going to want to make sure it doesn't get left behind when your spacecraft moves. As you can see from the quote above, even some of the foremost minds in science fiction simply hand-waved this detail away.

This can get particularly bothersome if you are on a planet. A basic 100 million ton black hole weighs, well, 100 million tons. Or about a trillion newtons of force. It's smaller than the nucleus of an atom. Any chemical bond will fail with a force of about 0.010 μN; the black hole will exert something like fourteen orders of magnitude more force than is needed to break any known force holding it to other atoms in matter. The pressure of all the force concentrated into such a tiny area means that nothing material could keep it from simply falling down. After which it will end up orbiting through the planet, mostly ignoring the matter in the way but gradually slowing down over geological time spans. If this happens and you wanted to do something other than geoengineering with your black hole, you're probably out of luck.

So how can you exert a force on a black hole?

By Newton's third law of motion, anything that gets gravitationally attracted to a black hole also exerts the same force back on a black hole. A black hole near something else massive will be tugged toward the massive thing as the massive thing pulls the black hole. So if that massive thing is made out of matter, you can pull the thing which can pull the black hole. Unfortunately, the resulting force is probably going to be really weak. If you had a 200 meter diameter ball of osmium (the densest material known) it would have a mass of 95 million tons. At the surface of the ball, it would attract a black hole with a gravitational acceleration of 0.63 mm/s2; about 1/15,500 that of Earth's gravity. The acceleration is pitiful, and you're going to have to be carrying around a lot of extra mass (whether it is a significant amount of extra mass compared to your black hole is another matter). But you can apply the acceleration continuously over long periods of time. If you use this to couple your black hole rocket to your spacecraft you can accelerate at 54 m/s per day; or a km/s every 20 days. Perhaps surprisingly, this is not entirely unworkable.

Note that this method does not provide overall propulsion. Conservation of momentum dictates that you still must use some kind of thruster than expels or exchanges momentum with the outside environment. Rather, this gives you the limits at which your black hole can be accelerated by whatever method you are using to move your spacecraft and the hole without the hole falling away.

You can also electrically charge the black hole. This will give it an electric field. If the black hole is also spinning, the combination of spin and charge will give it a magnetic field. You can then push or pull on the black hole with beefy capacitor plates or electromagnets. However, it can be challenging to give a black hole a large charge, or to have it keep its charge for long.

One problem is the electrical potential of the hole.
A black hole will have a capacitance of
<div align=center> C = 4 π ε0 rS</div>
where ε0 = 8.8541878188×10−12 F/m is the vacuum permittivity.
The potential &Vscr;, in volts, for a black hole with a charge Q in coulombs, is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"> &Vscr; = Q / C </div>
and the energy to charge the black hole up is
<div align=center> W = (1/2) C &Vscr;2.</div>
Generally, the charge you can achieve is limited by the voltage (or energy per particle, expressed in eV) you can get with your particle accelerator. For a given &Vscr;, this means the most charge you can put on your hole is
<div align=center> Q = C &Vscr;.</div>

With modern accelerators, we might get electrons up to an energy of 1 TeV (1×1012 eV), for a potential of &Vscr; = 1×1012 V.
For our example 100 million ton black hole, this gives a charge of Q = 1.65×10-14 C with a negligible charging energy. We can put this next to a highly charged capacitor plate to accelerate it. You can generate fields as high as the vacuum breakdown limit for the materials used to make your plate, which is typically about 𝓔 ~= 108 V/m. The force is F = Q 𝓔, or about (very roughly) 1 μN. Using F = M a, the acceleration a produced is a rather pathetic a ~= 10-17 m/s2, or about 10-18 g&oplus;. This is not going to get anyone anywhere in a reasonable time! But you can at least see the math needed to figure out how to move the hole so you can work other examples for yourself.

For electric containment, it is interesting to note that because rS</div> is proportional to the black hole mass, the capacitance is also proportional to the mass. So for a given attainable voltage the charge on the black hole is proportional to the mass. And consequently, for a given electric field the force on the black hole is proportional to the mass. With the final result that for a fixed voltage and electric field strength, the acceleration of the black hole you can get with electric methods is entirely independent of its mass.

If you have a charged rotating black hole, as described earlier it will have a magnetic moment. If you put a magnetic moment in a magnetic field gradient dB/dx the magnetic moment will experience a force F = m dB/dx. If we take our 100 million ton black hole charged up to a trillion volts from above, and give it enough spin that it becomes extremal, you will have an angular momentum of J = 2.2×10-8 kg m2/s. This gives it a magnetic dipole moment of m = 3.7×10-33 A m2. The highest magnetic field gradients we have managed to achieve have been about a GT/m<ref>[Zablotskii, V., Polyakova, T., Lunov, O. et al. How a High-Gradient Magnetic Field Could Affect Cell Life. Sci Rep 6, 37407 (2016). https://doi.org/10.1038/srep37407</ref>. Thus, we have a force of approximately 3.7×10-21 N and an acceleration of about 3.7×10-32 m/s2, which is many orders of magnitude worse than the already pathetic electric field case. But again, using these tools you can work out for yourself the best way to move your black hole if your black hole is not 100 million tons or is charged to a different potential. In particular, for a given voltage and magnetic field gradient, the acceleration should scale linearly with the black hole mass, thus favoring larger black holes.

But there is another issue to consider. If e &Vscr; / (TH kB), for e the fundamental charge, is not much less than 1, you will get significant discharging from the hawking radiation emitting unbalanced numbers of electrons and positrons. For e &Vscr; / (TH kB) much larger than 1 and for TH kB / (me c2) much larger than 1, the discharge rate is approximately e2 &Vscr; / &hbar;<ref name="Carter 1974"></ref>. In our previous example with a 100 million ton black hole, e &Vscr; / (TH kB) is about 10,000 and TH kB / (me c2) is about 200. Because these are much larger than 1 we can use our discharging estimate to find a discharge current of I = 24 million A. In a tiny fraction of a second, our charged black hole would be neutral again. Keeping it charged requires a power of P = I &Vscr; = 24 million terawatts from our particle accelerator.

But we have one more lever left to pull here. Momentum is conserved, so if we can get our black hole to consume matter moving at high speed the momentum of the matter the black hole eats will be transferred to the black hole. With a little bit of calculus you can find that for a Bondi-limited black hole, the optimum speed to shoot your mass stream at the black hole is v = √2 cs. The force on the black hole is v ṁBH.

Again for our example 100 million ton black hole, if we shoot it with a jet of thallium at 1157 m/s (the optimum for thallium's speed of sound) the black hole will experience a force of 2.7 N and an acceleration of 2.7×10-11 m/s2. This is still much less than the gravity tractor that was the first suggestion we floated for pulling a black hole; but at least it is much better than using electric or magnetic fields! Again, this is just one example. Black holes with different masses will get different results. In particular, because the Bondi accretion rate increases proportionally to the square of the mass, the acceleration you can get from shooting your black hole with a mass jet will increase linearly with its mass and thus favor larger black holes for more reasonable accelerations.

== Credit ==
Author: Luke Campbell

== References ==

[[Category:Engineering‏‎]][[Category:Physics & Engineering‏‎]][[Category:Physics & Math & Engineering]][[Category:Physics]][[Category:Astronomy & Cosmology]][[Category:Infrastructure]][[Category:Propulsion]]

Warp Drives

2026-07-31T18:50:40Z

Lwcamp: /* All the science, demonstration of the problem */

{{PageConstructionNotice}}

Science fiction often features spacecraft that can seemingly move across space and get between the place of departure and the destination much faster than light could have done. This appears to contradict the theory of relativity, which predicts unequivocally that nothing can move through space faster than light. Because relativity has been incredibly successful at describing nature, with its many other predictions regularly being confirmed to extraordinary accuracy and within the bounds of uncertainty of all the experiments that tested them, it gives confidence that relativity is a correct description of reality. Which seems to rather throw a wet towel on our hopes for rapid travel between stars.

However, while relativity does not allow things to move through space faster than light, it places no such restrictions on how fast space-time itself can expand, contract, or move around. This leads to the idea of a warp drive – the spacecraft remains stationary within a region of highly curved space-time, and that region moves at super-luminal speeds rather than the spacecraft.

== The Alcubierre warp drive ==

The first warp drive geometry that satisfied the Einstein field equations of relativity was proposed by Miguel Alcubierre<ref>M. Alcubierre, "The warp drive: hyper-fast travel within general relativity." Classical and Quantum Gravity. 11 (5): L73–L77 (1994). [https://arxiv.org/abs/gr-qc/0009013 arXiv:gr-qc/0009013]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1994CQGra..11L..73A 1994CQGra..11L..73A]. doi:[https://doi.org/10.1088%2F0264-9381%2F11%2F5%2F001 10.1088/0264-9381/11/5/001]. S2CID [https://api.semanticscholar.org/CorpusID:4797900 4797900].</ref>. In this geometry, a sphere of space-time moves at an arbitrary speed (potentially but not necessarily a speed much faster than light). Objects within the sphere are moved along with the sphere; an object at rest within the sphere would be moved along with the sphere indefinitely. Space is expanding at the rear boundary and contracting at the front boundary in order to keep the sphere moving. In order to satisfy the Einstein field equations, the boundary of the sphere must have a negative energy density. The challenges of space-time geometries with negative energy densities are described in our page on [[Wormholes#Exotic_energy_conditions|wormholes]], for our purposes it is enough to note that negative energy densities can pose problems if not handled carefully, there may be limits on how much negative energy you can have without nearby positive energy density, and it may not be possible to get enough negative energy to support a warp drive; although none of this is ruled out by physics – yet!

<table align=center>
<tr><td width=650>[[File:Energy_magnitude_of_Alcubierre_drive.png]]
<td width=650>[[File:Expansion_of_Alcubierre_drive.png]]
<tr><td>The magnitude of the energy of the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. (The energy density is negative everywhere in the shell, here only the magnitude is shown to aid visual comprehension.)
<td>The expansion of space in the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. Negative expansion means that space is contracting in that direction, positive expansion that it is expanding.
</table>

NOTE: What the? The energy distribution is symmetric! How does the drive know which way to go? There must be significant contributions of other components of the stress-energy tensor, and those have got to be asymmetric along the forward/backward axis. Check this when I get time.

(And what's with the ring in all the artwork, anyway?)

=== So what is going on? ===

What is it that makes a warp drive a warp drive, and how does it work?

A warp drive is a concept that comes out of the science of general relativity. General relativity collects space and time together into a single "structure" called spacetime. This structure is curved, and its curvature is determined by the material that is in it. Absent forces acting between matter, particles of matter move through spacetime on the shortest possible paths in this curved environment.
<blockquote>
"Spacetime tells matter how to move; matter tells spacetime how to curve." - John Archibald Wheeler
</blockquote>
All of this is encompassed in the Einstein field equation:
<div align=center>
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 8πG
<td rowspan="2" nowrap="nowrap" align="right"> T = R -  
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right"> R g
<tr>
<td style="border-top:solid 1px black;"> c4
<td style="border-top:solid 1px black;"> 2
</table>
</div>
Unfortunately, by convention, both the symbols G and R get used for a lot of different things. Here, symbols in bold are second rank tensors – constructs of arrays of 4×4 numbers where 4 is necessary because that is the number of dimensions of spacetime. Conventionally, these numbers are indexed by a value going from 0 to 3, with 0 being the part corresponding to the coordinate of time and 1 to 3 being the Cartesian coordinates that you might know as x, y, and z.

T is the stress-energy tensor: the (0,0) component represents the energy density at a given point; the (0,i) and (i, 0) components (with i going from 1 to 3) represent the momentum density; and the (i,j) components represent the stresses present, such as pressure or tension or shear.

The other term, R - R g/2, involves a partial description of the curvature. A full description of the curvature requires a construct of 4×4×4×4 components; too much for the stress energy tensor to fully constrain. Which is okay, because this allows curvature to exist away from where matter is, allowing gravity to have long range effects across empty space as well as for phenomena like gravitational waves. But it is enough that a given distribution of matter T gives enough constraints (along with specifying how the geometry behaves far away) that you can generally figure out what the actual gravity is.

This is the usual story that we are told. But there is an alternate way of looking at the same geometry such that the math works out the same, but it is described conceptually in a very different way. This divides spacetime up into "layers", called foliations, that represent slices of space and going from one foliation to the next is increasing in time. This is useful for computation because you can assign coordinates to your system. There are two additional constructs – the lapse function and the shift vector field. Starting at an event on one foliation, the lapse function at that position tells you that after a certain "proper time" (time perceived by an observer at that event) the time coordinate of the observer will be the proper time times the lapse function at that event. And the new spatial coordinates of the observer will be the proper time times the shift vector at that event. (Strictly speaking, this is only exactly true in the limit of the proper time approaching zero or when the lapse function and shift vectors are constant. But hopefully this gives a good enough insight to what these entities are so that we can go on to describe warp drives.)

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Lapse_shift.png]]
<tr><td>A folliation of space-time into slices of constant time coordinates (grey). At an event (black circle), the lapse function (yellow) shows how far into the future the point at that event will be projected in a unit amount of proper time experienced by an observer initially at the event. The shift vector (red) shows how that point changes coordinates after the unit amount of proper time. The cyan vector shows the new coordinates the point is projected onto. The lower black surface indicates a foliation of spacetime at the same time coordinate; after unit proper time, the events on the lower foliation get projected to the curved upper black foliation according to the shift vectors and lapse functions at each of their respective points.
</table>
</table>

Warp drives are generally restricted to spacetime geometries with a constant lapse function - all observers go forward by the same time coordinate after experiencing the same amount of proper time. Thus, all warp drives have no time dilation.
Far from the warp bubble, the shift vector vanishes – each spatial coordinate is mapped to the same spatial coordinate after any amount of proper time so long as the warp drive is not near. But the inside of the warp bubble is an extended region where the shift vector field is non-zero and constant. In this bubble, all spatial coordinates are mapped to a coordinate that shifts linearly in time with the proper time of the observer. Astute readers will see that this linear change in position with time is very similar to a velocity. This region of constant shift vector then moves with the shift vector velocity to keep stuff originally in the bubble so that it stays in the bubble. In between the inside of the bubble and the far away regions with zero shift vector, the shift vector field falls away to match one region to the other.

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Warp_shift.png]]
<tr><td>The shift vector of a one-dimensional slice through warp geometry, in red. At the edges there is no shift. At the center (inside the warp bubble) there is a region of uniform shift so that at a later time the inside of the warp bubble is rigidly translated to different spatial coordinates (in blue). The two foliations remain the same distance apart, because warp drives maintain a unit lapse function.
</table>
</table>

Traditionally, scientists would start with an interesting distribution of matter given by T and then find out what kind of curvature it created. This works well for planets and stars and black holes and cosmic strings and other things which are observed in the universe and you want to see what sort of effect they had. But starting in the 1980's, some physicists started looking at Einstein's field equations the other way around. They start with a desired geometry, find the curvature part of the field equation R - R g/2, and use that to find out what T has to be in order to get that curvature. This was originally done for [[Wormholes|wormholes]], but in 1994 Miguel Alcubierre used this same trick to figure out what could create a region of constant shift vector that moved with the shift vector - the warp drive. With this demand for a warp geometry, the required distribution of matter and energy and momentum and stress can be determined.

The original Alcubierre shift vector N→ produces a spherical warp bubble with an axi-symmetric distribution of, well, everything along the direction of motion with warp velocity vector v→.
<div align="center">
N→ = v→ f(rs)
</div>
Here, rs is the distance from the center of the warp bubble, and f(rs) can be any function that is 1 near rs = 0 and is 0 for rs well beyond the radius of the warp bubble.

=== Challenges and possible resolutions ===

Almost as soon as Alcubierre proposed his warp drive, others began picking it apart.
<ol>
<li>Pfenning and Ford<ref name="PfenningFord_2001">M. J. Pfenning and L. H. Ford, "The unphysical nature of 'Warp Drive'", Classical and Quantum Gravity. 14 (7): 1743–1751 (1997). arXiv:[https://arxiv.org/abs/gr-qc/9702026 gr-qc/9702026]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1997CQGra..14.1743P 1997CQGra..14.1743P]. doi:[https://doi.org/10.1088%2F0264-9381%2F14%2F7%2F011 10.1088/0264-9381/14/7/011]. S2CID [https://api.semanticscholar.org/CorpusID:15279207 15279207].</ref> found that if the negative energy density wall around the bubble obeys [[Wormholes#Quantum_energy_inequalities|quantum energy inequalities]], then for warp speeds of around light speed the shell thickness would be on the order of a hundred Planck lengths; a distance far smaller than any other known physical phenomenon or object.
<li>In the same work, Pfenning and Ford also found that for a bubble a hundred meters in radius, a warp speed of around the speed of light, and the required thickness of about 100 Planck lengths the energy in the bubble shell would be E ≈ -1063 kg c2. The magnitude of this latter value is ten orders of magnitude larger than the energy of the entire visible universe. They do note, however, that if quantum energy inequalities can be ignored then a warp drive with a hundred meters radius but a shell thickness of one meter would "only" have an energy magnitude of about a quarter solar mass.
<li>As with any method of faster than light travel, the warp drive could be used to make a time machine.
<li>Perhaps most seriously, if the warp drive is going faster than light speed, the negative energy regions on the outside of the shell won't be in the warp parts of the bubble. They will have to be moving through space at faster than light speed if the bubble is to maintain its integrity, which is the very problem that the warp drive was designed to avoid. The outside of the warp bubble shell that maintains the warp bubble would fall away and the warp bubble would quickly erode to sub-luminal speeds.<ref name="Van Den Broeck 2000">Chris Van Den Broeck, "On the (im)possibility of warp bubbles", arXiv:gr-qc/9906050 https://doi.org/10.48550/arXiv.gr-qc/9906050</ref>
<li>A more general problem that includes the previous one is that the geometry of the warp drive determines the distribution of energy, momentum, and stress (pressure, shear, and tension) by Einstein's field equation, which states that the stress energy tensor is directly proportional to a reduced order curvature tensor of space-time. However, the stress-energy tensor must also obey the continuity equation – basically, for the energy to change in a given volume of space that energy must come in or go out of the boundary of that volume and similarly for the momentum. So a warp drive can have the warp condition for one moment in time. But after that, the stress-energy that maintains the warp will start to move according to the forces on it and according to the natural paths that it takes through the warped space-time. There is no guarantee that the configuration of the stress-energy at later times will be sufficient to maintain the warp.
<blockquote>
From the perspective of simulating the warp drive dynamically, the key challenge is stability. One can straightforwardly generate the initial metric that describes the Alcubierre metric and infer the instantaneous matter configuration that must support it using the Einstein Equation, but a time-domain evolution of the coupled matter-gravity system necessitates that we specify the equation of state of the matter, which in turn determines how it evolves. In a perfect fluid, this involves specifying the relation between the fluid pressure and density, but in a more general case, the full relation between the
stress-energy components must be given. There is (to our knowledge) no known equation of state that would maintain the warp drive metric in a stable configuration over time - therefore, whilst one can require that initially, the warp bubble is constant, it will quickly evolve away from that state and, in most cases, the warp fluid and spacetime deformations will disperse or collapse into a central point.
 
– Clough, Dietrich, and Khan<ref>Katy Clough, Tim Dietrich, and Sebastian Khan, "What no one has seen before: gravitational waveforms from warp drive collapse", arXiv:2406.02466 [gr-qc] https://doi.org/10.48550/arXiv.2406.02466</ref>
</blockquote>
<li>In order to get the matter (or more generally, stress-energy) that forms the warp shell up to speeds that could allow the shell to exist, you need to be able to, well, accelerate that matter to warp speeds. Which gives a chicken-and-egg problem - in order to achieve superluminal travel you need to be able to get matter or energy moving superluminally in the first place<ref name="Bobrick and Martire">Alexey Bobrick and Gianni Martire, "Introducing Physical Warp Drives", arXiv:2102.06824 [gr-qc] https://doi.org/10.48550/arXiv.2102.06824</ref><ref>D. H. Coule, "No warp drive", Class. Quantum Grav. 15 (1998) 2523–2527. https://doi.org/10.1088/0264-9381/15/8/026</ref>, which rather makes warp drives unnecessary from the get-go.
<li>A warp bubble moving faster than light will not be able to get any signals in front of it as it will overtake its own signal. In fact, anyone in the interior of the bubble would not be able to send signals to the front surface of the bubble. In science-speak, a horizon forms at the front of the bubble<ref name="Natario">José Natário, "Warp drive with zero expansion", Classical and Quantum Gravity. 19 (6): 1157–1166 (2002). arXiv:[https://arxiv.org/abs/gr-qc/0110086 gr-qc/0110086]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2002CQGra..19.1157N 2002CQGra..19.1157N]. doi:[https://doi.org/10.1088%2F0264-9381%2F19%2F6%2F308 10.1088/0264-9381/19/6/308]. S2CID [https://api.semanticscholar.org/CorpusID:15859984 15859984].</ref>. As a consequence, the bubble cannot be controlled, steered, nor stopped from within the bubble itself.
<li>A Horizon will naturally emit Hawking radiation. It is estimated that the radiation will raise the temperature of the interior of the bubble to approximately 1032 K<ref name="Finazzi 2009">Stefano Finazzi, Stefano Liberati, and Carlos Barceló, "Semiclassical instability of dynamical warp drives", Phys. Rev. D 79, 124017 (12 June, 2009) https://doi.org/10.1103/PhysRevD.79.124017 , arXiv:0904.0141 [gr-qc] https://doi.org/10.48550/arXiv.0904.0141</ref> (at this level of accuracy, the result is the same if you replace temperatures in K by the same number in °C or °F). This temperature, however, may be lower if quantum inequalities do not have to be fulfilled.
<li>Semi-classical instabilities at the horizon are expected to lead to a divergence of the stress-energy tensor at the boundary in a short period of time, thus leading to the destruction of the warp bubble<ref name="Finazzi 2009"></ref>.
</ol>
Helpfully, Van Den Broeck<ref name="VanDenBroeck_2000">C. Van Den Broeck, "Alcubierre’s Warp Drive: Problems and Prospects" AIP Conference Proceedings. 504: 1105–1110 (2000). Bibcode:[https://ui.adsabs.harvard.edu/abs/2000AIPC..504.1105V 2000AIPC..504.1105V]. doi:[https://doi.org/10.1063%2F1.1290913 10.1063/1.1290913].</ref> suggested several solutions – or at least mitigations – for some of these problems
<ol>
<li>Quantum inequalities had not been shown to be true in general for highly curved space-times.
<li>A proposed warp drive geometry<ref name="VanDenBroeck_1999">C. Van Den Broeck, "A 'warp drive' with more reasonable total energy requirements". Classical and Quantum Gravity. 16 (12): 3973–3979 (1999). arXiv:[https://arxiv.org/abs/gr-qc/9905084 gr-qc/9905084]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1999CQGra..16.3973V 1999CQGra..16.3973V]. doi:[https://doi.org/10.1088%2F0264-9381%2F16%2F12%2F314 10.1088/0264-9381/16/12/314]. S2CID [https://api.semanticscholar.org/CorpusID:15466313 15466313]. </ref> (see the van Den Broeck drive, below) would be able to minimize the magnitude of the energy required to about that of the mass of our sun. While still large, it is not unphysically large.
<li>Time travel is always going to be a worry with faster than light travel. There's no neat solution to this one.
<li>It may be possible for the negative energy outside of the superluminal region to compress into the superluminal region to form a shock in space-time. The front surface of the bubble would then be a singularity. It is not clear if such a sharp jump in physical properties is possible, but neither is sure that it is impossible, either. However, while this might help with the negative energy in front of the bubble getting swept up because it cannot move faster than light, it does little to help with the material at the back of the bubble on the outside getting left behind because it cannot keep up.
</ol>
Other mitigating research has also shown
<ol start=9>
<li>Semi-classical divergence only occurs in warp drives with only one spatial dimensions. Those with two or more spatial dimensions (as in our universe) only contain points where divergences occur, and these divergences can be managed. In fact, their effects can be reduced by making the warp shape more elongated<ref>Carlos Barceló, Valentin Boyanov, Luis J. Garay, Eduardo Martín-Martínez,
and Jose M. Sánchez Velázquez, "Warp drive aerodynamics", JHEP08 (2022) 288 https://doi.org/10.1007/JHEP08(2022)288 , arXiv:2207.06458 [gr-qc] https://doi.org/10.48550/arXiv.2207.06458</ref>.
</ol>

=== Alcubierre warp interactions with light and matter ===

Space is not entirely empty. It is filled with a diffuse plasma in the form of the interstellar medium, as well as cosmic radiation, light from stars, and cosmic microwave background radiation. A warp drive propagating through space will encounter this stuff. What happens when this matter and radiation have a warp bubble pass across them?

The first analysis of matter encountering a warp bubble was performed by Pfenning and Ford<ref name="PfenningFord_2001"></ref>. They looked at the warp bubble interaction with a massive object at rest with the frame of reference of the warp drive (keep in mind that the rest frame is the same inside and outside the warp bubble; in this rest frame objects inside the bubble have no momentum even though they are, in some sense, changing location rapidly with time). When the warp bubble passes, the object experiences an acceleration in the direction of the warp bubble motion. When the warp shell passes and the object is inside the bubble, it will be moving with approximately the speed of the bubble. Pfenning and Ford analyzed this problem with a continuous distribution of shell energy that, strictly speaking, never falls to zero except at the bubble center and at spatial infinity, so unless the object passes through the center in Pfenning and Ford's description it will never quite get up to the bubble's speed. In this case, the object will move almost as fast as the object but will pass through the bubble in a finite time, after which it will again be at rest with respect to the reference frame but displaced along the direction of the warp bubble motion by some distance. With this description of the warp bubble, a spacecraft of finite size will always be moving a little slower than the warp bubble and would have to use rockets to keep up with it. in addition, the spacecraft would experience tidal forces that would cause stress on the spacecraft's structure.

Pfenning and Ford also examined cases where the shell is of a finite (possibly infinitesimal) width and falls to zero both inside and outside the bubble. In this case any matter encountering the bubble would be collected at the bow of the bubble and thereafter move along with it.

McMonigal et al. analyzed the situation for both massive particles and light moving along the axis of travel of the bubble<ref name="McMonigalEtAl_2012">B. McMonigal, G. F. Lewis, and P. O'Byrne, "Alcubierre warp drive: On the matter of matter". Physical Review D. 85 (6) 064024 (20 March 2012). arXiv:[https://arxiv.org/abs/1202.5708 1202.5708]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2012PhRvD..85f4024M 2012PhRvD..85f4024M]. doi:[https://doi.org/10.1103%2FPhysRevD.85.064024 10.1103/PhysRevD.85.064024]. S2CID [https://api.semanticscholar.org/CorpusID:3993148 3993148].</ref>. They found that light moving opposite the warp direction passed through the bubble without incident, being only somewhat delayed by passing through the bubble. A warp bubble that is warping at sub-luminal speeds can have light catch up from behind it. This light is able to pass through the bubble, and is somewhat advanced in its path by the speed of the bubble. For super-luminal warp bubbles, however, the situation is different. The bubble will catch up to light moving it its own direction that is originally in front of it. This light cannot escape forward, the bubble being too fast. Nor can it escape backward, as the light is propagating forward and the interior of the bubble is at rest. Thus, the light gets caught at the bow of the warp bubble, unable to escape for so long as the warp bubble is active. This light is strongly blue shifted to extremely energetic x-rays and gamma rays. The space behind the bubble is swept clear of forward-moving light.

The situation for matter over-run by a super-luminal warp bubble is similar. Matter moving backward passes through the bubble, being only swept a ways along the bubble's path as sign of its passing but otherwise continuing on their way. Matter at rest behaves the same way as Pfenning and Ford discovered. Meanwhile, matter moving in the same direction of the warp is overtaken and collects at the bow of the bubble, unable to leave for so long as the bubble is warping. This matter "surfing" on the bubble bow is highly accelerated to relativistic speeds, experiencing extreme time dilation.

For a sub-luminal bubble, there is a speed faster than the warp speed where particles coming up from behind overtake the bubble and pass through, out the front. Particles below this speed but still faster than the warp speed still overtake the bubble, but then bounce backwards out of the bubble from behind. They are still moving forward, but are now moving more slowly than the warp speed. Particles moving forward but slower than the bubble will be overtaken and then bounced out the front with a speed higher than the warp speed. Those particles moving backward will pass through the bubble and exit with their initial speed.

The spacecraft inside the bubble will observe that light moving backward compared to the warp direction is blue-shifted, and matter moving backward passes through with increased energy. Meanwhile, a sub-luminal bubble will see light that was moving forward as red-shifted and matter catching up to the bubble from behind will pass by with reduced energy (superluminal bubbles, of course, do not have any matter or radiation catching up to them from behind).

When a superluminal warp bubble that has been collecting matter and radiation for the duration of its journey and blue-shifting it to much higher energies is turned off, all that matter and energy is released as a blast of radiation in the direction the bubble was warping.

McMonigal et al. conclude by noting that any spacecraft in the warp bubble would need shielding to protect against the blue-shifted radiation and matter of increased energy. In addition the destination would be "blasted into oblivion" by the release of matter and radiation that had been caught in the bubble during the trip.

Clark et al.<ref>Chad Clark, William A. Hiscock, and Shane L. Larson, "Null geodesics in the Alcubierre warp drive spacetime: the view from the bridge", arXiv:gr-qc/9907019 (2018) https://doi.org/10.48550/arXiv.gr-qc/9907019</ref> examine what it would look like from within an operating warp drive. They find that light moving at 90 degrees to the direction of motion will look like it is still coming from 90 degrees. However, light from the front hemisphere is squished progressively toward the forward point the faster you go. For a warp bubble moving faster than the speed of light, there is a cone behind the warp bubble where no light can reach it. However, you don't see a black spot behind you – light from the regions in the backward hemisphere from outside that cone that can reach you is distorted toward the direction behind you to fill in that spot. Light from the forward direction is blue-shifted and light from behind is red-shifted. At speeds of around 200 times light speed, the cosmic microwave background would be blue shifted to energies similar to that of the solar photosphere, heating anything in the warp bubble as if they were close to the sun.

== Other warp geometries ==

The Alcubierre warp drive is the first and best known of the warp drives. It has been around long enough that there have been many studies done on it. The warp geometry is particularly simple, making analysis easier. But in the time that the Alcubierre warp was proposed, many other researchers have come up with other warp designs, and even generalized the idea to include a wide array of additional possibilities.

Most of the general statements about the challenges of the Alcubierre drive hold for these other drives. Many of the details about interactions with matter are also likely to hold, although these other styles of warp drive have not yet been subject to such rigorous investigation.

=== Van Den Broeck warp drive ===

If an Alcubierre warp bubble a hundred meters across requires a magnitude of energy greater than the entire energy in the observable universe, one option to reduce the magnitude of energy used is to make the warp bubble smaller. Much smaller. Pfenning and Ford<ref name="PfenningFord_2001"></ref> suggested making the warp bubble smaller than an atom. Of course, that brings up the problem of how to stuff a spacecraft in there. Van Den Broeck proposed a solution<ref name="VanDenBroeck_1999"></ref>: make the warp bubble only about the size of an atomic nucleus but expand the space inside the bubble enormously so that a spacecraft could fit in. This is somewhat like a nuclear sized wormhole that leads to a pocket universe that holds the spacecraft. Unfortunately, the metric doesn't fit neatly into any [[Wormholes#Three_dimensions|embedding diagram]] that I can figure out, but the math works out so that spatial coordinates inside the bubble are expanded by a factor of 1017 and a spacecraft would have a few hundred meters of bubble interior to putz around in. This trick manages to reduce the magnitude of energy to only about the mass energy of a few stars similar to our own. Other than that, it is otherwise a normal Alcubierre drive.

Further work<ref>S. Krasnikov, "The quantum inequalities do not forbid spacetime shortcuts", Physical Review D. 67 (10) 104013. https://doi.org/10.1103%2FPhysRevD.67.104013, arXiv:gr-qc/0207057 (2003) https://doi.org/10.48550/arXiv.gr-qc/0207057 </ref> has managed to use Van Den Broeck's trick to reduce the magnitude of the negative energy to only about a milligram.

=== Natário warp drive ===

The Alcubierre drive is not the only way to construct a warp drive. José Natário<ref name="Natario"></ref> showed that it was just one example of an entire class of warp drives. These are space-time geometries with a smooth and localized shift vector field and a unit lapse function.

Natário then went on to find in that class a set of warp drives that do not have any expansion or contraction of space at all. Instead, space encountering the front boundary of the warp bubble instead "slides" around the outside of the bubble until it gets to the corresponding place on the back and is left behind there. Another way to think of it is that space entering the bubble shell is compressed in the radial direction of the shell but is simultaneously expanded in the tangential direction so that there is no net change in volume. When it gets to the back, the opposite occurs. In the language of mathematics, the shift vector field is divergenceless.

<table align=center>
<tr><td width=360>[[File:Natario_drive_flow_of_space.png]]
<td width=445>[[File:Natario_warp_drive_horizons.png]]
<tr><td style="vertical-align: top;">A representation of the motion of space in a zero-expansion Natário drive with a vanishingly thin warp shell around the bubble.
<td>The horizons of the generalized Natário class of warp drives (including the Alcubierre warp drive and all fell-Heisenberg drives) at superluminal speed in the direction of the green arrow. No event inside the bubble can affect events outside the bubble in front of the blue line. No events outside the bubble behind the red line can affect events inside the bubble. The lines have an angle α with respect to the direction of motion with sin(α) = 1/vs for vs the speed of the bubble relative to light speed. This is analogous to the Mach cone of supersonic objects.
</table>

NOTE: There are two different things that can be meant by the term "Natário warp drive". The generalized Natário warp drive is the general theory of all warp drives that includes within it the Alcubierre, Fell-Heisenberg, and Lentz warp drives, among infinitely many others. The zero-expansion Natário warp drive is the specific implementation of the generalized Natário warp drive that has a divergenceless shift vector field.

The generalized class of warp drives developed by Natário (including both the Alcubierre and this zero-expansion warp drive) are shown to always have regions in the warp shell where the energy density is negative to at least some observers. You can't get away from it – to warp, you need negative energy density.

The Natário class of warp drives blueshift light coming in from the front and redshift light catching up from the back. For the case of a super-luminal warp bubble, there will of course be a horizon at the back of the bubble and light will not be able to get through from behind. For a bubble speed vs relative to the speed of light, light coming from straight ahead will be blueshifted up in frequency and photon energy by a factor of 1 + vs. For sub-luminal travel, light coming from behind will be redshifted by a factor 1 - vs. In general for n as the unit direction along which the light is propagating, the blueshift factor will be 1 + vs ⋅ n. If the warp shell is infinitesimally thin, light from outside that reaches the center of the bubble will not be distorted in direction although it will be frequency shifted, but observers still inside the bubble but displaced from the center will see the field of view distorted as well as frequency shifted.

Most of the difficulties that apply to the Alcubierre warp drive also apply to the generalized Natário warp drive (and thus to the zero-expansion Natário drive and Fell-Heisenberg drive as well). In particular, quantum energy inequalities, high magnitudes of the energy required, formation of time machines, instabilities related to the stress-energy making up the warp shell being unable to keep up with the bubble, the chicken-and-egg problem of needing FTL to achieve FTL, the formation of horizons making warp bubbles uncontrollable, and Hawking radiation at the horizons, are all still concerns for generic warp drives.

=== Fell-Heisenberg warp drives ===

S. D. B. Fell and L. Heisenberg investigated what sort of warp structure is necessary to produce a warp drive with positive energy density everywhere<ref name="Fell-Heisenberg">S. D. B. Fell and L> Heisenberg, "Positive energy warp drive from hidden geometric structures", Classical and Quantum Gravity 38 155020 (2021) https://doi.org/10.1088/1361-6382/ac0e47 https://arxiv.org/abs/2104.06488</ref>. They found that warp drives of the generic Natário form with irrotational shift vectors had a simple expression for the energy density and vanishing momentum (irrotational means that the shift vector can be described as the gradient of a scalar potential field or, equivalently, that the curl of the shift vector field vanishes everywhere). Fell and Heisenberg proposed several potential fields that either had positive energy density everywhere except at discontinuities, or that were continuous but had regions that violated the weak energy condition (i. e. negative energy densities in some frames of reference). One specific example they provided of a warp field had an energy that is about 10,000 times less than the mass-energy of our sun. Or only about half the mass-energy of Jupiter. This is a significant improvement over many other proposed drives.

Despite the introduction discussing warp drive configurations that satisfy the various energy conditions, the configurations described in the paper are shown to locally violate the weak and strong energy conditions. Nonetheless, the energy density is still mostly positive. Santiago, Schuster, and Visser's<ref name="Santiago Schuster Visser 2022">J. Santiago, S. Schuster, and M. Visser, "Generic warp drives violate the null energy condition", Physical Review D 105, 064038 (2022) https://doi.org/10.1103/PhysRevD.105.064038</ref> work shows that the various energy conditions must still be violated by this warp drive; to not violate these, energy density must be positive in all reference frames not just those of the co-moving observer.

Any vector field can be described by its divergence, its curl, and a constant. The constant part of a warp drive's shift vector can be discarded because then the warp drive would not be localized. It is interesting, therefore, that the Natário zero-expansion drive has zero divergence and can be described only by its curl, while the Fell-Heisenberg drive has zero curl and can be described only by its divergence. These two classes of drives thus rely on opposite choices of setting one of the two different parts describing a vector field to zero. As might be expected, the Fell-Heisenberg drive is not zero expansion – space-time will change in volume as it passes through the warp bubble (although after going all the way through it will return to its original volume). However, objects in space-time will retain their original orientation as they pass through the warp bubble.

Any vector field with zero curl can be represented as the gradient of a suitable scalar function. Thus, the Fell-Heisenberg drive can be completely described by just scalar field (one number at all points of space and time) rather than a vector field.

Previous examples of warp drives, such as the Alcubierre and zero-expansion Natário drives, tend toward a flat space-time as you go far enough away faster than would occur if there were any mass present in the warp drive. That is, the ADM mass of such drives is zero. However, one of the Fell-Heisenberg examples described a spacetime where the geometry approaches that of what you would get for a massive object at the warp location, giving one of the first examples of a warp drive with mass.

In the Fell-Heisenberg drive the momentum vanishes everywhere. Yet the energy occupies regions where the shift vector is varying rapidly. The lack of momentum means that the energy will not move to keep up with the differential expansion and movement of the spacetime. As a consequence, at later times the energy will have a different distribution than what is necessary to maintain the given warp configuration. The exact time evolution is not solved but this likely leads to collapse of warp bubble. Note that this is not a problem unique to the Fell-Heisenberg drive. Assuming a geometry that describes a warp produces a specific distribution of the stress-energy tensor that satisfies Einstein's equation. But there is no guarantee that this stress energy tensor satisfies the continuity equation, which describes how the distribution of matter and momentum change with time. The issue of stability of any warp drive is not well understood, but most warp drive geometries will be unstable in the sense that their matter that generates the warp fields will not be able to keep up with the motion of the warp. The Fell-Heisenberg drive is unique only in that, with vanishing momentum, there is no configuration that allows stable propagation.

(n. b. The Heisenberg here is Lavinia Heisenberg, not the Werner Heisenberg of quantum physics and uncertainty principle fame.)

=== Lentz warp drive ===

E. Lentz<ref name="Lentz">E. W. Lentz, "Breaking the warp barrier: hyper-fast solitons in Einstein–Maxwell-plasma theory", Classical and Quantum Gravity. 38 075015 (2021). arXiv:[https://arxiv.org/abs/2006.07125 2006.07125]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2021CQGra..38g5015L 2021CQGra..38g5015L]. doi:[https://doi.org/10.1088%2F1361-6382%2Fabe692 10.1088/1361-6382/abe692]. ISSN [https://search.worldcat.org/issn/0264-9381 0264-9381]. S2CID [https://api.semanticscholar.org/CorpusID:219635854 219635854].</ref> investigated a particular configuration of the Fell-Heisenberg class of warp drives, where the scalar field forms a selection of rhomboid regions, resulting in an energy density that is everywhere positive and a shift vector with vanishing tidal force at the center for a comfortable ride.

However, Santiago, Schuster, and Visser<ref name="Santiago Schuster Visser 2022"></ref> dispute claims that the Lentz drive satisfies the energy conditions, noting that everywhere positive energy density in one frame of reference is insufficient to establish that the energy density is positive in all reference frames; knowledge of the Cauchy stress tensor is also needed. They show that any generic warp drive will violate the strong energy condition, null energy condition, and weak energy condition.

== Conservation laws ==

=== General introduction ===

All the warp drives investigated so far have geometries that quickly fall off from the highly curved region near the warp bubble to a region of spacetime that is almost flat; and becomes even more flat the farther away from the warp bubble you get. There is a word for geometries of this kind – they are called asymptotically flat. Basically, you can always go far enough away in any direction to reach a place where the geometry is sufficiently flat to meet any flatness criterion you choose.

When you have an asymptotically flat geometry, then in the far away mostly flat regions gravity is well described by a linearized theory of gravity – where the perturbations to the geometry are linear in the stress energy tensor. This means that gravity is basically Newtonian, along with a few other bells and whistles like gravitational waves and frame dragging.

When linearized gravity is a good approximation, then it is known that energy, momentum, and linear momentum are conserved. Although parts of a warp geometry are highly curved and non-linear, you can measure the conserved energy and momentum and angular momentum from far away in the linear region. This ADM energy and etcetera is the quantity that is conserved when you are worrying about conserving such things, not necessarily the energy and so forth you get in the highly curved spacetime region. In fact, it is not even simple to define where the energy and such actually is withing the highly curved spacetime region; it is non-local and different observers will disagree on how it is distributed. But everyone agrees on the ADM conserved quantities from out in nearly flat spacetime (up to the usual changes due to changing speeds, which are trivial and come up with normal objects as well).

And now we run into a bit of a conundrum: most of the warp geometries so far proposed have zero ADM mass<ref>Sebastian Schuster, Jessica Santiago,and Matt Visser, "ADM mass in warp drive spacetimes" General Relativity and Gravitation (2023) 55:14 https://doi.org/10.1007/s10714-022-03061-9</ref>, meaning that things like the Alcubierre drive technically have no energy or momentum or angular momentum at all. Despite all of the extremes of negative energy regions within the wall of the warp bubble, the actual energy that you need to spend to make one is zero. This can lead to a bit of a problem. If you have a spacecraft that has mass, that mass-energy has to be conserved. So when you turn on your warp drive and the total warp mass is zero, where does that energy go?

One resolution is just that the mass of the spacecraft is negligible and the small amount of curvature it causes can be ignored. In this case, the warp geometries discussed so far are merely an approximation that neglects this effect. In fact, some of the proposed Fell-Heisenberg drives map not onto entirely flat spacetime geometries at large distances, but onto the Schwarzschild geometry that is appropriate for systems with mass at the center. If this is the case, then the warping bubble and spacecraft system will need to worry about conservation of all their conserved properties.

=== All the science, demonstration of the problem ===

Let's look at what the conservation laws mean for a spacecraft equipped with a warp drive, assuming that no matter or radiation are emitted by activating or deactivating the warp drive.

We can start with an observer floating in space. We'll call her Alice. And, approaching Alice at 100 m/s is the starship Enterprise with a mass of a cool 200,000 tons (2×108 kg). from this we know that the Enterprise has, in Alice's frame of reference, a kinetic energy of 1 TJ, a total energy (including mass and kinetic energy) of 1.8×1025 J, a momentum of 2×1010 kg m/s, and – because it is headed straight toward Alice – an angular momentum of zero. All of this is without any warp drive activated.

On the bridge of the Enterprise, Spock picks up Alice on the sensors and tells Captain Kirk. Kirk does not want to run into Alice (he'd rather date her), so he orders helm to turn on the Enterprise's warp drive and move them 1 km up so that their course takes them 1 km from Alice instead of straight into her. Sulu warps the Enterprise according to these commands, and then turns off the warp drive.

After this maneuver, keeping the Enterprise's total energy and momentum the same means that the Enterprise still has a mass of 200,000 tons and a velocity of 100 m/s in its original direction. So we can see that the warp drive must preserve the velocity vector of the thing in it (at least in flat spacetime where outside forces are not acting on it). The problem comes from angular momentum. With a known mass of 200,000 tons, a speed of 100 m/s, and a distance of closest approach of 1 km; from Alice's frame of reference the Enterprise now has an angular momentum of 2×1013 kg m2/s after the warp maneuver, while before the maneuver it has an angular momentum of zero.

From this we can see that warp drives cannot be reactionless. Arbitrary displacements using warp without emitting propellant result in violation of the law of conservation of angular momentum. As long as we restrict ourselves to accepted science based on what we know of general relativity, and assuming that we are in an asymptotically flat spacetime (so not near a cosmic string or other cosmic object with extreme mass or other components of the stress energy tensor that is infinite in extent in at least one dimension), to move a warp drive with mass you need to accelerate it or decelerate it with rockets or other normal means of delivering momentum (such as light sails, mag sails, or pushing off of a planet). Warp drives must be inertial to comply with physics. And all of this was analyzed only from times when the warp drive was turned off; it does not depend on any assumptions about what happens when the warp is on.

=== A zero mass warp "solution" ===

So let's start picking at our assumptions. First, what if the warp bubble actually does have zero mass? Now, while in warp, there is no momentum or angular momentum or energy to be conserved. A spacecraft in warp can bounce around like a ferret on stimulants and the universe doesn't care.

But there's a problem, and it's probably pretty obvious. Our Starship Enterprise has a lot of mass when it hasn't activated its warp bubble. So, 200,000 tons × c2 = 1.8×1025 J of energy before the warp is turned on, and 0 J of energy afterward. This seems like a flagrant violation of conservation of energy.

Well, there's a solution. Maybe turning on the warp drive somehow dumps all that energy. You could get a pulse of gravitational and electromagnetic radiation rippling away from the turn-on point at the speed of light, maybe with various kinds of particle radiation thrown in. Or maybe a 200,000 ton black hole is pooped out behind the Enterprise when the warp bubble is turned on. Anything that dumps the energy and leaves it behind.

Okay, so, the good ship Enterprise zips off to Starbase 9 with in a zero mass warp bubble. But now they're in a bit of a pickle. In order to come out of warp so they can visit the starbase, they somehow need to acquire 200,000 tons worth of mass-energy from somewhere.

One option is that the Enterprise rams its warp bubble into some random asteroid and grabs 200,000 tons of it, or skims 200,000 tons of gas from the top of the atmosphere of a nearby world, or maybe the starbase keeps around millions of tons of ballast to give to visiting starships so they can de-warp. It's a perfectly reasonable way to go about conserving all the conserved things the universe seems to care about.

But their is an even weirder option. It's pretty speculative – all of this is speculative, but this is even more so – but it is fun so it's worth mentioning. Normally, a source of radiation produces waves that radiate out from it into the universe that expand with time. But the time reverse of this process is also allowed by the laws of physics, with waves converging from the far reaches of the universe, interacting and scattering off of matter and black holes and other stuff in such a way so as to make an ever-shrinking concentric wave that collapses on the source. This solution is generally rejected because it doesn't mesh with the second law of thermodynamics. But if you have faster than light warp drive, causality goes out the window, causes and effects can be reversed depending on your frame of reference, and strange coincidences can occur because they must occur to maintain consistency. So what if dropping out of warp always just happens to coincide with being at the center of a converging sphere of radiation that gives you exactly enough energy to go from a zero energy warp bubble back to your full-mass spacecraft?

This would have interesting consequence, because your arrival would always be heralded by a converging wave-front. You could build warp sensors that would alert you to when and where someone is going to come out of warp. No longer can an entire invasion fleet suddenly appear over your planet and demand your surrender, you'll have advance warning and can mount a defense.

(If angular momentum conservation is ignored, discuss how momentum & energy are affected by outside forces. In a gravitational field equivalent to inertial frame ... like being in an accelerated elevator; will acquire momentum buildup while staying "at rest".)

== Warp Railroads ==

Van Den Broeck<ref name="VanDenBroeck_2000"></ref> suggested a warp railroad,

You can set up devices ahead of time along the path of the bubble that produce the negative energy regions for the warp bubble at the appropriate time without needing the negative energy to ever move with the bubble at all. At least for bubbles on a predictable schedule and route.

helps with:

-stability (elaborate)

-conservation laws (elaborate)

Mention Krasnikov<ref name="Krasnikov 1998">S. V. Krasnikov, "Hyperfast travel in general relativity" Phys. Rev. D 57 4760</ref>

Hyperwave<ref name="Pieri 2024">Lorenzo Pieri, "Hyperwave: hyper-Fast Communication within General Relativity", arXiv:2311.12069 [gr-qc] https://doi.org/10.48550/arXiv.2311.12069</ref> makes use of this warp railroad idea.

== Quantum effects ==

(quantum stuff here<ref name="Hiscock1997">W. A. Hiscock, "Quantum effects in the Alcubierre warp drive spacetime", Classical and Quantum Gravity 14 L183 https://doi.org/10.1088/0264-9381/14/11/002 https://arxiv.org/abs/gr-qc/9707024</ref><ref name="Finazzi et al 2009">S. Finazzi, S. Liberati, C. Barceló, "Semiclassical instability of dynamical warp drives", Physical Review D 79, 124017 (2009)https://doi.org/10.1103/PhysRevD.79.124017 https://arxiv.org/abs/0904.0141</ref>)

From<ref name="Pieri 2024"></ref>
<blockquote>
It should be emphasised that Quantum Inequalities have not been proven for realistic interacting quantum fields, where the they are expected to be weaker if they exist [Cadamuro, 2019], and they often require Minkowskian conditions at infinity, a condition which is not appropriate for fields inside bounded regions, as in a Casimir devices.
</blockquote>

== Warp drives and black holes ==

What happens if you drive your warp drive into a black hole?

The pop-sci reason that nothing can escape a black hole is that at the event horizon the escape velocity is faster than light and nothing can go faster than light. A warp drive can go faster than light.

The more detailed reason from general relativity is that at the event horizon space and time are rotated sufficiently that "inward" becomes "forward in time." You can no more go farther away from the singularity in a black hole once you are inside the event horizon than you can go backward in time. Any super-luminal travel can go backward in time, and warp drives can engage in super-luminal travel.

The Hawking area theorem that says that black holes can only grow, never shrink relies on the condition that the energy is everywhere positive. Warp drives have negative energy density.

So in principle, there's nothing that prevents a warp drive from taking a dip into a black hole and coming back out again.

Garattini and Zatrimaylov looked into the problem of a warp drive near (and in) a black hole<ref name="Garattini and Zatrimaylov">R. Garattini and K. Zatrimaylov, "Black holes, warp drives, and energy conditions", Physics Letters B 856 138910 (2024) https://doi.org/10.1016/j.physletb.2024.138910</ref>. The found that the black hole reduces the negative energy requirement for the warp bubble to move inward toward the black hole singularity, but increases the negative energy needed to move away from the center of the black hole. In addition, a warp drive parked at the event horizon would allow light from inside the horizon to pass through the bubble and escape back outside.

== Credit ==
Author: Luke Campbell

== References ==

[[Category:Physics]][[Category:Infrastructure]][[Category:Physics & Engineering‏‎]][[Category:Physics & Math & Engineering]][[Category:Transportation & Infrastructure‏‎]][[Category:Metric Engineering]]

Warp Drives

2026-07-31T18:45:42Z

Lwcamp: /* A zero mass warp "solution" */

{{PageConstructionNotice}}

Science fiction often features spacecraft that can seemingly move across space and get between the place of departure and the destination much faster than light could have done. This appears to contradict the theory of relativity, which predicts unequivocally that nothing can move through space faster than light. Because relativity has been incredibly successful at describing nature, with its many other predictions regularly being confirmed to extraordinary accuracy and within the bounds of uncertainty of all the experiments that tested them, it gives confidence that relativity is a correct description of reality. Which seems to rather throw a wet towel on our hopes for rapid travel between stars.

However, while relativity does not allow things to move through space faster than light, it places no such restrictions on how fast space-time itself can expand, contract, or move around. This leads to the idea of a warp drive – the spacecraft remains stationary within a region of highly curved space-time, and that region moves at super-luminal speeds rather than the spacecraft.

== The Alcubierre warp drive ==

The first warp drive geometry that satisfied the Einstein field equations of relativity was proposed by Miguel Alcubierre<ref>M. Alcubierre, "The warp drive: hyper-fast travel within general relativity." Classical and Quantum Gravity. 11 (5): L73–L77 (1994). [https://arxiv.org/abs/gr-qc/0009013 arXiv:gr-qc/0009013]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1994CQGra..11L..73A 1994CQGra..11L..73A]. doi:[https://doi.org/10.1088%2F0264-9381%2F11%2F5%2F001 10.1088/0264-9381/11/5/001]. S2CID [https://api.semanticscholar.org/CorpusID:4797900 4797900].</ref>. In this geometry, a sphere of space-time moves at an arbitrary speed (potentially but not necessarily a speed much faster than light). Objects within the sphere are moved along with the sphere; an object at rest within the sphere would be moved along with the sphere indefinitely. Space is expanding at the rear boundary and contracting at the front boundary in order to keep the sphere moving. In order to satisfy the Einstein field equations, the boundary of the sphere must have a negative energy density. The challenges of space-time geometries with negative energy densities are described in our page on [[Wormholes#Exotic_energy_conditions|wormholes]], for our purposes it is enough to note that negative energy densities can pose problems if not handled carefully, there may be limits on how much negative energy you can have without nearby positive energy density, and it may not be possible to get enough negative energy to support a warp drive; although none of this is ruled out by physics – yet!

<table align=center>
<tr><td width=650>[[File:Energy_magnitude_of_Alcubierre_drive.png]]
<td width=650>[[File:Expansion_of_Alcubierre_drive.png]]
<tr><td>The magnitude of the energy of the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. (The energy density is negative everywhere in the shell, here only the magnitude is shown to aid visual comprehension.)
<td>The expansion of space in the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. Negative expansion means that space is contracting in that direction, positive expansion that it is expanding.
</table>

NOTE: What the? The energy distribution is symmetric! How does the drive know which way to go? There must be significant contributions of other components of the stress-energy tensor, and those have got to be asymmetric along the forward/backward axis. Check this when I get time.

(And what's with the ring in all the artwork, anyway?)

=== So what is going on? ===

What is it that makes a warp drive a warp drive, and how does it work?

A warp drive is a concept that comes out of the science of general relativity. General relativity collects space and time together into a single "structure" called spacetime. This structure is curved, and its curvature is determined by the material that is in it. Absent forces acting between matter, particles of matter move through spacetime on the shortest possible paths in this curved environment.
<blockquote>
"Spacetime tells matter how to move; matter tells spacetime how to curve." - John Archibald Wheeler
</blockquote>
All of this is encompassed in the Einstein field equation:
<div align=center>
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 8πG
<td rowspan="2" nowrap="nowrap" align="right"> T = R -  
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right"> R g
<tr>
<td style="border-top:solid 1px black;"> c4
<td style="border-top:solid 1px black;"> 2
</table>
</div>
Unfortunately, by convention, both the symbols G and R get used for a lot of different things. Here, symbols in bold are second rank tensors – constructs of arrays of 4×4 numbers where 4 is necessary because that is the number of dimensions of spacetime. Conventionally, these numbers are indexed by a value going from 0 to 3, with 0 being the part corresponding to the coordinate of time and 1 to 3 being the Cartesian coordinates that you might know as x, y, and z.

T is the stress-energy tensor: the (0,0) component represents the energy density at a given point; the (0,i) and (i, 0) components (with i going from 1 to 3) represent the momentum density; and the (i,j) components represent the stresses present, such as pressure or tension or shear.

The other term, R - R g/2, involves a partial description of the curvature. A full description of the curvature requires a construct of 4×4×4×4 components; too much for the stress energy tensor to fully constrain. Which is okay, because this allows curvature to exist away from where matter is, allowing gravity to have long range effects across empty space as well as for phenomena like gravitational waves. But it is enough that a given distribution of matter T gives enough constraints (along with specifying how the geometry behaves far away) that you can generally figure out what the actual gravity is.

This is the usual story that we are told. But there is an alternate way of looking at the same geometry such that the math works out the same, but it is described conceptually in a very different way. This divides spacetime up into "layers", called foliations, that represent slices of space and going from one foliation to the next is increasing in time. This is useful for computation because you can assign coordinates to your system. There are two additional constructs – the lapse function and the shift vector field. Starting at an event on one foliation, the lapse function at that position tells you that after a certain "proper time" (time perceived by an observer at that event) the time coordinate of the observer will be the proper time times the lapse function at that event. And the new spatial coordinates of the observer will be the proper time times the shift vector at that event. (Strictly speaking, this is only exactly true in the limit of the proper time approaching zero or when the lapse function and shift vectors are constant. But hopefully this gives a good enough insight to what these entities are so that we can go on to describe warp drives.)

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Lapse_shift.png]]
<tr><td>A folliation of space-time into slices of constant time coordinates (grey). At an event (black circle), the lapse function (yellow) shows how far into the future the point at that event will be projected in a unit amount of proper time experienced by an observer initially at the event. The shift vector (red) shows how that point changes coordinates after the unit amount of proper time. The cyan vector shows the new coordinates the point is projected onto. The lower black surface indicates a foliation of spacetime at the same time coordinate; after unit proper time, the events on the lower foliation get projected to the curved upper black foliation according to the shift vectors and lapse functions at each of their respective points.
</table>
</table>

Warp drives are generally restricted to spacetime geometries with a constant lapse function - all observers go forward by the same time coordinate after experiencing the same amount of proper time. Thus, all warp drives have no time dilation.
Far from the warp bubble, the shift vector vanishes – each spatial coordinate is mapped to the same spatial coordinate after any amount of proper time so long as the warp drive is not near. But the inside of the warp bubble is an extended region where the shift vector field is non-zero and constant. In this bubble, all spatial coordinates are mapped to a coordinate that shifts linearly in time with the proper time of the observer. Astute readers will see that this linear change in position with time is very similar to a velocity. This region of constant shift vector then moves with the shift vector velocity to keep stuff originally in the bubble so that it stays in the bubble. In between the inside of the bubble and the far away regions with zero shift vector, the shift vector field falls away to match one region to the other.

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Warp_shift.png]]
<tr><td>The shift vector of a one-dimensional slice through warp geometry, in red. At the edges there is no shift. At the center (inside the warp bubble) there is a region of uniform shift so that at a later time the inside of the warp bubble is rigidly translated to different spatial coordinates (in blue). The two foliations remain the same distance apart, because warp drives maintain a unit lapse function.
</table>
</table>

Traditionally, scientists would start with an interesting distribution of matter given by T and then find out what kind of curvature it created. This works well for planets and stars and black holes and cosmic strings and other things which are observed in the universe and you want to see what sort of effect they had. But starting in the 1980's, some physicists started looking at Einstein's field equations the other way around. They start with a desired geometry, find the curvature part of the field equation R - R g/2, and use that to find out what T has to be in order to get that curvature. This was originally done for [[Wormholes|wormholes]], but in 1994 Miguel Alcubierre used this same trick to figure out what could create a region of constant shift vector that moved with the shift vector - the warp drive. With this demand for a warp geometry, the required distribution of matter and energy and momentum and stress can be determined.

The original Alcubierre shift vector N→ produces a spherical warp bubble with an axi-symmetric distribution of, well, everything along the direction of motion with warp velocity vector v→.
<div align="center">
N→ = v→ f(rs)
</div>
Here, rs is the distance from the center of the warp bubble, and f(rs) can be any function that is 1 near rs = 0 and is 0 for rs well beyond the radius of the warp bubble.

=== Challenges and possible resolutions ===

Almost as soon as Alcubierre proposed his warp drive, others began picking it apart.
<ol>
<li>Pfenning and Ford<ref name="PfenningFord_2001">M. J. Pfenning and L. H. Ford, "The unphysical nature of 'Warp Drive'", Classical and Quantum Gravity. 14 (7): 1743–1751 (1997). arXiv:[https://arxiv.org/abs/gr-qc/9702026 gr-qc/9702026]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1997CQGra..14.1743P 1997CQGra..14.1743P]. doi:[https://doi.org/10.1088%2F0264-9381%2F14%2F7%2F011 10.1088/0264-9381/14/7/011]. S2CID [https://api.semanticscholar.org/CorpusID:15279207 15279207].</ref> found that if the negative energy density wall around the bubble obeys [[Wormholes#Quantum_energy_inequalities|quantum energy inequalities]], then for warp speeds of around light speed the shell thickness would be on the order of a hundred Planck lengths; a distance far smaller than any other known physical phenomenon or object.
<li>In the same work, Pfenning and Ford also found that for a bubble a hundred meters in radius, a warp speed of around the speed of light, and the required thickness of about 100 Planck lengths the energy in the bubble shell would be E ≈ -1063 kg c2. The magnitude of this latter value is ten orders of magnitude larger than the energy of the entire visible universe. They do note, however, that if quantum energy inequalities can be ignored then a warp drive with a hundred meters radius but a shell thickness of one meter would "only" have an energy magnitude of about a quarter solar mass.
<li>As with any method of faster than light travel, the warp drive could be used to make a time machine.
<li>Perhaps most seriously, if the warp drive is going faster than light speed, the negative energy regions on the outside of the shell won't be in the warp parts of the bubble. They will have to be moving through space at faster than light speed if the bubble is to maintain its integrity, which is the very problem that the warp drive was designed to avoid. The outside of the warp bubble shell that maintains the warp bubble would fall away and the warp bubble would quickly erode to sub-luminal speeds.<ref name="Van Den Broeck 2000">Chris Van Den Broeck, "On the (im)possibility of warp bubbles", arXiv:gr-qc/9906050 https://doi.org/10.48550/arXiv.gr-qc/9906050</ref>
<li>A more general problem that includes the previous one is that the geometry of the warp drive determines the distribution of energy, momentum, and stress (pressure, shear, and tension) by Einstein's field equation, which states that the stress energy tensor is directly proportional to a reduced order curvature tensor of space-time. However, the stress-energy tensor must also obey the continuity equation – basically, for the energy to change in a given volume of space that energy must come in or go out of the boundary of that volume and similarly for the momentum. So a warp drive can have the warp condition for one moment in time. But after that, the stress-energy that maintains the warp will start to move according to the forces on it and according to the natural paths that it takes through the warped space-time. There is no guarantee that the configuration of the stress-energy at later times will be sufficient to maintain the warp.
<blockquote>
From the perspective of simulating the warp drive dynamically, the key challenge is stability. One can straightforwardly generate the initial metric that describes the Alcubierre metric and infer the instantaneous matter configuration that must support it using the Einstein Equation, but a time-domain evolution of the coupled matter-gravity system necessitates that we specify the equation of state of the matter, which in turn determines how it evolves. In a perfect fluid, this involves specifying the relation between the fluid pressure and density, but in a more general case, the full relation between the
stress-energy components must be given. There is (to our knowledge) no known equation of state that would maintain the warp drive metric in a stable configuration over time - therefore, whilst one can require that initially, the warp bubble is constant, it will quickly evolve away from that state and, in most cases, the warp fluid and spacetime deformations will disperse or collapse into a central point.
 
– Clough, Dietrich, and Khan<ref>Katy Clough, Tim Dietrich, and Sebastian Khan, "What no one has seen before: gravitational waveforms from warp drive collapse", arXiv:2406.02466 [gr-qc] https://doi.org/10.48550/arXiv.2406.02466</ref>
</blockquote>
<li>In order to get the matter (or more generally, stress-energy) that forms the warp shell up to speeds that could allow the shell to exist, you need to be able to, well, accelerate that matter to warp speeds. Which gives a chicken-and-egg problem - in order to achieve superluminal travel you need to be able to get matter or energy moving superluminally in the first place<ref name="Bobrick and Martire">Alexey Bobrick and Gianni Martire, "Introducing Physical Warp Drives", arXiv:2102.06824 [gr-qc] https://doi.org/10.48550/arXiv.2102.06824</ref><ref>D. H. Coule, "No warp drive", Class. Quantum Grav. 15 (1998) 2523–2527. https://doi.org/10.1088/0264-9381/15/8/026</ref>, which rather makes warp drives unnecessary from the get-go.
<li>A warp bubble moving faster than light will not be able to get any signals in front of it as it will overtake its own signal. In fact, anyone in the interior of the bubble would not be able to send signals to the front surface of the bubble. In science-speak, a horizon forms at the front of the bubble<ref name="Natario">José Natário, "Warp drive with zero expansion", Classical and Quantum Gravity. 19 (6): 1157–1166 (2002). arXiv:[https://arxiv.org/abs/gr-qc/0110086 gr-qc/0110086]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2002CQGra..19.1157N 2002CQGra..19.1157N]. doi:[https://doi.org/10.1088%2F0264-9381%2F19%2F6%2F308 10.1088/0264-9381/19/6/308]. S2CID [https://api.semanticscholar.org/CorpusID:15859984 15859984].</ref>. As a consequence, the bubble cannot be controlled, steered, nor stopped from within the bubble itself.
<li>A Horizon will naturally emit Hawking radiation. It is estimated that the radiation will raise the temperature of the interior of the bubble to approximately 1032 K<ref name="Finazzi 2009">Stefano Finazzi, Stefano Liberati, and Carlos Barceló, "Semiclassical instability of dynamical warp drives", Phys. Rev. D 79, 124017 (12 June, 2009) https://doi.org/10.1103/PhysRevD.79.124017 , arXiv:0904.0141 [gr-qc] https://doi.org/10.48550/arXiv.0904.0141</ref> (at this level of accuracy, the result is the same if you replace temperatures in K by the same number in °C or °F). This temperature, however, may be lower if quantum inequalities do not have to be fulfilled.
<li>Semi-classical instabilities at the horizon are expected to lead to a divergence of the stress-energy tensor at the boundary in a short period of time, thus leading to the destruction of the warp bubble<ref name="Finazzi 2009"></ref>.
</ol>
Helpfully, Van Den Broeck<ref name="VanDenBroeck_2000">C. Van Den Broeck, "Alcubierre’s Warp Drive: Problems and Prospects" AIP Conference Proceedings. 504: 1105–1110 (2000). Bibcode:[https://ui.adsabs.harvard.edu/abs/2000AIPC..504.1105V 2000AIPC..504.1105V]. doi:[https://doi.org/10.1063%2F1.1290913 10.1063/1.1290913].</ref> suggested several solutions – or at least mitigations – for some of these problems
<ol>
<li>Quantum inequalities had not been shown to be true in general for highly curved space-times.
<li>A proposed warp drive geometry<ref name="VanDenBroeck_1999">C. Van Den Broeck, "A 'warp drive' with more reasonable total energy requirements". Classical and Quantum Gravity. 16 (12): 3973–3979 (1999). arXiv:[https://arxiv.org/abs/gr-qc/9905084 gr-qc/9905084]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1999CQGra..16.3973V 1999CQGra..16.3973V]. doi:[https://doi.org/10.1088%2F0264-9381%2F16%2F12%2F314 10.1088/0264-9381/16/12/314]. S2CID [https://api.semanticscholar.org/CorpusID:15466313 15466313]. </ref> (see the van Den Broeck drive, below) would be able to minimize the magnitude of the energy required to about that of the mass of our sun. While still large, it is not unphysically large.
<li>Time travel is always going to be a worry with faster than light travel. There's no neat solution to this one.
<li>It may be possible for the negative energy outside of the superluminal region to compress into the superluminal region to form a shock in space-time. The front surface of the bubble would then be a singularity. It is not clear if such a sharp jump in physical properties is possible, but neither is sure that it is impossible, either. However, while this might help with the negative energy in front of the bubble getting swept up because it cannot move faster than light, it does little to help with the material at the back of the bubble on the outside getting left behind because it cannot keep up.
</ol>
Other mitigating research has also shown
<ol start=9>
<li>Semi-classical divergence only occurs in warp drives with only one spatial dimensions. Those with two or more spatial dimensions (as in our universe) only contain points where divergences occur, and these divergences can be managed. In fact, their effects can be reduced by making the warp shape more elongated<ref>Carlos Barceló, Valentin Boyanov, Luis J. Garay, Eduardo Martín-Martínez,
and Jose M. Sánchez Velázquez, "Warp drive aerodynamics", JHEP08 (2022) 288 https://doi.org/10.1007/JHEP08(2022)288 , arXiv:2207.06458 [gr-qc] https://doi.org/10.48550/arXiv.2207.06458</ref>.
</ol>

=== Alcubierre warp interactions with light and matter ===

Space is not entirely empty. It is filled with a diffuse plasma in the form of the interstellar medium, as well as cosmic radiation, light from stars, and cosmic microwave background radiation. A warp drive propagating through space will encounter this stuff. What happens when this matter and radiation have a warp bubble pass across them?

The first analysis of matter encountering a warp bubble was performed by Pfenning and Ford<ref name="PfenningFord_2001"></ref>. They looked at the warp bubble interaction with a massive object at rest with the frame of reference of the warp drive (keep in mind that the rest frame is the same inside and outside the warp bubble; in this rest frame objects inside the bubble have no momentum even though they are, in some sense, changing location rapidly with time). When the warp bubble passes, the object experiences an acceleration in the direction of the warp bubble motion. When the warp shell passes and the object is inside the bubble, it will be moving with approximately the speed of the bubble. Pfenning and Ford analyzed this problem with a continuous distribution of shell energy that, strictly speaking, never falls to zero except at the bubble center and at spatial infinity, so unless the object passes through the center in Pfenning and Ford's description it will never quite get up to the bubble's speed. In this case, the object will move almost as fast as the object but will pass through the bubble in a finite time, after which it will again be at rest with respect to the reference frame but displaced along the direction of the warp bubble motion by some distance. With this description of the warp bubble, a spacecraft of finite size will always be moving a little slower than the warp bubble and would have to use rockets to keep up with it. in addition, the spacecraft would experience tidal forces that would cause stress on the spacecraft's structure.

Pfenning and Ford also examined cases where the shell is of a finite (possibly infinitesimal) width and falls to zero both inside and outside the bubble. In this case any matter encountering the bubble would be collected at the bow of the bubble and thereafter move along with it.

McMonigal et al. analyzed the situation for both massive particles and light moving along the axis of travel of the bubble<ref name="McMonigalEtAl_2012">B. McMonigal, G. F. Lewis, and P. O'Byrne, "Alcubierre warp drive: On the matter of matter". Physical Review D. 85 (6) 064024 (20 March 2012). arXiv:[https://arxiv.org/abs/1202.5708 1202.5708]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2012PhRvD..85f4024M 2012PhRvD..85f4024M]. doi:[https://doi.org/10.1103%2FPhysRevD.85.064024 10.1103/PhysRevD.85.064024]. S2CID [https://api.semanticscholar.org/CorpusID:3993148 3993148].</ref>. They found that light moving opposite the warp direction passed through the bubble without incident, being only somewhat delayed by passing through the bubble. A warp bubble that is warping at sub-luminal speeds can have light catch up from behind it. This light is able to pass through the bubble, and is somewhat advanced in its path by the speed of the bubble. For super-luminal warp bubbles, however, the situation is different. The bubble will catch up to light moving it its own direction that is originally in front of it. This light cannot escape forward, the bubble being too fast. Nor can it escape backward, as the light is propagating forward and the interior of the bubble is at rest. Thus, the light gets caught at the bow of the warp bubble, unable to escape for so long as the warp bubble is active. This light is strongly blue shifted to extremely energetic x-rays and gamma rays. The space behind the bubble is swept clear of forward-moving light.

The situation for matter over-run by a super-luminal warp bubble is similar. Matter moving backward passes through the bubble, being only swept a ways along the bubble's path as sign of its passing but otherwise continuing on their way. Matter at rest behaves the same way as Pfenning and Ford discovered. Meanwhile, matter moving in the same direction of the warp is overtaken and collects at the bow of the bubble, unable to leave for so long as the bubble is warping. This matter "surfing" on the bubble bow is highly accelerated to relativistic speeds, experiencing extreme time dilation.

For a sub-luminal bubble, there is a speed faster than the warp speed where particles coming up from behind overtake the bubble and pass through, out the front. Particles below this speed but still faster than the warp speed still overtake the bubble, but then bounce backwards out of the bubble from behind. They are still moving forward, but are now moving more slowly than the warp speed. Particles moving forward but slower than the bubble will be overtaken and then bounced out the front with a speed higher than the warp speed. Those particles moving backward will pass through the bubble and exit with their initial speed.

The spacecraft inside the bubble will observe that light moving backward compared to the warp direction is blue-shifted, and matter moving backward passes through with increased energy. Meanwhile, a sub-luminal bubble will see light that was moving forward as red-shifted and matter catching up to the bubble from behind will pass by with reduced energy (superluminal bubbles, of course, do not have any matter or radiation catching up to them from behind).

When a superluminal warp bubble that has been collecting matter and radiation for the duration of its journey and blue-shifting it to much higher energies is turned off, all that matter and energy is released as a blast of radiation in the direction the bubble was warping.

McMonigal et al. conclude by noting that any spacecraft in the warp bubble would need shielding to protect against the blue-shifted radiation and matter of increased energy. In addition the destination would be "blasted into oblivion" by the release of matter and radiation that had been caught in the bubble during the trip.

Clark et al.<ref>Chad Clark, William A. Hiscock, and Shane L. Larson, "Null geodesics in the Alcubierre warp drive spacetime: the view from the bridge", arXiv:gr-qc/9907019 (2018) https://doi.org/10.48550/arXiv.gr-qc/9907019</ref> examine what it would look like from within an operating warp drive. They find that light moving at 90 degrees to the direction of motion will look like it is still coming from 90 degrees. However, light from the front hemisphere is squished progressively toward the forward point the faster you go. For a warp bubble moving faster than the speed of light, there is a cone behind the warp bubble where no light can reach it. However, you don't see a black spot behind you – light from the regions in the backward hemisphere from outside that cone that can reach you is distorted toward the direction behind you to fill in that spot. Light from the forward direction is blue-shifted and light from behind is red-shifted. At speeds of around 200 times light speed, the cosmic microwave background would be blue shifted to energies similar to that of the solar photosphere, heating anything in the warp bubble as if they were close to the sun.

== Other warp geometries ==

The Alcubierre warp drive is the first and best known of the warp drives. It has been around long enough that there have been many studies done on it. The warp geometry is particularly simple, making analysis easier. But in the time that the Alcubierre warp was proposed, many other researchers have come up with other warp designs, and even generalized the idea to include a wide array of additional possibilities.

Most of the general statements about the challenges of the Alcubierre drive hold for these other drives. Many of the details about interactions with matter are also likely to hold, although these other styles of warp drive have not yet been subject to such rigorous investigation.

=== Van Den Broeck warp drive ===

If an Alcubierre warp bubble a hundred meters across requires a magnitude of energy greater than the entire energy in the observable universe, one option to reduce the magnitude of energy used is to make the warp bubble smaller. Much smaller. Pfenning and Ford<ref name="PfenningFord_2001"></ref> suggested making the warp bubble smaller than an atom. Of course, that brings up the problem of how to stuff a spacecraft in there. Van Den Broeck proposed a solution<ref name="VanDenBroeck_1999"></ref>: make the warp bubble only about the size of an atomic nucleus but expand the space inside the bubble enormously so that a spacecraft could fit in. This is somewhat like a nuclear sized wormhole that leads to a pocket universe that holds the spacecraft. Unfortunately, the metric doesn't fit neatly into any [[Wormholes#Three_dimensions|embedding diagram]] that I can figure out, but the math works out so that spatial coordinates inside the bubble are expanded by a factor of 1017 and a spacecraft would have a few hundred meters of bubble interior to putz around in. This trick manages to reduce the magnitude of energy to only about the mass energy of a few stars similar to our own. Other than that, it is otherwise a normal Alcubierre drive.

Further work<ref>S. Krasnikov, "The quantum inequalities do not forbid spacetime shortcuts", Physical Review D. 67 (10) 104013. https://doi.org/10.1103%2FPhysRevD.67.104013, arXiv:gr-qc/0207057 (2003) https://doi.org/10.48550/arXiv.gr-qc/0207057 </ref> has managed to use Van Den Broeck's trick to reduce the magnitude of the negative energy to only about a milligram.

=== Natário warp drive ===

The Alcubierre drive is not the only way to construct a warp drive. José Natário<ref name="Natario"></ref> showed that it was just one example of an entire class of warp drives. These are space-time geometries with a smooth and localized shift vector field and a unit lapse function.

Natário then went on to find in that class a set of warp drives that do not have any expansion or contraction of space at all. Instead, space encountering the front boundary of the warp bubble instead "slides" around the outside of the bubble until it gets to the corresponding place on the back and is left behind there. Another way to think of it is that space entering the bubble shell is compressed in the radial direction of the shell but is simultaneously expanded in the tangential direction so that there is no net change in volume. When it gets to the back, the opposite occurs. In the language of mathematics, the shift vector field is divergenceless.

<table align=center>
<tr><td width=360>[[File:Natario_drive_flow_of_space.png]]
<td width=445>[[File:Natario_warp_drive_horizons.png]]
<tr><td style="vertical-align: top;">A representation of the motion of space in a zero-expansion Natário drive with a vanishingly thin warp shell around the bubble.
<td>The horizons of the generalized Natário class of warp drives (including the Alcubierre warp drive and all fell-Heisenberg drives) at superluminal speed in the direction of the green arrow. No event inside the bubble can affect events outside the bubble in front of the blue line. No events outside the bubble behind the red line can affect events inside the bubble. The lines have an angle α with respect to the direction of motion with sin(α) = 1/vs for vs the speed of the bubble relative to light speed. This is analogous to the Mach cone of supersonic objects.
</table>

NOTE: There are two different things that can be meant by the term "Natário warp drive". The generalized Natário warp drive is the general theory of all warp drives that includes within it the Alcubierre, Fell-Heisenberg, and Lentz warp drives, among infinitely many others. The zero-expansion Natário warp drive is the specific implementation of the generalized Natário warp drive that has a divergenceless shift vector field.

The generalized class of warp drives developed by Natário (including both the Alcubierre and this zero-expansion warp drive) are shown to always have regions in the warp shell where the energy density is negative to at least some observers. You can't get away from it – to warp, you need negative energy density.

The Natário class of warp drives blueshift light coming in from the front and redshift light catching up from the back. For the case of a super-luminal warp bubble, there will of course be a horizon at the back of the bubble and light will not be able to get through from behind. For a bubble speed vs relative to the speed of light, light coming from straight ahead will be blueshifted up in frequency and photon energy by a factor of 1 + vs. For sub-luminal travel, light coming from behind will be redshifted by a factor 1 - vs. In general for n as the unit direction along which the light is propagating, the blueshift factor will be 1 + vs ⋅ n. If the warp shell is infinitesimally thin, light from outside that reaches the center of the bubble will not be distorted in direction although it will be frequency shifted, but observers still inside the bubble but displaced from the center will see the field of view distorted as well as frequency shifted.

Most of the difficulties that apply to the Alcubierre warp drive also apply to the generalized Natário warp drive (and thus to the zero-expansion Natário drive and Fell-Heisenberg drive as well). In particular, quantum energy inequalities, high magnitudes of the energy required, formation of time machines, instabilities related to the stress-energy making up the warp shell being unable to keep up with the bubble, the chicken-and-egg problem of needing FTL to achieve FTL, the formation of horizons making warp bubbles uncontrollable, and Hawking radiation at the horizons, are all still concerns for generic warp drives.

=== Fell-Heisenberg warp drives ===

S. D. B. Fell and L. Heisenberg investigated what sort of warp structure is necessary to produce a warp drive with positive energy density everywhere<ref name="Fell-Heisenberg">S. D. B. Fell and L> Heisenberg, "Positive energy warp drive from hidden geometric structures", Classical and Quantum Gravity 38 155020 (2021) https://doi.org/10.1088/1361-6382/ac0e47 https://arxiv.org/abs/2104.06488</ref>. They found that warp drives of the generic Natário form with irrotational shift vectors had a simple expression for the energy density and vanishing momentum (irrotational means that the shift vector can be described as the gradient of a scalar potential field or, equivalently, that the curl of the shift vector field vanishes everywhere). Fell and Heisenberg proposed several potential fields that either had positive energy density everywhere except at discontinuities, or that were continuous but had regions that violated the weak energy condition (i. e. negative energy densities in some frames of reference). One specific example they provided of a warp field had an energy that is about 10,000 times less than the mass-energy of our sun. Or only about half the mass-energy of Jupiter. This is a significant improvement over many other proposed drives.

Despite the introduction discussing warp drive configurations that satisfy the various energy conditions, the configurations described in the paper are shown to locally violate the weak and strong energy conditions. Nonetheless, the energy density is still mostly positive. Santiago, Schuster, and Visser's<ref name="Santiago Schuster Visser 2022">J. Santiago, S. Schuster, and M. Visser, "Generic warp drives violate the null energy condition", Physical Review D 105, 064038 (2022) https://doi.org/10.1103/PhysRevD.105.064038</ref> work shows that the various energy conditions must still be violated by this warp drive; to not violate these, energy density must be positive in all reference frames not just those of the co-moving observer.

Any vector field can be described by its divergence, its curl, and a constant. The constant part of a warp drive's shift vector can be discarded because then the warp drive would not be localized. It is interesting, therefore, that the Natário zero-expansion drive has zero divergence and can be described only by its curl, while the Fell-Heisenberg drive has zero curl and can be described only by its divergence. These two classes of drives thus rely on opposite choices of setting one of the two different parts describing a vector field to zero. As might be expected, the Fell-Heisenberg drive is not zero expansion – space-time will change in volume as it passes through the warp bubble (although after going all the way through it will return to its original volume). However, objects in space-time will retain their original orientation as they pass through the warp bubble.

Any vector field with zero curl can be represented as the gradient of a suitable scalar function. Thus, the Fell-Heisenberg drive can be completely described by just scalar field (one number at all points of space and time) rather than a vector field.

Previous examples of warp drives, such as the Alcubierre and zero-expansion Natário drives, tend toward a flat space-time as you go far enough away faster than would occur if there were any mass present in the warp drive. That is, the ADM mass of such drives is zero. However, one of the Fell-Heisenberg examples described a spacetime where the geometry approaches that of what you would get for a massive object at the warp location, giving one of the first examples of a warp drive with mass.

In the Fell-Heisenberg drive the momentum vanishes everywhere. Yet the energy occupies regions where the shift vector is varying rapidly. The lack of momentum means that the energy will not move to keep up with the differential expansion and movement of the spacetime. As a consequence, at later times the energy will have a different distribution than what is necessary to maintain the given warp configuration. The exact time evolution is not solved but this likely leads to collapse of warp bubble. Note that this is not a problem unique to the Fell-Heisenberg drive. Assuming a geometry that describes a warp produces a specific distribution of the stress-energy tensor that satisfies Einstein's equation. But there is no guarantee that this stress energy tensor satisfies the continuity equation, which describes how the distribution of matter and momentum change with time. The issue of stability of any warp drive is not well understood, but most warp drive geometries will be unstable in the sense that their matter that generates the warp fields will not be able to keep up with the motion of the warp. The Fell-Heisenberg drive is unique only in that, with vanishing momentum, there is no configuration that allows stable propagation.

(n. b. The Heisenberg here is Lavinia Heisenberg, not the Werner Heisenberg of quantum physics and uncertainty principle fame.)

=== Lentz warp drive ===

E. Lentz<ref name="Lentz">E. W. Lentz, "Breaking the warp barrier: hyper-fast solitons in Einstein–Maxwell-plasma theory", Classical and Quantum Gravity. 38 075015 (2021). arXiv:[https://arxiv.org/abs/2006.07125 2006.07125]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2021CQGra..38g5015L 2021CQGra..38g5015L]. doi:[https://doi.org/10.1088%2F1361-6382%2Fabe692 10.1088/1361-6382/abe692]. ISSN [https://search.worldcat.org/issn/0264-9381 0264-9381]. S2CID [https://api.semanticscholar.org/CorpusID:219635854 219635854].</ref> investigated a particular configuration of the Fell-Heisenberg class of warp drives, where the scalar field forms a selection of rhomboid regions, resulting in an energy density that is everywhere positive and a shift vector with vanishing tidal force at the center for a comfortable ride.

However, Santiago, Schuster, and Visser<ref name="Santiago Schuster Visser 2022"></ref> dispute claims that the Lentz drive satisfies the energy conditions, noting that everywhere positive energy density in one frame of reference is insufficient to establish that the energy density is positive in all reference frames; knowledge of the Cauchy stress tensor is also needed. They show that any generic warp drive will violate the strong energy condition, null energy condition, and weak energy condition.

== Conservation laws ==

=== General introduction ===

All the warp drives investigated so far have geometries that quickly fall off from the highly curved region near the warp bubble to a region of spacetime that is almost flat; and becomes even more flat the farther away from the warp bubble you get. There is a word for geometries of this kind – they are called asymptotically flat. Basically, you can always go far enough away in any direction to reach a place where the geometry is sufficiently flat to meet any flatness criterion you choose.

When you have an asymptotically flat geometry, then in the far away mostly flat regions gravity is well described by a linearized theory of gravity – where the perturbations to the geometry are linear in the stress energy tensor. This means that gravity is basically Newtonian, along with a few other bells and whistles like gravitational waves and frame dragging.

When linearized gravity is a good approximation, then it is known that energy, momentum, and linear momentum are conserved. Although parts of a warp geometry are highly curved and non-linear, you can measure the conserved energy and momentum and angular momentum from far away in the linear region. This ADM energy and etcetera is the quantity that is conserved when you are worrying about conserving such things, not necessarily the energy and so forth you get in the highly curved spacetime region. In fact, it is not even simple to define where the energy and such actually is withing the highly curved spacetime region; it is non-local and different observers will disagree on how it is distributed. But everyone agrees on the ADM conserved quantities from out in nearly flat spacetime (up to the usual changes due to changing speeds, which are trivial and come up with normal objects as well).

And now we run into a bit of a conundrum: most of the warp geometries so far proposed have zero ADM mass<ref>Sebastian Schuster, Jessica Santiago,and Matt Visser, "ADM mass in warp drive spacetimes" General Relativity and Gravitation (2023) 55:14 https://doi.org/10.1007/s10714-022-03061-9</ref>, meaning that things like the Alcubierre drive technically have no energy or momentum or angular momentum at all. Despite all of the extremes of negative energy regions within the wall of the warp bubble, the actual energy that you need to spend to make one is zero. This can lead to a bit of a problem. If you have a spacecraft that has mass, that mass-energy has to be conserved. So when you turn on your warp drive and the total warp mass is zero, where does that energy go?

One resolution is just that the mass of the spacecraft is negligible and the small amount of curvature it causes can be ignored. In this case, the warp geometries discussed so far are merely an approximation that neglects this effect. In fact, some of the proposed Fell-Heisenberg drives map not onto entirely flat spacetime geometries at large distances, but onto the Schwarzschild geometry that is appropriate for systems with mass at the center. If this is the case, then the warping bubble and spacecraft system will need to worry about conservation of all their conserved properties.

=== All the science, demonstration of the problem ===

Let's look at what the conservation laws mean for a spacecraft equipped with a warp drive, assuming that no matter or radiation are emitted by activating or deactivating the warp drive.

We can start with an observer floating in space. We'll call her Alice. And, approaching Alice at 100 m/s is the starship Enterprise with a mass of a cool 200,000 tons (2×108 kg). from this we know that the Enterprise has, in Alice's frame of reference, a kinetic energy of 1 TJ, a total energy (including mass and kinetic energy) of 1.8×1025 J, a momentum of 2×1010 kg m/s, and – because it is headed straight toward Alice – an angular momentum of zero. All of this is without any warp drive activated.

On the bridge of the Enterprise, Spock picks up Alice on the sensors and tells Captain Kirk. Kirk does not want to run into Alice (he'd rather date her), so he orders helm to turn on the Enterprise's warp drive and move them 1 km up so that their course takes them 1 km from Alice instead of straight into her. Sulu warps the Enterprise according to these commands, and then turns off the warp drive.

After this maneuver, keeping the Enterprise's total energy and momentum the same means that the Enterprise still has a mass of 200,000 tons and a velocity of 100 m/s in its original direction. So we can see that the warp drive must preserve the velocity vector of the thing in it (at least in flat spacetime where outside forces are not acting on it). The problem comes from angular momentum. With a known mass of 200,000 tons, a speed of 100 m/s, and a distance of closest approach of 1 km; from Alice's frame of reference the Enterprise now has an angular momentum of 2×1013 kg m2/s after the warp maneuver, while before the maneuver it has an angular momentum of zero.

From this we can see that warp drives cannot be reactionless. Arbitrary displacements using warp without emitting propellant result in violation of the law of conservation of angular momentum. As long as we restrict ourselves to accepted science based on what we know of general relativity, to move a warp drive with mass you need to accelerate it or decelerate it with rockets or other normal means of delivering momentum (such as light sails, mag sails, or pushing off of a planet). Warp drives must be inertial to comply with physics. And all of this was analyzed only from times when the warp drive was turned off; it does not depend on any assumptions about what happens when the warp is on.

=== A zero mass warp "solution" ===

So let's start picking at our assumptions. First, what if the warp bubble actually does have zero mass? Now, while in warp, there is no momentum or angular momentum or energy to be conserved. A spacecraft in warp can bounce around like a ferret on stimulants and the universe doesn't care.

But there's a problem, and it's probably pretty obvious. Our Starship Enterprise has a lot of mass when it hasn't activated its warp bubble. So, 200,000 tons × c2 = 1.8×1025 J of energy before the warp is turned on, and 0 J of energy afterward. This seems like a flagrant violation of conservation of energy.

Well, there's a solution. Maybe turning on the warp drive somehow dumps all that energy. You could get a pulse of gravitational and electromagnetic radiation rippling away from the turn-on point at the speed of light, maybe with various kinds of particle radiation thrown in. Or maybe a 200,000 ton black hole is pooped out behind the Enterprise when the warp bubble is turned on. Anything that dumps the energy and leaves it behind.

Okay, so, the good ship Enterprise zips off to Starbase 9 with in a zero mass warp bubble. But now they're in a bit of a pickle. In order to come out of warp so they can visit the starbase, they somehow need to acquire 200,000 tons worth of mass-energy from somewhere.

One option is that the Enterprise rams its warp bubble into some random asteroid and grabs 200,000 tons of it, or skims 200,000 tons of gas from the top of the atmosphere of a nearby world, or maybe the starbase keeps around millions of tons of ballast to give to visiting starships so they can de-warp. It's a perfectly reasonable way to go about conserving all the conserved things the universe seems to care about.

But their is an even weirder option. It's pretty speculative – all of this is speculative, but this is even more so – but it is fun so it's worth mentioning. Normally, a source of radiation produces waves that radiate out from it into the universe that expand with time. But the time reverse of this process is also allowed by the laws of physics, with waves converging from the far reaches of the universe, interacting and scattering off of matter and black holes and other stuff in such a way so as to make an ever-shrinking concentric wave that collapses on the source. This solution is generally rejected because it doesn't mesh with the second law of thermodynamics. But if you have faster than light warp drive, causality goes out the window, causes and effects can be reversed depending on your frame of reference, and strange coincidences can occur because they must occur to maintain consistency. So what if dropping out of warp always just happens to coincide with being at the center of a converging sphere of radiation that gives you exactly enough energy to go from a zero energy warp bubble back to your full-mass spacecraft?

This would have interesting consequence, because your arrival would always be heralded by a converging wave-front. You could build warp sensors that would alert you to when and where someone is going to come out of warp. No longer can an entire invasion fleet suddenly appear over your planet and demand your surrender, you'll have advance warning and can mount a defense.

(If angular momentum conservation is ignored, discuss how momentum & energy are affected by outside forces. In a gravitational field equivalent to inertial frame ... like being in an accelerated elevator; will acquire momentum buildup while staying "at rest".)

== Warp Railroads ==

Van Den Broeck<ref name="VanDenBroeck_2000"></ref> suggested a warp railroad,

You can set up devices ahead of time along the path of the bubble that produce the negative energy regions for the warp bubble at the appropriate time without needing the negative energy to ever move with the bubble at all. At least for bubbles on a predictable schedule and route.

helps with:

-stability (elaborate)

-conservation laws (elaborate)

Mention Krasnikov<ref name="Krasnikov 1998">S. V. Krasnikov, "Hyperfast travel in general relativity" Phys. Rev. D 57 4760</ref>

Hyperwave<ref name="Pieri 2024">Lorenzo Pieri, "Hyperwave: hyper-Fast Communication within General Relativity", arXiv:2311.12069 [gr-qc] https://doi.org/10.48550/arXiv.2311.12069</ref> makes use of this warp railroad idea.

== Quantum effects ==

(quantum stuff here<ref name="Hiscock1997">W. A. Hiscock, "Quantum effects in the Alcubierre warp drive spacetime", Classical and Quantum Gravity 14 L183 https://doi.org/10.1088/0264-9381/14/11/002 https://arxiv.org/abs/gr-qc/9707024</ref><ref name="Finazzi et al 2009">S. Finazzi, S. Liberati, C. Barceló, "Semiclassical instability of dynamical warp drives", Physical Review D 79, 124017 (2009)https://doi.org/10.1103/PhysRevD.79.124017 https://arxiv.org/abs/0904.0141</ref>)

From<ref name="Pieri 2024"></ref>
<blockquote>
It should be emphasised that Quantum Inequalities have not been proven for realistic interacting quantum fields, where the they are expected to be weaker if they exist [Cadamuro, 2019], and they often require Minkowskian conditions at infinity, a condition which is not appropriate for fields inside bounded regions, as in a Casimir devices.
</blockquote>

== Warp drives and black holes ==

What happens if you drive your warp drive into a black hole?

The pop-sci reason that nothing can escape a black hole is that at the event horizon the escape velocity is faster than light and nothing can go faster than light. A warp drive can go faster than light.

The more detailed reason from general relativity is that at the event horizon space and time are rotated sufficiently that "inward" becomes "forward in time." You can no more go farther away from the singularity in a black hole once you are inside the event horizon than you can go backward in time. Any super-luminal travel can go backward in time, and warp drives can engage in super-luminal travel.

The Hawking area theorem that says that black holes can only grow, never shrink relies on the condition that the energy is everywhere positive. Warp drives have negative energy density.

So in principle, there's nothing that prevents a warp drive from taking a dip into a black hole and coming back out again.

Garattini and Zatrimaylov looked into the problem of a warp drive near (and in) a black hole<ref name="Garattini and Zatrimaylov">R. Garattini and K. Zatrimaylov, "Black holes, warp drives, and energy conditions", Physics Letters B 856 138910 (2024) https://doi.org/10.1016/j.physletb.2024.138910</ref>. The found that the black hole reduces the negative energy requirement for the warp bubble to move inward toward the black hole singularity, but increases the negative energy needed to move away from the center of the black hole. In addition, a warp drive parked at the event horizon would allow light from inside the horizon to pass through the bubble and escape back outside.

== Credit ==
Author: Luke Campbell

== References ==

[[Category:Physics]][[Category:Infrastructure]][[Category:Physics & Engineering‏‎]][[Category:Physics & Math & Engineering]][[Category:Transportation & Infrastructure‏‎]][[Category:Metric Engineering]]

Warp Drives

2026-07-31T18:45:22Z

Lwcamp: /* A zero mass warp "solution" */

{{PageConstructionNotice}}

Science fiction often features spacecraft that can seemingly move across space and get between the place of departure and the destination much faster than light could have done. This appears to contradict the theory of relativity, which predicts unequivocally that nothing can move through space faster than light. Because relativity has been incredibly successful at describing nature, with its many other predictions regularly being confirmed to extraordinary accuracy and within the bounds of uncertainty of all the experiments that tested them, it gives confidence that relativity is a correct description of reality. Which seems to rather throw a wet towel on our hopes for rapid travel between stars.

However, while relativity does not allow things to move through space faster than light, it places no such restrictions on how fast space-time itself can expand, contract, or move around. This leads to the idea of a warp drive – the spacecraft remains stationary within a region of highly curved space-time, and that region moves at super-luminal speeds rather than the spacecraft.

== The Alcubierre warp drive ==

The first warp drive geometry that satisfied the Einstein field equations of relativity was proposed by Miguel Alcubierre<ref>M. Alcubierre, "The warp drive: hyper-fast travel within general relativity." Classical and Quantum Gravity. 11 (5): L73–L77 (1994). [https://arxiv.org/abs/gr-qc/0009013 arXiv:gr-qc/0009013]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1994CQGra..11L..73A 1994CQGra..11L..73A]. doi:[https://doi.org/10.1088%2F0264-9381%2F11%2F5%2F001 10.1088/0264-9381/11/5/001]. S2CID [https://api.semanticscholar.org/CorpusID:4797900 4797900].</ref>. In this geometry, a sphere of space-time moves at an arbitrary speed (potentially but not necessarily a speed much faster than light). Objects within the sphere are moved along with the sphere; an object at rest within the sphere would be moved along with the sphere indefinitely. Space is expanding at the rear boundary and contracting at the front boundary in order to keep the sphere moving. In order to satisfy the Einstein field equations, the boundary of the sphere must have a negative energy density. The challenges of space-time geometries with negative energy densities are described in our page on [[Wormholes#Exotic_energy_conditions|wormholes]], for our purposes it is enough to note that negative energy densities can pose problems if not handled carefully, there may be limits on how much negative energy you can have without nearby positive energy density, and it may not be possible to get enough negative energy to support a warp drive; although none of this is ruled out by physics – yet!

<table align=center>
<tr><td width=650>[[File:Energy_magnitude_of_Alcubierre_drive.png]]
<td width=650>[[File:Expansion_of_Alcubierre_drive.png]]
<tr><td>The magnitude of the energy of the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. (The energy density is negative everywhere in the shell, here only the magnitude is shown to aid visual comprehension.)
<td>The expansion of space in the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. Negative expansion means that space is contracting in that direction, positive expansion that it is expanding.
</table>

NOTE: What the? The energy distribution is symmetric! How does the drive know which way to go? There must be significant contributions of other components of the stress-energy tensor, and those have got to be asymmetric along the forward/backward axis. Check this when I get time.

(And what's with the ring in all the artwork, anyway?)

=== So what is going on? ===

What is it that makes a warp drive a warp drive, and how does it work?

A warp drive is a concept that comes out of the science of general relativity. General relativity collects space and time together into a single "structure" called spacetime. This structure is curved, and its curvature is determined by the material that is in it. Absent forces acting between matter, particles of matter move through spacetime on the shortest possible paths in this curved environment.
<blockquote>
"Spacetime tells matter how to move; matter tells spacetime how to curve." - John Archibald Wheeler
</blockquote>
All of this is encompassed in the Einstein field equation:
<div align=center>
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 8πG
<td rowspan="2" nowrap="nowrap" align="right"> T = R -  
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right"> R g
<tr>
<td style="border-top:solid 1px black;"> c4
<td style="border-top:solid 1px black;"> 2
</table>
</div>
Unfortunately, by convention, both the symbols G and R get used for a lot of different things. Here, symbols in bold are second rank tensors – constructs of arrays of 4×4 numbers where 4 is necessary because that is the number of dimensions of spacetime. Conventionally, these numbers are indexed by a value going from 0 to 3, with 0 being the part corresponding to the coordinate of time and 1 to 3 being the Cartesian coordinates that you might know as x, y, and z.

T is the stress-energy tensor: the (0,0) component represents the energy density at a given point; the (0,i) and (i, 0) components (with i going from 1 to 3) represent the momentum density; and the (i,j) components represent the stresses present, such as pressure or tension or shear.

The other term, R - R g/2, involves a partial description of the curvature. A full description of the curvature requires a construct of 4×4×4×4 components; too much for the stress energy tensor to fully constrain. Which is okay, because this allows curvature to exist away from where matter is, allowing gravity to have long range effects across empty space as well as for phenomena like gravitational waves. But it is enough that a given distribution of matter T gives enough constraints (along with specifying how the geometry behaves far away) that you can generally figure out what the actual gravity is.

This is the usual story that we are told. But there is an alternate way of looking at the same geometry such that the math works out the same, but it is described conceptually in a very different way. This divides spacetime up into "layers", called foliations, that represent slices of space and going from one foliation to the next is increasing in time. This is useful for computation because you can assign coordinates to your system. There are two additional constructs – the lapse function and the shift vector field. Starting at an event on one foliation, the lapse function at that position tells you that after a certain "proper time" (time perceived by an observer at that event) the time coordinate of the observer will be the proper time times the lapse function at that event. And the new spatial coordinates of the observer will be the proper time times the shift vector at that event. (Strictly speaking, this is only exactly true in the limit of the proper time approaching zero or when the lapse function and shift vectors are constant. But hopefully this gives a good enough insight to what these entities are so that we can go on to describe warp drives.)

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Lapse_shift.png]]
<tr><td>A folliation of space-time into slices of constant time coordinates (grey). At an event (black circle), the lapse function (yellow) shows how far into the future the point at that event will be projected in a unit amount of proper time experienced by an observer initially at the event. The shift vector (red) shows how that point changes coordinates after the unit amount of proper time. The cyan vector shows the new coordinates the point is projected onto. The lower black surface indicates a foliation of spacetime at the same time coordinate; after unit proper time, the events on the lower foliation get projected to the curved upper black foliation according to the shift vectors and lapse functions at each of their respective points.
</table>
</table>

Warp drives are generally restricted to spacetime geometries with a constant lapse function - all observers go forward by the same time coordinate after experiencing the same amount of proper time. Thus, all warp drives have no time dilation.
Far from the warp bubble, the shift vector vanishes – each spatial coordinate is mapped to the same spatial coordinate after any amount of proper time so long as the warp drive is not near. But the inside of the warp bubble is an extended region where the shift vector field is non-zero and constant. In this bubble, all spatial coordinates are mapped to a coordinate that shifts linearly in time with the proper time of the observer. Astute readers will see that this linear change in position with time is very similar to a velocity. This region of constant shift vector then moves with the shift vector velocity to keep stuff originally in the bubble so that it stays in the bubble. In between the inside of the bubble and the far away regions with zero shift vector, the shift vector field falls away to match one region to the other.

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Warp_shift.png]]
<tr><td>The shift vector of a one-dimensional slice through warp geometry, in red. At the edges there is no shift. At the center (inside the warp bubble) there is a region of uniform shift so that at a later time the inside of the warp bubble is rigidly translated to different spatial coordinates (in blue). The two foliations remain the same distance apart, because warp drives maintain a unit lapse function.
</table>
</table>

Traditionally, scientists would start with an interesting distribution of matter given by T and then find out what kind of curvature it created. This works well for planets and stars and black holes and cosmic strings and other things which are observed in the universe and you want to see what sort of effect they had. But starting in the 1980's, some physicists started looking at Einstein's field equations the other way around. They start with a desired geometry, find the curvature part of the field equation R - R g/2, and use that to find out what T has to be in order to get that curvature. This was originally done for [[Wormholes|wormholes]], but in 1994 Miguel Alcubierre used this same trick to figure out what could create a region of constant shift vector that moved with the shift vector - the warp drive. With this demand for a warp geometry, the required distribution of matter and energy and momentum and stress can be determined.

The original Alcubierre shift vector N→ produces a spherical warp bubble with an axi-symmetric distribution of, well, everything along the direction of motion with warp velocity vector v→.
<div align="center">
N→ = v→ f(rs)
</div>
Here, rs is the distance from the center of the warp bubble, and f(rs) can be any function that is 1 near rs = 0 and is 0 for rs well beyond the radius of the warp bubble.

=== Challenges and possible resolutions ===

Almost as soon as Alcubierre proposed his warp drive, others began picking it apart.
<ol>
<li>Pfenning and Ford<ref name="PfenningFord_2001">M. J. Pfenning and L. H. Ford, "The unphysical nature of 'Warp Drive'", Classical and Quantum Gravity. 14 (7): 1743–1751 (1997). arXiv:[https://arxiv.org/abs/gr-qc/9702026 gr-qc/9702026]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1997CQGra..14.1743P 1997CQGra..14.1743P]. doi:[https://doi.org/10.1088%2F0264-9381%2F14%2F7%2F011 10.1088/0264-9381/14/7/011]. S2CID [https://api.semanticscholar.org/CorpusID:15279207 15279207].</ref> found that if the negative energy density wall around the bubble obeys [[Wormholes#Quantum_energy_inequalities|quantum energy inequalities]], then for warp speeds of around light speed the shell thickness would be on the order of a hundred Planck lengths; a distance far smaller than any other known physical phenomenon or object.
<li>In the same work, Pfenning and Ford also found that for a bubble a hundred meters in radius, a warp speed of around the speed of light, and the required thickness of about 100 Planck lengths the energy in the bubble shell would be E ≈ -1063 kg c2. The magnitude of this latter value is ten orders of magnitude larger than the energy of the entire visible universe. They do note, however, that if quantum energy inequalities can be ignored then a warp drive with a hundred meters radius but a shell thickness of one meter would "only" have an energy magnitude of about a quarter solar mass.
<li>As with any method of faster than light travel, the warp drive could be used to make a time machine.
<li>Perhaps most seriously, if the warp drive is going faster than light speed, the negative energy regions on the outside of the shell won't be in the warp parts of the bubble. They will have to be moving through space at faster than light speed if the bubble is to maintain its integrity, which is the very problem that the warp drive was designed to avoid. The outside of the warp bubble shell that maintains the warp bubble would fall away and the warp bubble would quickly erode to sub-luminal speeds.<ref name="Van Den Broeck 2000">Chris Van Den Broeck, "On the (im)possibility of warp bubbles", arXiv:gr-qc/9906050 https://doi.org/10.48550/arXiv.gr-qc/9906050</ref>
<li>A more general problem that includes the previous one is that the geometry of the warp drive determines the distribution of energy, momentum, and stress (pressure, shear, and tension) by Einstein's field equation, which states that the stress energy tensor is directly proportional to a reduced order curvature tensor of space-time. However, the stress-energy tensor must also obey the continuity equation – basically, for the energy to change in a given volume of space that energy must come in or go out of the boundary of that volume and similarly for the momentum. So a warp drive can have the warp condition for one moment in time. But after that, the stress-energy that maintains the warp will start to move according to the forces on it and according to the natural paths that it takes through the warped space-time. There is no guarantee that the configuration of the stress-energy at later times will be sufficient to maintain the warp.
<blockquote>
From the perspective of simulating the warp drive dynamically, the key challenge is stability. One can straightforwardly generate the initial metric that describes the Alcubierre metric and infer the instantaneous matter configuration that must support it using the Einstein Equation, but a time-domain evolution of the coupled matter-gravity system necessitates that we specify the equation of state of the matter, which in turn determines how it evolves. In a perfect fluid, this involves specifying the relation between the fluid pressure and density, but in a more general case, the full relation between the
stress-energy components must be given. There is (to our knowledge) no known equation of state that would maintain the warp drive metric in a stable configuration over time - therefore, whilst one can require that initially, the warp bubble is constant, it will quickly evolve away from that state and, in most cases, the warp fluid and spacetime deformations will disperse or collapse into a central point.
 
– Clough, Dietrich, and Khan<ref>Katy Clough, Tim Dietrich, and Sebastian Khan, "What no one has seen before: gravitational waveforms from warp drive collapse", arXiv:2406.02466 [gr-qc] https://doi.org/10.48550/arXiv.2406.02466</ref>
</blockquote>
<li>In order to get the matter (or more generally, stress-energy) that forms the warp shell up to speeds that could allow the shell to exist, you need to be able to, well, accelerate that matter to warp speeds. Which gives a chicken-and-egg problem - in order to achieve superluminal travel you need to be able to get matter or energy moving superluminally in the first place<ref name="Bobrick and Martire">Alexey Bobrick and Gianni Martire, "Introducing Physical Warp Drives", arXiv:2102.06824 [gr-qc] https://doi.org/10.48550/arXiv.2102.06824</ref><ref>D. H. Coule, "No warp drive", Class. Quantum Grav. 15 (1998) 2523–2527. https://doi.org/10.1088/0264-9381/15/8/026</ref>, which rather makes warp drives unnecessary from the get-go.
<li>A warp bubble moving faster than light will not be able to get any signals in front of it as it will overtake its own signal. In fact, anyone in the interior of the bubble would not be able to send signals to the front surface of the bubble. In science-speak, a horizon forms at the front of the bubble<ref name="Natario">José Natário, "Warp drive with zero expansion", Classical and Quantum Gravity. 19 (6): 1157–1166 (2002). arXiv:[https://arxiv.org/abs/gr-qc/0110086 gr-qc/0110086]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2002CQGra..19.1157N 2002CQGra..19.1157N]. doi:[https://doi.org/10.1088%2F0264-9381%2F19%2F6%2F308 10.1088/0264-9381/19/6/308]. S2CID [https://api.semanticscholar.org/CorpusID:15859984 15859984].</ref>. As a consequence, the bubble cannot be controlled, steered, nor stopped from within the bubble itself.
<li>A Horizon will naturally emit Hawking radiation. It is estimated that the radiation will raise the temperature of the interior of the bubble to approximately 1032 K<ref name="Finazzi 2009">Stefano Finazzi, Stefano Liberati, and Carlos Barceló, "Semiclassical instability of dynamical warp drives", Phys. Rev. D 79, 124017 (12 June, 2009) https://doi.org/10.1103/PhysRevD.79.124017 , arXiv:0904.0141 [gr-qc] https://doi.org/10.48550/arXiv.0904.0141</ref> (at this level of accuracy, the result is the same if you replace temperatures in K by the same number in °C or °F). This temperature, however, may be lower if quantum inequalities do not have to be fulfilled.
<li>Semi-classical instabilities at the horizon are expected to lead to a divergence of the stress-energy tensor at the boundary in a short period of time, thus leading to the destruction of the warp bubble<ref name="Finazzi 2009"></ref>.
</ol>
Helpfully, Van Den Broeck<ref name="VanDenBroeck_2000">C. Van Den Broeck, "Alcubierre’s Warp Drive: Problems and Prospects" AIP Conference Proceedings. 504: 1105–1110 (2000). Bibcode:[https://ui.adsabs.harvard.edu/abs/2000AIPC..504.1105V 2000AIPC..504.1105V]. doi:[https://doi.org/10.1063%2F1.1290913 10.1063/1.1290913].</ref> suggested several solutions – or at least mitigations – for some of these problems
<ol>
<li>Quantum inequalities had not been shown to be true in general for highly curved space-times.
<li>A proposed warp drive geometry<ref name="VanDenBroeck_1999">C. Van Den Broeck, "A 'warp drive' with more reasonable total energy requirements". Classical and Quantum Gravity. 16 (12): 3973–3979 (1999). arXiv:[https://arxiv.org/abs/gr-qc/9905084 gr-qc/9905084]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1999CQGra..16.3973V 1999CQGra..16.3973V]. doi:[https://doi.org/10.1088%2F0264-9381%2F16%2F12%2F314 10.1088/0264-9381/16/12/314]. S2CID [https://api.semanticscholar.org/CorpusID:15466313 15466313]. </ref> (see the van Den Broeck drive, below) would be able to minimize the magnitude of the energy required to about that of the mass of our sun. While still large, it is not unphysically large.
<li>Time travel is always going to be a worry with faster than light travel. There's no neat solution to this one.
<li>It may be possible for the negative energy outside of the superluminal region to compress into the superluminal region to form a shock in space-time. The front surface of the bubble would then be a singularity. It is not clear if such a sharp jump in physical properties is possible, but neither is sure that it is impossible, either. However, while this might help with the negative energy in front of the bubble getting swept up because it cannot move faster than light, it does little to help with the material at the back of the bubble on the outside getting left behind because it cannot keep up.
</ol>
Other mitigating research has also shown
<ol start=9>
<li>Semi-classical divergence only occurs in warp drives with only one spatial dimensions. Those with two or more spatial dimensions (as in our universe) only contain points where divergences occur, and these divergences can be managed. In fact, their effects can be reduced by making the warp shape more elongated<ref>Carlos Barceló, Valentin Boyanov, Luis J. Garay, Eduardo Martín-Martínez,
and Jose M. Sánchez Velázquez, "Warp drive aerodynamics", JHEP08 (2022) 288 https://doi.org/10.1007/JHEP08(2022)288 , arXiv:2207.06458 [gr-qc] https://doi.org/10.48550/arXiv.2207.06458</ref>.
</ol>

=== Alcubierre warp interactions with light and matter ===

Space is not entirely empty. It is filled with a diffuse plasma in the form of the interstellar medium, as well as cosmic radiation, light from stars, and cosmic microwave background radiation. A warp drive propagating through space will encounter this stuff. What happens when this matter and radiation have a warp bubble pass across them?

The first analysis of matter encountering a warp bubble was performed by Pfenning and Ford<ref name="PfenningFord_2001"></ref>. They looked at the warp bubble interaction with a massive object at rest with the frame of reference of the warp drive (keep in mind that the rest frame is the same inside and outside the warp bubble; in this rest frame objects inside the bubble have no momentum even though they are, in some sense, changing location rapidly with time). When the warp bubble passes, the object experiences an acceleration in the direction of the warp bubble motion. When the warp shell passes and the object is inside the bubble, it will be moving with approximately the speed of the bubble. Pfenning and Ford analyzed this problem with a continuous distribution of shell energy that, strictly speaking, never falls to zero except at the bubble center and at spatial infinity, so unless the object passes through the center in Pfenning and Ford's description it will never quite get up to the bubble's speed. In this case, the object will move almost as fast as the object but will pass through the bubble in a finite time, after which it will again be at rest with respect to the reference frame but displaced along the direction of the warp bubble motion by some distance. With this description of the warp bubble, a spacecraft of finite size will always be moving a little slower than the warp bubble and would have to use rockets to keep up with it. in addition, the spacecraft would experience tidal forces that would cause stress on the spacecraft's structure.

Pfenning and Ford also examined cases where the shell is of a finite (possibly infinitesimal) width and falls to zero both inside and outside the bubble. In this case any matter encountering the bubble would be collected at the bow of the bubble and thereafter move along with it.

McMonigal et al. analyzed the situation for both massive particles and light moving along the axis of travel of the bubble<ref name="McMonigalEtAl_2012">B. McMonigal, G. F. Lewis, and P. O'Byrne, "Alcubierre warp drive: On the matter of matter". Physical Review D. 85 (6) 064024 (20 March 2012). arXiv:[https://arxiv.org/abs/1202.5708 1202.5708]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2012PhRvD..85f4024M 2012PhRvD..85f4024M]. doi:[https://doi.org/10.1103%2FPhysRevD.85.064024 10.1103/PhysRevD.85.064024]. S2CID [https://api.semanticscholar.org/CorpusID:3993148 3993148].</ref>. They found that light moving opposite the warp direction passed through the bubble without incident, being only somewhat delayed by passing through the bubble. A warp bubble that is warping at sub-luminal speeds can have light catch up from behind it. This light is able to pass through the bubble, and is somewhat advanced in its path by the speed of the bubble. For super-luminal warp bubbles, however, the situation is different. The bubble will catch up to light moving it its own direction that is originally in front of it. This light cannot escape forward, the bubble being too fast. Nor can it escape backward, as the light is propagating forward and the interior of the bubble is at rest. Thus, the light gets caught at the bow of the warp bubble, unable to escape for so long as the warp bubble is active. This light is strongly blue shifted to extremely energetic x-rays and gamma rays. The space behind the bubble is swept clear of forward-moving light.

The situation for matter over-run by a super-luminal warp bubble is similar. Matter moving backward passes through the bubble, being only swept a ways along the bubble's path as sign of its passing but otherwise continuing on their way. Matter at rest behaves the same way as Pfenning and Ford discovered. Meanwhile, matter moving in the same direction of the warp is overtaken and collects at the bow of the bubble, unable to leave for so long as the bubble is warping. This matter "surfing" on the bubble bow is highly accelerated to relativistic speeds, experiencing extreme time dilation.

For a sub-luminal bubble, there is a speed faster than the warp speed where particles coming up from behind overtake the bubble and pass through, out the front. Particles below this speed but still faster than the warp speed still overtake the bubble, but then bounce backwards out of the bubble from behind. They are still moving forward, but are now moving more slowly than the warp speed. Particles moving forward but slower than the bubble will be overtaken and then bounced out the front with a speed higher than the warp speed. Those particles moving backward will pass through the bubble and exit with their initial speed.

The spacecraft inside the bubble will observe that light moving backward compared to the warp direction is blue-shifted, and matter moving backward passes through with increased energy. Meanwhile, a sub-luminal bubble will see light that was moving forward as red-shifted and matter catching up to the bubble from behind will pass by with reduced energy (superluminal bubbles, of course, do not have any matter or radiation catching up to them from behind).

When a superluminal warp bubble that has been collecting matter and radiation for the duration of its journey and blue-shifting it to much higher energies is turned off, all that matter and energy is released as a blast of radiation in the direction the bubble was warping.

McMonigal et al. conclude by noting that any spacecraft in the warp bubble would need shielding to protect against the blue-shifted radiation and matter of increased energy. In addition the destination would be "blasted into oblivion" by the release of matter and radiation that had been caught in the bubble during the trip.

Clark et al.<ref>Chad Clark, William A. Hiscock, and Shane L. Larson, "Null geodesics in the Alcubierre warp drive spacetime: the view from the bridge", arXiv:gr-qc/9907019 (2018) https://doi.org/10.48550/arXiv.gr-qc/9907019</ref> examine what it would look like from within an operating warp drive. They find that light moving at 90 degrees to the direction of motion will look like it is still coming from 90 degrees. However, light from the front hemisphere is squished progressively toward the forward point the faster you go. For a warp bubble moving faster than the speed of light, there is a cone behind the warp bubble where no light can reach it. However, you don't see a black spot behind you – light from the regions in the backward hemisphere from outside that cone that can reach you is distorted toward the direction behind you to fill in that spot. Light from the forward direction is blue-shifted and light from behind is red-shifted. At speeds of around 200 times light speed, the cosmic microwave background would be blue shifted to energies similar to that of the solar photosphere, heating anything in the warp bubble as if they were close to the sun.

== Other warp geometries ==

The Alcubierre warp drive is the first and best known of the warp drives. It has been around long enough that there have been many studies done on it. The warp geometry is particularly simple, making analysis easier. But in the time that the Alcubierre warp was proposed, many other researchers have come up with other warp designs, and even generalized the idea to include a wide array of additional possibilities.

Most of the general statements about the challenges of the Alcubierre drive hold for these other drives. Many of the details about interactions with matter are also likely to hold, although these other styles of warp drive have not yet been subject to such rigorous investigation.

=== Van Den Broeck warp drive ===

If an Alcubierre warp bubble a hundred meters across requires a magnitude of energy greater than the entire energy in the observable universe, one option to reduce the magnitude of energy used is to make the warp bubble smaller. Much smaller. Pfenning and Ford<ref name="PfenningFord_2001"></ref> suggested making the warp bubble smaller than an atom. Of course, that brings up the problem of how to stuff a spacecraft in there. Van Den Broeck proposed a solution<ref name="VanDenBroeck_1999"></ref>: make the warp bubble only about the size of an atomic nucleus but expand the space inside the bubble enormously so that a spacecraft could fit in. This is somewhat like a nuclear sized wormhole that leads to a pocket universe that holds the spacecraft. Unfortunately, the metric doesn't fit neatly into any [[Wormholes#Three_dimensions|embedding diagram]] that I can figure out, but the math works out so that spatial coordinates inside the bubble are expanded by a factor of 1017 and a spacecraft would have a few hundred meters of bubble interior to putz around in. This trick manages to reduce the magnitude of energy to only about the mass energy of a few stars similar to our own. Other than that, it is otherwise a normal Alcubierre drive.

Further work<ref>S. Krasnikov, "The quantum inequalities do not forbid spacetime shortcuts", Physical Review D. 67 (10) 104013. https://doi.org/10.1103%2FPhysRevD.67.104013, arXiv:gr-qc/0207057 (2003) https://doi.org/10.48550/arXiv.gr-qc/0207057 </ref> has managed to use Van Den Broeck's trick to reduce the magnitude of the negative energy to only about a milligram.

=== Natário warp drive ===

The Alcubierre drive is not the only way to construct a warp drive. José Natário<ref name="Natario"></ref> showed that it was just one example of an entire class of warp drives. These are space-time geometries with a smooth and localized shift vector field and a unit lapse function.

Natário then went on to find in that class a set of warp drives that do not have any expansion or contraction of space at all. Instead, space encountering the front boundary of the warp bubble instead "slides" around the outside of the bubble until it gets to the corresponding place on the back and is left behind there. Another way to think of it is that space entering the bubble shell is compressed in the radial direction of the shell but is simultaneously expanded in the tangential direction so that there is no net change in volume. When it gets to the back, the opposite occurs. In the language of mathematics, the shift vector field is divergenceless.

<table align=center>
<tr><td width=360>[[File:Natario_drive_flow_of_space.png]]
<td width=445>[[File:Natario_warp_drive_horizons.png]]
<tr><td style="vertical-align: top;">A representation of the motion of space in a zero-expansion Natário drive with a vanishingly thin warp shell around the bubble.
<td>The horizons of the generalized Natário class of warp drives (including the Alcubierre warp drive and all fell-Heisenberg drives) at superluminal speed in the direction of the green arrow. No event inside the bubble can affect events outside the bubble in front of the blue line. No events outside the bubble behind the red line can affect events inside the bubble. The lines have an angle α with respect to the direction of motion with sin(α) = 1/vs for vs the speed of the bubble relative to light speed. This is analogous to the Mach cone of supersonic objects.
</table>

NOTE: There are two different things that can be meant by the term "Natário warp drive". The generalized Natário warp drive is the general theory of all warp drives that includes within it the Alcubierre, Fell-Heisenberg, and Lentz warp drives, among infinitely many others. The zero-expansion Natário warp drive is the specific implementation of the generalized Natário warp drive that has a divergenceless shift vector field.

The generalized class of warp drives developed by Natário (including both the Alcubierre and this zero-expansion warp drive) are shown to always have regions in the warp shell where the energy density is negative to at least some observers. You can't get away from it – to warp, you need negative energy density.

The Natário class of warp drives blueshift light coming in from the front and redshift light catching up from the back. For the case of a super-luminal warp bubble, there will of course be a horizon at the back of the bubble and light will not be able to get through from behind. For a bubble speed vs relative to the speed of light, light coming from straight ahead will be blueshifted up in frequency and photon energy by a factor of 1 + vs. For sub-luminal travel, light coming from behind will be redshifted by a factor 1 - vs. In general for n as the unit direction along which the light is propagating, the blueshift factor will be 1 + vs ⋅ n. If the warp shell is infinitesimally thin, light from outside that reaches the center of the bubble will not be distorted in direction although it will be frequency shifted, but observers still inside the bubble but displaced from the center will see the field of view distorted as well as frequency shifted.

Most of the difficulties that apply to the Alcubierre warp drive also apply to the generalized Natário warp drive (and thus to the zero-expansion Natário drive and Fell-Heisenberg drive as well). In particular, quantum energy inequalities, high magnitudes of the energy required, formation of time machines, instabilities related to the stress-energy making up the warp shell being unable to keep up with the bubble, the chicken-and-egg problem of needing FTL to achieve FTL, the formation of horizons making warp bubbles uncontrollable, and Hawking radiation at the horizons, are all still concerns for generic warp drives.

=== Fell-Heisenberg warp drives ===

S. D. B. Fell and L. Heisenberg investigated what sort of warp structure is necessary to produce a warp drive with positive energy density everywhere<ref name="Fell-Heisenberg">S. D. B. Fell and L> Heisenberg, "Positive energy warp drive from hidden geometric structures", Classical and Quantum Gravity 38 155020 (2021) https://doi.org/10.1088/1361-6382/ac0e47 https://arxiv.org/abs/2104.06488</ref>. They found that warp drives of the generic Natário form with irrotational shift vectors had a simple expression for the energy density and vanishing momentum (irrotational means that the shift vector can be described as the gradient of a scalar potential field or, equivalently, that the curl of the shift vector field vanishes everywhere). Fell and Heisenberg proposed several potential fields that either had positive energy density everywhere except at discontinuities, or that were continuous but had regions that violated the weak energy condition (i. e. negative energy densities in some frames of reference). One specific example they provided of a warp field had an energy that is about 10,000 times less than the mass-energy of our sun. Or only about half the mass-energy of Jupiter. This is a significant improvement over many other proposed drives.

Despite the introduction discussing warp drive configurations that satisfy the various energy conditions, the configurations described in the paper are shown to locally violate the weak and strong energy conditions. Nonetheless, the energy density is still mostly positive. Santiago, Schuster, and Visser's<ref name="Santiago Schuster Visser 2022">J. Santiago, S. Schuster, and M. Visser, "Generic warp drives violate the null energy condition", Physical Review D 105, 064038 (2022) https://doi.org/10.1103/PhysRevD.105.064038</ref> work shows that the various energy conditions must still be violated by this warp drive; to not violate these, energy density must be positive in all reference frames not just those of the co-moving observer.

Any vector field can be described by its divergence, its curl, and a constant. The constant part of a warp drive's shift vector can be discarded because then the warp drive would not be localized. It is interesting, therefore, that the Natário zero-expansion drive has zero divergence and can be described only by its curl, while the Fell-Heisenberg drive has zero curl and can be described only by its divergence. These two classes of drives thus rely on opposite choices of setting one of the two different parts describing a vector field to zero. As might be expected, the Fell-Heisenberg drive is not zero expansion – space-time will change in volume as it passes through the warp bubble (although after going all the way through it will return to its original volume). However, objects in space-time will retain their original orientation as they pass through the warp bubble.

Any vector field with zero curl can be represented as the gradient of a suitable scalar function. Thus, the Fell-Heisenberg drive can be completely described by just scalar field (one number at all points of space and time) rather than a vector field.

Previous examples of warp drives, such as the Alcubierre and zero-expansion Natário drives, tend toward a flat space-time as you go far enough away faster than would occur if there were any mass present in the warp drive. That is, the ADM mass of such drives is zero. However, one of the Fell-Heisenberg examples described a spacetime where the geometry approaches that of what you would get for a massive object at the warp location, giving one of the first examples of a warp drive with mass.

In the Fell-Heisenberg drive the momentum vanishes everywhere. Yet the energy occupies regions where the shift vector is varying rapidly. The lack of momentum means that the energy will not move to keep up with the differential expansion and movement of the spacetime. As a consequence, at later times the energy will have a different distribution than what is necessary to maintain the given warp configuration. The exact time evolution is not solved but this likely leads to collapse of warp bubble. Note that this is not a problem unique to the Fell-Heisenberg drive. Assuming a geometry that describes a warp produces a specific distribution of the stress-energy tensor that satisfies Einstein's equation. But there is no guarantee that this stress energy tensor satisfies the continuity equation, which describes how the distribution of matter and momentum change with time. The issue of stability of any warp drive is not well understood, but most warp drive geometries will be unstable in the sense that their matter that generates the warp fields will not be able to keep up with the motion of the warp. The Fell-Heisenberg drive is unique only in that, with vanishing momentum, there is no configuration that allows stable propagation.

(n. b. The Heisenberg here is Lavinia Heisenberg, not the Werner Heisenberg of quantum physics and uncertainty principle fame.)

=== Lentz warp drive ===

E. Lentz<ref name="Lentz">E. W. Lentz, "Breaking the warp barrier: hyper-fast solitons in Einstein–Maxwell-plasma theory", Classical and Quantum Gravity. 38 075015 (2021). arXiv:[https://arxiv.org/abs/2006.07125 2006.07125]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2021CQGra..38g5015L 2021CQGra..38g5015L]. doi:[https://doi.org/10.1088%2F1361-6382%2Fabe692 10.1088/1361-6382/abe692]. ISSN [https://search.worldcat.org/issn/0264-9381 0264-9381]. S2CID [https://api.semanticscholar.org/CorpusID:219635854 219635854].</ref> investigated a particular configuration of the Fell-Heisenberg class of warp drives, where the scalar field forms a selection of rhomboid regions, resulting in an energy density that is everywhere positive and a shift vector with vanishing tidal force at the center for a comfortable ride.

However, Santiago, Schuster, and Visser<ref name="Santiago Schuster Visser 2022"></ref> dispute claims that the Lentz drive satisfies the energy conditions, noting that everywhere positive energy density in one frame of reference is insufficient to establish that the energy density is positive in all reference frames; knowledge of the Cauchy stress tensor is also needed. They show that any generic warp drive will violate the strong energy condition, null energy condition, and weak energy condition.

== Conservation laws ==

=== General introduction ===

All the warp drives investigated so far have geometries that quickly fall off from the highly curved region near the warp bubble to a region of spacetime that is almost flat; and becomes even more flat the farther away from the warp bubble you get. There is a word for geometries of this kind – they are called asymptotically flat. Basically, you can always go far enough away in any direction to reach a place where the geometry is sufficiently flat to meet any flatness criterion you choose.

When you have an asymptotically flat geometry, then in the far away mostly flat regions gravity is well described by a linearized theory of gravity – where the perturbations to the geometry are linear in the stress energy tensor. This means that gravity is basically Newtonian, along with a few other bells and whistles like gravitational waves and frame dragging.

When linearized gravity is a good approximation, then it is known that energy, momentum, and linear momentum are conserved. Although parts of a warp geometry are highly curved and non-linear, you can measure the conserved energy and momentum and angular momentum from far away in the linear region. This ADM energy and etcetera is the quantity that is conserved when you are worrying about conserving such things, not necessarily the energy and so forth you get in the highly curved spacetime region. In fact, it is not even simple to define where the energy and such actually is withing the highly curved spacetime region; it is non-local and different observers will disagree on how it is distributed. But everyone agrees on the ADM conserved quantities from out in nearly flat spacetime (up to the usual changes due to changing speeds, which are trivial and come up with normal objects as well).

And now we run into a bit of a conundrum: most of the warp geometries so far proposed have zero ADM mass<ref>Sebastian Schuster, Jessica Santiago,and Matt Visser, "ADM mass in warp drive spacetimes" General Relativity and Gravitation (2023) 55:14 https://doi.org/10.1007/s10714-022-03061-9</ref>, meaning that things like the Alcubierre drive technically have no energy or momentum or angular momentum at all. Despite all of the extremes of negative energy regions within the wall of the warp bubble, the actual energy that you need to spend to make one is zero. This can lead to a bit of a problem. If you have a spacecraft that has mass, that mass-energy has to be conserved. So when you turn on your warp drive and the total warp mass is zero, where does that energy go?

One resolution is just that the mass of the spacecraft is negligible and the small amount of curvature it causes can be ignored. In this case, the warp geometries discussed so far are merely an approximation that neglects this effect. In fact, some of the proposed Fell-Heisenberg drives map not onto entirely flat spacetime geometries at large distances, but onto the Schwarzschild geometry that is appropriate for systems with mass at the center. If this is the case, then the warping bubble and spacecraft system will need to worry about conservation of all their conserved properties.

=== All the science, demonstration of the problem ===

Let's look at what the conservation laws mean for a spacecraft equipped with a warp drive, assuming that no matter or radiation are emitted by activating or deactivating the warp drive.

We can start with an observer floating in space. We'll call her Alice. And, approaching Alice at 100 m/s is the starship Enterprise with a mass of a cool 200,000 tons (2×108 kg). from this we know that the Enterprise has, in Alice's frame of reference, a kinetic energy of 1 TJ, a total energy (including mass and kinetic energy) of 1.8×1025 J, a momentum of 2×1010 kg m/s, and – because it is headed straight toward Alice – an angular momentum of zero. All of this is without any warp drive activated.

On the bridge of the Enterprise, Spock picks up Alice on the sensors and tells Captain Kirk. Kirk does not want to run into Alice (he'd rather date her), so he orders helm to turn on the Enterprise's warp drive and move them 1 km up so that their course takes them 1 km from Alice instead of straight into her. Sulu warps the Enterprise according to these commands, and then turns off the warp drive.

After this maneuver, keeping the Enterprise's total energy and momentum the same means that the Enterprise still has a mass of 200,000 tons and a velocity of 100 m/s in its original direction. So we can see that the warp drive must preserve the velocity vector of the thing in it (at least in flat spacetime where outside forces are not acting on it). The problem comes from angular momentum. With a known mass of 200,000 tons, a speed of 100 m/s, and a distance of closest approach of 1 km; from Alice's frame of reference the Enterprise now has an angular momentum of 2×1013 kg m2/s after the warp maneuver, while before the maneuver it has an angular momentum of zero.

From this we can see that warp drives cannot be reactionless. Arbitrary displacements using warp without emitting propellant result in violation of the law of conservation of angular momentum. As long as we restrict ourselves to accepted science based on what we know of general relativity, to move a warp drive with mass you need to accelerate it or decelerate it with rockets or other normal means of delivering momentum (such as light sails, mag sails, or pushing off of a planet). Warp drives must be inertial to comply with physics. And all of this was analyzed only from times when the warp drive was turned off; it does not depend on any assumptions about what happens when the warp is on.

=== A zero mass warp "solution" ===

So let's start picking at our assumptions. First, what if the warp bubble actually does have zero mass? Now, while in warp, there is no momentum or angular momentum or energy to be conserved. A spacecraft in warp can bounce around like a ferret on stimulants and the universe doesn't care.

But there's a problem, and it's probably pretty obvious. Our Starship Enterprise has a lot of mass when it hasn't activated its warp bubble. So, 200,000 tons × c2 = 1.8×1025 J of energy before the warp is turned on, and 0 J of energy afterward. This seems like a flagrant violation of conservation of energy.

Well, there's a solution. Maybe turning on the warp drive somehow dumps all that energy. You could get a pulse of gravitational and electromagnetic radiation rippling away from the turn-on point at the speed of light, maybe with various kinds of particle radiation thrown in. Or maybe a 200,000 ton black hole is pooped out behind the Enterprise when the warp bubble is turned on. Anything that dumps the energy and leaves it behind.

Okay, so, the good ship Enterprise zips off to Starbase 9 with in a zero mass warp bubble. But now they're in a bit of a pickle. In order to come out of warp so they can visit the starbase, they somehow need to acquire 200,000 tons worth of mass-energy from somewhere.

One option is that the Enterprise rams its warp bubble into some random asteroid and grabs 200,000 tons of it, or skims 200,000 tons of gas from the top of the atmosphere of a nearby world, or maybe the starbase keeps around millions of tons of ballast to give to visiting starships so they can de-warp. It's a perfectly reasonable way to go about conserving all the conserved things the universe seems to care about.

But their is an even weirder option. It's pretty speculative – all of this is speculative, but this is even more so – but it is fun so it's worth mentioning. Normally, a source of radiation produces waves that radiate out from it into the universe that expand with time. But the time reverse of this process is also allowed by the laws of physics, with waves converging from the far reaches of the universe, interacting and scattering off of matter and black holes and other stuff in such a way so as to make an ever-shrinking concentric wave that collapses on the source. This solution is generally rejected because it doesn't mesh with the second law of thermodynamics. But if you have faster than light warp drive, causality goes out the window, causes and effects can be reversed depending on your frame of reference, and strange coincidences can occur because they must occur to maintain consistency. So what if dropping out of warp always just happens to coincide with being at the center of a converging sphere of radiation that gives you exactly enough energy to go from a zero energy warp bubble back to your full-mass spacecraft?

This would have interesting consequence, because your arrival would always be heralded by a converging wave-front. You could build warp sensors that would alert you to when and where someone is going to come out of warp. No longer can an entire invasion fleet suddenly appear over your planet and demand your surrender, you'll have advance warning and can mount a defense.

(Zero mass warp drives, relaxing conservation requirements, and activation/deactivation)

(If angular momentum conservation is ignored, discuss how momentum & energy are affected by outside forces. In a gravitational field equivalent to inertial frame ... like being in an accelerated elevator; will acquire momentum buildup while staying "at rest".)

== Warp Railroads ==

Van Den Broeck<ref name="VanDenBroeck_2000"></ref> suggested a warp railroad,

You can set up devices ahead of time along the path of the bubble that produce the negative energy regions for the warp bubble at the appropriate time without needing the negative energy to ever move with the bubble at all. At least for bubbles on a predictable schedule and route.

helps with:

-stability (elaborate)

-conservation laws (elaborate)

Mention Krasnikov<ref name="Krasnikov 1998">S. V. Krasnikov, "Hyperfast travel in general relativity" Phys. Rev. D 57 4760</ref>

Hyperwave<ref name="Pieri 2024">Lorenzo Pieri, "Hyperwave: hyper-Fast Communication within General Relativity", arXiv:2311.12069 [gr-qc] https://doi.org/10.48550/arXiv.2311.12069</ref> makes use of this warp railroad idea.

== Quantum effects ==

(quantum stuff here<ref name="Hiscock1997">W. A. Hiscock, "Quantum effects in the Alcubierre warp drive spacetime", Classical and Quantum Gravity 14 L183 https://doi.org/10.1088/0264-9381/14/11/002 https://arxiv.org/abs/gr-qc/9707024</ref><ref name="Finazzi et al 2009">S. Finazzi, S. Liberati, C. Barceló, "Semiclassical instability of dynamical warp drives", Physical Review D 79, 124017 (2009)https://doi.org/10.1103/PhysRevD.79.124017 https://arxiv.org/abs/0904.0141</ref>)

From<ref name="Pieri 2024"></ref>
<blockquote>
It should be emphasised that Quantum Inequalities have not been proven for realistic interacting quantum fields, where the they are expected to be weaker if they exist [Cadamuro, 2019], and they often require Minkowskian conditions at infinity, a condition which is not appropriate for fields inside bounded regions, as in a Casimir devices.
</blockquote>

== Warp drives and black holes ==

What happens if you drive your warp drive into a black hole?

The pop-sci reason that nothing can escape a black hole is that at the event horizon the escape velocity is faster than light and nothing can go faster than light. A warp drive can go faster than light.

The more detailed reason from general relativity is that at the event horizon space and time are rotated sufficiently that "inward" becomes "forward in time." You can no more go farther away from the singularity in a black hole once you are inside the event horizon than you can go backward in time. Any super-luminal travel can go backward in time, and warp drives can engage in super-luminal travel.

The Hawking area theorem that says that black holes can only grow, never shrink relies on the condition that the energy is everywhere positive. Warp drives have negative energy density.

So in principle, there's nothing that prevents a warp drive from taking a dip into a black hole and coming back out again.

Garattini and Zatrimaylov looked into the problem of a warp drive near (and in) a black hole<ref name="Garattini and Zatrimaylov">R. Garattini and K. Zatrimaylov, "Black holes, warp drives, and energy conditions", Physics Letters B 856 138910 (2024) https://doi.org/10.1016/j.physletb.2024.138910</ref>. The found that the black hole reduces the negative energy requirement for the warp bubble to move inward toward the black hole singularity, but increases the negative energy needed to move away from the center of the black hole. In addition, a warp drive parked at the event horizon would allow light from inside the horizon to pass through the bubble and escape back outside.

== Credit ==
Author: Luke Campbell

== References ==

[[Category:Physics]][[Category:Infrastructure]][[Category:Physics & Engineering‏‎]][[Category:Physics & Math & Engineering]][[Category:Transportation & Infrastructure‏‎]][[Category:Metric Engineering]]

Warp Drives

2026-07-31T15:20:20Z

Lwcamp: /* Conservation laws */

{{PageConstructionNotice}}

Science fiction often features spacecraft that can seemingly move across space and get between the place of departure and the destination much faster than light could have done. This appears to contradict the theory of relativity, which predicts unequivocally that nothing can move through space faster than light. Because relativity has been incredibly successful at describing nature, with its many other predictions regularly being confirmed to extraordinary accuracy and within the bounds of uncertainty of all the experiments that tested them, it gives confidence that relativity is a correct description of reality. Which seems to rather throw a wet towel on our hopes for rapid travel between stars.

However, while relativity does not allow things to move through space faster than light, it places no such restrictions on how fast space-time itself can expand, contract, or move around. This leads to the idea of a warp drive – the spacecraft remains stationary within a region of highly curved space-time, and that region moves at super-luminal speeds rather than the spacecraft.

== The Alcubierre warp drive ==

The first warp drive geometry that satisfied the Einstein field equations of relativity was proposed by Miguel Alcubierre<ref>M. Alcubierre, "The warp drive: hyper-fast travel within general relativity." Classical and Quantum Gravity. 11 (5): L73–L77 (1994). [https://arxiv.org/abs/gr-qc/0009013 arXiv:gr-qc/0009013]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1994CQGra..11L..73A 1994CQGra..11L..73A]. doi:[https://doi.org/10.1088%2F0264-9381%2F11%2F5%2F001 10.1088/0264-9381/11/5/001]. S2CID [https://api.semanticscholar.org/CorpusID:4797900 4797900].</ref>. In this geometry, a sphere of space-time moves at an arbitrary speed (potentially but not necessarily a speed much faster than light). Objects within the sphere are moved along with the sphere; an object at rest within the sphere would be moved along with the sphere indefinitely. Space is expanding at the rear boundary and contracting at the front boundary in order to keep the sphere moving. In order to satisfy the Einstein field equations, the boundary of the sphere must have a negative energy density. The challenges of space-time geometries with negative energy densities are described in our page on [[Wormholes#Exotic_energy_conditions|wormholes]], for our purposes it is enough to note that negative energy densities can pose problems if not handled carefully, there may be limits on how much negative energy you can have without nearby positive energy density, and it may not be possible to get enough negative energy to support a warp drive; although none of this is ruled out by physics – yet!

<table align=center>
<tr><td width=650>[[File:Energy_magnitude_of_Alcubierre_drive.png]]
<td width=650>[[File:Expansion_of_Alcubierre_drive.png]]
<tr><td>The magnitude of the energy of the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. (The energy density is negative everywhere in the shell, here only the magnitude is shown to aid visual comprehension.)
<td>The expansion of space in the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. Negative expansion means that space is contracting in that direction, positive expansion that it is expanding.
</table>

NOTE: What the? The energy distribution is symmetric! How does the drive know which way to go? There must be significant contributions of other components of the stress-energy tensor, and those have got to be asymmetric along the forward/backward axis. Check this when I get time.

(And what's with the ring in all the artwork, anyway?)

=== So what is going on? ===

What is it that makes a warp drive a warp drive, and how does it work?

A warp drive is a concept that comes out of the science of general relativity. General relativity collects space and time together into a single "structure" called spacetime. This structure is curved, and its curvature is determined by the material that is in it. Absent forces acting between matter, particles of matter move through spacetime on the shortest possible paths in this curved environment.
<blockquote>
"Spacetime tells matter how to move; matter tells spacetime how to curve." - John Archibald Wheeler
</blockquote>
All of this is encompassed in the Einstein field equation:
<div align=center>
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 8πG
<td rowspan="2" nowrap="nowrap" align="right"> T = R -  
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right"> R g
<tr>
<td style="border-top:solid 1px black;"> c4
<td style="border-top:solid 1px black;"> 2
</table>
</div>
Unfortunately, by convention, both the symbols G and R get used for a lot of different things. Here, symbols in bold are second rank tensors – constructs of arrays of 4×4 numbers where 4 is necessary because that is the number of dimensions of spacetime. Conventionally, these numbers are indexed by a value going from 0 to 3, with 0 being the part corresponding to the coordinate of time and 1 to 3 being the Cartesian coordinates that you might know as x, y, and z.

T is the stress-energy tensor: the (0,0) component represents the energy density at a given point; the (0,i) and (i, 0) components (with i going from 1 to 3) represent the momentum density; and the (i,j) components represent the stresses present, such as pressure or tension or shear.

The other term, R - R g/2, involves a partial description of the curvature. A full description of the curvature requires a construct of 4×4×4×4 components; too much for the stress energy tensor to fully constrain. Which is okay, because this allows curvature to exist away from where matter is, allowing gravity to have long range effects across empty space as well as for phenomena like gravitational waves. But it is enough that a given distribution of matter T gives enough constraints (along with specifying how the geometry behaves far away) that you can generally figure out what the actual gravity is.

This is the usual story that we are told. But there is an alternate way of looking at the same geometry such that the math works out the same, but it is described conceptually in a very different way. This divides spacetime up into "layers", called foliations, that represent slices of space and going from one foliation to the next is increasing in time. This is useful for computation because you can assign coordinates to your system. There are two additional constructs – the lapse function and the shift vector field. Starting at an event on one foliation, the lapse function at that position tells you that after a certain "proper time" (time perceived by an observer at that event) the time coordinate of the observer will be the proper time times the lapse function at that event. And the new spatial coordinates of the observer will be the proper time times the shift vector at that event. (Strictly speaking, this is only exactly true in the limit of the proper time approaching zero or when the lapse function and shift vectors are constant. But hopefully this gives a good enough insight to what these entities are so that we can go on to describe warp drives.)

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Lapse_shift.png]]
<tr><td>A folliation of space-time into slices of constant time coordinates (grey). At an event (black circle), the lapse function (yellow) shows how far into the future the point at that event will be projected in a unit amount of proper time experienced by an observer initially at the event. The shift vector (red) shows how that point changes coordinates after the unit amount of proper time. The cyan vector shows the new coordinates the point is projected onto. The lower black surface indicates a foliation of spacetime at the same time coordinate; after unit proper time, the events on the lower foliation get projected to the curved upper black foliation according to the shift vectors and lapse functions at each of their respective points.
</table>
</table>

Warp drives are generally restricted to spacetime geometries with a constant lapse function - all observers go forward by the same time coordinate after experiencing the same amount of proper time. Thus, all warp drives have no time dilation.
Far from the warp bubble, the shift vector vanishes – each spatial coordinate is mapped to the same spatial coordinate after any amount of proper time so long as the warp drive is not near. But the inside of the warp bubble is an extended region where the shift vector field is non-zero and constant. In this bubble, all spatial coordinates are mapped to a coordinate that shifts linearly in time with the proper time of the observer. Astute readers will see that this linear change in position with time is very similar to a velocity. This region of constant shift vector then moves with the shift vector velocity to keep stuff originally in the bubble so that it stays in the bubble. In between the inside of the bubble and the far away regions with zero shift vector, the shift vector field falls away to match one region to the other.

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Warp_shift.png]]
<tr><td>The shift vector of a one-dimensional slice through warp geometry, in red. At the edges there is no shift. At the center (inside the warp bubble) there is a region of uniform shift so that at a later time the inside of the warp bubble is rigidly translated to different spatial coordinates (in blue). The two foliations remain the same distance apart, because warp drives maintain a unit lapse function.
</table>
</table>

Traditionally, scientists would start with an interesting distribution of matter given by T and then find out what kind of curvature it created. This works well for planets and stars and black holes and cosmic strings and other things which are observed in the universe and you want to see what sort of effect they had. But starting in the 1980's, some physicists started looking at Einstein's field equations the other way around. They start with a desired geometry, find the curvature part of the field equation R - R g/2, and use that to find out what T has to be in order to get that curvature. This was originally done for [[Wormholes|wormholes]], but in 1994 Miguel Alcubierre used this same trick to figure out what could create a region of constant shift vector that moved with the shift vector - the warp drive. With this demand for a warp geometry, the required distribution of matter and energy and momentum and stress can be determined.

The original Alcubierre shift vector N→ produces a spherical warp bubble with an axi-symmetric distribution of, well, everything along the direction of motion with warp velocity vector v→.
<div align="center">
N→ = v→ f(rs)
</div>
Here, rs is the distance from the center of the warp bubble, and f(rs) can be any function that is 1 near rs = 0 and is 0 for rs well beyond the radius of the warp bubble.

=== Challenges and possible resolutions ===

Almost as soon as Alcubierre proposed his warp drive, others began picking it apart.
<ol>
<li>Pfenning and Ford<ref name="PfenningFord_2001">M. J. Pfenning and L. H. Ford, "The unphysical nature of 'Warp Drive'", Classical and Quantum Gravity. 14 (7): 1743–1751 (1997). arXiv:[https://arxiv.org/abs/gr-qc/9702026 gr-qc/9702026]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1997CQGra..14.1743P 1997CQGra..14.1743P]. doi:[https://doi.org/10.1088%2F0264-9381%2F14%2F7%2F011 10.1088/0264-9381/14/7/011]. S2CID [https://api.semanticscholar.org/CorpusID:15279207 15279207].</ref> found that if the negative energy density wall around the bubble obeys [[Wormholes#Quantum_energy_inequalities|quantum energy inequalities]], then for warp speeds of around light speed the shell thickness would be on the order of a hundred Planck lengths; a distance far smaller than any other known physical phenomenon or object.
<li>In the same work, Pfenning and Ford also found that for a bubble a hundred meters in radius, a warp speed of around the speed of light, and the required thickness of about 100 Planck lengths the energy in the bubble shell would be E ≈ -1063 kg c2. The magnitude of this latter value is ten orders of magnitude larger than the energy of the entire visible universe. They do note, however, that if quantum energy inequalities can be ignored then a warp drive with a hundred meters radius but a shell thickness of one meter would "only" have an energy magnitude of about a quarter solar mass.
<li>As with any method of faster than light travel, the warp drive could be used to make a time machine.
<li>Perhaps most seriously, if the warp drive is going faster than light speed, the negative energy regions on the outside of the shell won't be in the warp parts of the bubble. They will have to be moving through space at faster than light speed if the bubble is to maintain its integrity, which is the very problem that the warp drive was designed to avoid. The outside of the warp bubble shell that maintains the warp bubble would fall away and the warp bubble would quickly erode to sub-luminal speeds.<ref name="Van Den Broeck 2000">Chris Van Den Broeck, "On the (im)possibility of warp bubbles", arXiv:gr-qc/9906050 https://doi.org/10.48550/arXiv.gr-qc/9906050</ref>
<li>A more general problem that includes the previous one is that the geometry of the warp drive determines the distribution of energy, momentum, and stress (pressure, shear, and tension) by Einstein's field equation, which states that the stress energy tensor is directly proportional to a reduced order curvature tensor of space-time. However, the stress-energy tensor must also obey the continuity equation – basically, for the energy to change in a given volume of space that energy must come in or go out of the boundary of that volume and similarly for the momentum. So a warp drive can have the warp condition for one moment in time. But after that, the stress-energy that maintains the warp will start to move according to the forces on it and according to the natural paths that it takes through the warped space-time. There is no guarantee that the configuration of the stress-energy at later times will be sufficient to maintain the warp.
<blockquote>
From the perspective of simulating the warp drive dynamically, the key challenge is stability. One can straightforwardly generate the initial metric that describes the Alcubierre metric and infer the instantaneous matter configuration that must support it using the Einstein Equation, but a time-domain evolution of the coupled matter-gravity system necessitates that we specify the equation of state of the matter, which in turn determines how it evolves. In a perfect fluid, this involves specifying the relation between the fluid pressure and density, but in a more general case, the full relation between the
stress-energy components must be given. There is (to our knowledge) no known equation of state that would maintain the warp drive metric in a stable configuration over time - therefore, whilst one can require that initially, the warp bubble is constant, it will quickly evolve away from that state and, in most cases, the warp fluid and spacetime deformations will disperse or collapse into a central point.
 
– Clough, Dietrich, and Khan<ref>Katy Clough, Tim Dietrich, and Sebastian Khan, "What no one has seen before: gravitational waveforms from warp drive collapse", arXiv:2406.02466 [gr-qc] https://doi.org/10.48550/arXiv.2406.02466</ref>
</blockquote>
<li>In order to get the matter (or more generally, stress-energy) that forms the warp shell up to speeds that could allow the shell to exist, you need to be able to, well, accelerate that matter to warp speeds. Which gives a chicken-and-egg problem - in order to achieve superluminal travel you need to be able to get matter or energy moving superluminally in the first place<ref name="Bobrick and Martire">Alexey Bobrick and Gianni Martire, "Introducing Physical Warp Drives", arXiv:2102.06824 [gr-qc] https://doi.org/10.48550/arXiv.2102.06824</ref><ref>D. H. Coule, "No warp drive", Class. Quantum Grav. 15 (1998) 2523–2527. https://doi.org/10.1088/0264-9381/15/8/026</ref>, which rather makes warp drives unnecessary from the get-go.
<li>A warp bubble moving faster than light will not be able to get any signals in front of it as it will overtake its own signal. In fact, anyone in the interior of the bubble would not be able to send signals to the front surface of the bubble. In science-speak, a horizon forms at the front of the bubble<ref name="Natario">José Natário, "Warp drive with zero expansion", Classical and Quantum Gravity. 19 (6): 1157–1166 (2002). arXiv:[https://arxiv.org/abs/gr-qc/0110086 gr-qc/0110086]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2002CQGra..19.1157N 2002CQGra..19.1157N]. doi:[https://doi.org/10.1088%2F0264-9381%2F19%2F6%2F308 10.1088/0264-9381/19/6/308]. S2CID [https://api.semanticscholar.org/CorpusID:15859984 15859984].</ref>. As a consequence, the bubble cannot be controlled, steered, nor stopped from within the bubble itself.
<li>A Horizon will naturally emit Hawking radiation. It is estimated that the radiation will raise the temperature of the interior of the bubble to approximately 1032 K<ref name="Finazzi 2009">Stefano Finazzi, Stefano Liberati, and Carlos Barceló, "Semiclassical instability of dynamical warp drives", Phys. Rev. D 79, 124017 (12 June, 2009) https://doi.org/10.1103/PhysRevD.79.124017 , arXiv:0904.0141 [gr-qc] https://doi.org/10.48550/arXiv.0904.0141</ref> (at this level of accuracy, the result is the same if you replace temperatures in K by the same number in °C or °F). This temperature, however, may be lower if quantum inequalities do not have to be fulfilled.
<li>Semi-classical instabilities at the horizon are expected to lead to a divergence of the stress-energy tensor at the boundary in a short period of time, thus leading to the destruction of the warp bubble<ref name="Finazzi 2009"></ref>.
</ol>
Helpfully, Van Den Broeck<ref name="VanDenBroeck_2000">C. Van Den Broeck, "Alcubierre’s Warp Drive: Problems and Prospects" AIP Conference Proceedings. 504: 1105–1110 (2000). Bibcode:[https://ui.adsabs.harvard.edu/abs/2000AIPC..504.1105V 2000AIPC..504.1105V]. doi:[https://doi.org/10.1063%2F1.1290913 10.1063/1.1290913].</ref> suggested several solutions – or at least mitigations – for some of these problems
<ol>
<li>Quantum inequalities had not been shown to be true in general for highly curved space-times.
<li>A proposed warp drive geometry<ref name="VanDenBroeck_1999">C. Van Den Broeck, "A 'warp drive' with more reasonable total energy requirements". Classical and Quantum Gravity. 16 (12): 3973–3979 (1999). arXiv:[https://arxiv.org/abs/gr-qc/9905084 gr-qc/9905084]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1999CQGra..16.3973V 1999CQGra..16.3973V]. doi:[https://doi.org/10.1088%2F0264-9381%2F16%2F12%2F314 10.1088/0264-9381/16/12/314]. S2CID [https://api.semanticscholar.org/CorpusID:15466313 15466313]. </ref> (see the van Den Broeck drive, below) would be able to minimize the magnitude of the energy required to about that of the mass of our sun. While still large, it is not unphysically large.
<li>Time travel is always going to be a worry with faster than light travel. There's no neat solution to this one.
<li>It may be possible for the negative energy outside of the superluminal region to compress into the superluminal region to form a shock in space-time. The front surface of the bubble would then be a singularity. It is not clear if such a sharp jump in physical properties is possible, but neither is sure that it is impossible, either. However, while this might help with the negative energy in front of the bubble getting swept up because it cannot move faster than light, it does little to help with the material at the back of the bubble on the outside getting left behind because it cannot keep up.
</ol>
Other mitigating research has also shown
<ol start=9>
<li>Semi-classical divergence only occurs in warp drives with only one spatial dimensions. Those with two or more spatial dimensions (as in our universe) only contain points where divergences occur, and these divergences can be managed. In fact, their effects can be reduced by making the warp shape more elongated<ref>Carlos Barceló, Valentin Boyanov, Luis J. Garay, Eduardo Martín-Martínez,
and Jose M. Sánchez Velázquez, "Warp drive aerodynamics", JHEP08 (2022) 288 https://doi.org/10.1007/JHEP08(2022)288 , arXiv:2207.06458 [gr-qc] https://doi.org/10.48550/arXiv.2207.06458</ref>.
</ol>

=== Alcubierre warp interactions with light and matter ===

Space is not entirely empty. It is filled with a diffuse plasma in the form of the interstellar medium, as well as cosmic radiation, light from stars, and cosmic microwave background radiation. A warp drive propagating through space will encounter this stuff. What happens when this matter and radiation have a warp bubble pass across them?

The first analysis of matter encountering a warp bubble was performed by Pfenning and Ford<ref name="PfenningFord_2001"></ref>. They looked at the warp bubble interaction with a massive object at rest with the frame of reference of the warp drive (keep in mind that the rest frame is the same inside and outside the warp bubble; in this rest frame objects inside the bubble have no momentum even though they are, in some sense, changing location rapidly with time). When the warp bubble passes, the object experiences an acceleration in the direction of the warp bubble motion. When the warp shell passes and the object is inside the bubble, it will be moving with approximately the speed of the bubble. Pfenning and Ford analyzed this problem with a continuous distribution of shell energy that, strictly speaking, never falls to zero except at the bubble center and at spatial infinity, so unless the object passes through the center in Pfenning and Ford's description it will never quite get up to the bubble's speed. In this case, the object will move almost as fast as the object but will pass through the bubble in a finite time, after which it will again be at rest with respect to the reference frame but displaced along the direction of the warp bubble motion by some distance. With this description of the warp bubble, a spacecraft of finite size will always be moving a little slower than the warp bubble and would have to use rockets to keep up with it. in addition, the spacecraft would experience tidal forces that would cause stress on the spacecraft's structure.

Pfenning and Ford also examined cases where the shell is of a finite (possibly infinitesimal) width and falls to zero both inside and outside the bubble. In this case any matter encountering the bubble would be collected at the bow of the bubble and thereafter move along with it.

McMonigal et al. analyzed the situation for both massive particles and light moving along the axis of travel of the bubble<ref name="McMonigalEtAl_2012">B. McMonigal, G. F. Lewis, and P. O'Byrne, "Alcubierre warp drive: On the matter of matter". Physical Review D. 85 (6) 064024 (20 March 2012). arXiv:[https://arxiv.org/abs/1202.5708 1202.5708]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2012PhRvD..85f4024M 2012PhRvD..85f4024M]. doi:[https://doi.org/10.1103%2FPhysRevD.85.064024 10.1103/PhysRevD.85.064024]. S2CID [https://api.semanticscholar.org/CorpusID:3993148 3993148].</ref>. They found that light moving opposite the warp direction passed through the bubble without incident, being only somewhat delayed by passing through the bubble. A warp bubble that is warping at sub-luminal speeds can have light catch up from behind it. This light is able to pass through the bubble, and is somewhat advanced in its path by the speed of the bubble. For super-luminal warp bubbles, however, the situation is different. The bubble will catch up to light moving it its own direction that is originally in front of it. This light cannot escape forward, the bubble being too fast. Nor can it escape backward, as the light is propagating forward and the interior of the bubble is at rest. Thus, the light gets caught at the bow of the warp bubble, unable to escape for so long as the warp bubble is active. This light is strongly blue shifted to extremely energetic x-rays and gamma rays. The space behind the bubble is swept clear of forward-moving light.

The situation for matter over-run by a super-luminal warp bubble is similar. Matter moving backward passes through the bubble, being only swept a ways along the bubble's path as sign of its passing but otherwise continuing on their way. Matter at rest behaves the same way as Pfenning and Ford discovered. Meanwhile, matter moving in the same direction of the warp is overtaken and collects at the bow of the bubble, unable to leave for so long as the bubble is warping. This matter "surfing" on the bubble bow is highly accelerated to relativistic speeds, experiencing extreme time dilation.

For a sub-luminal bubble, there is a speed faster than the warp speed where particles coming up from behind overtake the bubble and pass through, out the front. Particles below this speed but still faster than the warp speed still overtake the bubble, but then bounce backwards out of the bubble from behind. They are still moving forward, but are now moving more slowly than the warp speed. Particles moving forward but slower than the bubble will be overtaken and then bounced out the front with a speed higher than the warp speed. Those particles moving backward will pass through the bubble and exit with their initial speed.

The spacecraft inside the bubble will observe that light moving backward compared to the warp direction is blue-shifted, and matter moving backward passes through with increased energy. Meanwhile, a sub-luminal bubble will see light that was moving forward as red-shifted and matter catching up to the bubble from behind will pass by with reduced energy (superluminal bubbles, of course, do not have any matter or radiation catching up to them from behind).

When a superluminal warp bubble that has been collecting matter and radiation for the duration of its journey and blue-shifting it to much higher energies is turned off, all that matter and energy is released as a blast of radiation in the direction the bubble was warping.

McMonigal et al. conclude by noting that any spacecraft in the warp bubble would need shielding to protect against the blue-shifted radiation and matter of increased energy. In addition the destination would be "blasted into oblivion" by the release of matter and radiation that had been caught in the bubble during the trip.

Clark et al.<ref>Chad Clark, William A. Hiscock, and Shane L. Larson, "Null geodesics in the Alcubierre warp drive spacetime: the view from the bridge", arXiv:gr-qc/9907019 (2018) https://doi.org/10.48550/arXiv.gr-qc/9907019</ref> examine what it would look like from within an operating warp drive. They find that light moving at 90 degrees to the direction of motion will look like it is still coming from 90 degrees. However, light from the front hemisphere is squished progressively toward the forward point the faster you go. For a warp bubble moving faster than the speed of light, there is a cone behind the warp bubble where no light can reach it. However, you don't see a black spot behind you – light from the regions in the backward hemisphere from outside that cone that can reach you is distorted toward the direction behind you to fill in that spot. Light from the forward direction is blue-shifted and light from behind is red-shifted. At speeds of around 200 times light speed, the cosmic microwave background would be blue shifted to energies similar to that of the solar photosphere, heating anything in the warp bubble as if they were close to the sun.

== Other warp geometries ==

The Alcubierre warp drive is the first and best known of the warp drives. It has been around long enough that there have been many studies done on it. The warp geometry is particularly simple, making analysis easier. But in the time that the Alcubierre warp was proposed, many other researchers have come up with other warp designs, and even generalized the idea to include a wide array of additional possibilities.

Most of the general statements about the challenges of the Alcubierre drive hold for these other drives. Many of the details about interactions with matter are also likely to hold, although these other styles of warp drive have not yet been subject to such rigorous investigation.

=== Van Den Broeck warp drive ===

If an Alcubierre warp bubble a hundred meters across requires a magnitude of energy greater than the entire energy in the observable universe, one option to reduce the magnitude of energy used is to make the warp bubble smaller. Much smaller. Pfenning and Ford<ref name="PfenningFord_2001"></ref> suggested making the warp bubble smaller than an atom. Of course, that brings up the problem of how to stuff a spacecraft in there. Van Den Broeck proposed a solution<ref name="VanDenBroeck_1999"></ref>: make the warp bubble only about the size of an atomic nucleus but expand the space inside the bubble enormously so that a spacecraft could fit in. This is somewhat like a nuclear sized wormhole that leads to a pocket universe that holds the spacecraft. Unfortunately, the metric doesn't fit neatly into any [[Wormholes#Three_dimensions|embedding diagram]] that I can figure out, but the math works out so that spatial coordinates inside the bubble are expanded by a factor of 1017 and a spacecraft would have a few hundred meters of bubble interior to putz around in. This trick manages to reduce the magnitude of energy to only about the mass energy of a few stars similar to our own. Other than that, it is otherwise a normal Alcubierre drive.

Further work<ref>S. Krasnikov, "The quantum inequalities do not forbid spacetime shortcuts", Physical Review D. 67 (10) 104013. https://doi.org/10.1103%2FPhysRevD.67.104013, arXiv:gr-qc/0207057 (2003) https://doi.org/10.48550/arXiv.gr-qc/0207057 </ref> has managed to use Van Den Broeck's trick to reduce the magnitude of the negative energy to only about a milligram.

=== Natário warp drive ===

The Alcubierre drive is not the only way to construct a warp drive. José Natário<ref name="Natario"></ref> showed that it was just one example of an entire class of warp drives. These are space-time geometries with a smooth and localized shift vector field and a unit lapse function.

Natário then went on to find in that class a set of warp drives that do not have any expansion or contraction of space at all. Instead, space encountering the front boundary of the warp bubble instead "slides" around the outside of the bubble until it gets to the corresponding place on the back and is left behind there. Another way to think of it is that space entering the bubble shell is compressed in the radial direction of the shell but is simultaneously expanded in the tangential direction so that there is no net change in volume. When it gets to the back, the opposite occurs. In the language of mathematics, the shift vector field is divergenceless.

<table align=center>
<tr><td width=360>[[File:Natario_drive_flow_of_space.png]]
<td width=445>[[File:Natario_warp_drive_horizons.png]]
<tr><td style="vertical-align: top;">A representation of the motion of space in a zero-expansion Natário drive with a vanishingly thin warp shell around the bubble.
<td>The horizons of the generalized Natário class of warp drives (including the Alcubierre warp drive and all fell-Heisenberg drives) at superluminal speed in the direction of the green arrow. No event inside the bubble can affect events outside the bubble in front of the blue line. No events outside the bubble behind the red line can affect events inside the bubble. The lines have an angle α with respect to the direction of motion with sin(α) = 1/vs for vs the speed of the bubble relative to light speed. This is analogous to the Mach cone of supersonic objects.
</table>

NOTE: There are two different things that can be meant by the term "Natário warp drive". The generalized Natário warp drive is the general theory of all warp drives that includes within it the Alcubierre, Fell-Heisenberg, and Lentz warp drives, among infinitely many others. The zero-expansion Natário warp drive is the specific implementation of the generalized Natário warp drive that has a divergenceless shift vector field.

The generalized class of warp drives developed by Natário (including both the Alcubierre and this zero-expansion warp drive) are shown to always have regions in the warp shell where the energy density is negative to at least some observers. You can't get away from it – to warp, you need negative energy density.

The Natário class of warp drives blueshift light coming in from the front and redshift light catching up from the back. For the case of a super-luminal warp bubble, there will of course be a horizon at the back of the bubble and light will not be able to get through from behind. For a bubble speed vs relative to the speed of light, light coming from straight ahead will be blueshifted up in frequency and photon energy by a factor of 1 + vs. For sub-luminal travel, light coming from behind will be redshifted by a factor 1 - vs. In general for n as the unit direction along which the light is propagating, the blueshift factor will be 1 + vs ⋅ n. If the warp shell is infinitesimally thin, light from outside that reaches the center of the bubble will not be distorted in direction although it will be frequency shifted, but observers still inside the bubble but displaced from the center will see the field of view distorted as well as frequency shifted.

Most of the difficulties that apply to the Alcubierre warp drive also apply to the generalized Natário warp drive (and thus to the zero-expansion Natário drive and Fell-Heisenberg drive as well). In particular, quantum energy inequalities, high magnitudes of the energy required, formation of time machines, instabilities related to the stress-energy making up the warp shell being unable to keep up with the bubble, the chicken-and-egg problem of needing FTL to achieve FTL, the formation of horizons making warp bubbles uncontrollable, and Hawking radiation at the horizons, are all still concerns for generic warp drives.

=== Fell-Heisenberg warp drives ===

S. D. B. Fell and L. Heisenberg investigated what sort of warp structure is necessary to produce a warp drive with positive energy density everywhere<ref name="Fell-Heisenberg">S. D. B. Fell and L> Heisenberg, "Positive energy warp drive from hidden geometric structures", Classical and Quantum Gravity 38 155020 (2021) https://doi.org/10.1088/1361-6382/ac0e47 https://arxiv.org/abs/2104.06488</ref>. They found that warp drives of the generic Natário form with irrotational shift vectors had a simple expression for the energy density and vanishing momentum (irrotational means that the shift vector can be described as the gradient of a scalar potential field or, equivalently, that the curl of the shift vector field vanishes everywhere). Fell and Heisenberg proposed several potential fields that either had positive energy density everywhere except at discontinuities, or that were continuous but had regions that violated the weak energy condition (i. e. negative energy densities in some frames of reference). One specific example they provided of a warp field had an energy that is about 10,000 times less than the mass-energy of our sun. Or only about half the mass-energy of Jupiter. This is a significant improvement over many other proposed drives.

Despite the introduction discussing warp drive configurations that satisfy the various energy conditions, the configurations described in the paper are shown to locally violate the weak and strong energy conditions. Nonetheless, the energy density is still mostly positive. Santiago, Schuster, and Visser's<ref name="Santiago Schuster Visser 2022">J. Santiago, S. Schuster, and M. Visser, "Generic warp drives violate the null energy condition", Physical Review D 105, 064038 (2022) https://doi.org/10.1103/PhysRevD.105.064038</ref> work shows that the various energy conditions must still be violated by this warp drive; to not violate these, energy density must be positive in all reference frames not just those of the co-moving observer.

Any vector field can be described by its divergence, its curl, and a constant. The constant part of a warp drive's shift vector can be discarded because then the warp drive would not be localized. It is interesting, therefore, that the Natário zero-expansion drive has zero divergence and can be described only by its curl, while the Fell-Heisenberg drive has zero curl and can be described only by its divergence. These two classes of drives thus rely on opposite choices of setting one of the two different parts describing a vector field to zero. As might be expected, the Fell-Heisenberg drive is not zero expansion – space-time will change in volume as it passes through the warp bubble (although after going all the way through it will return to its original volume). However, objects in space-time will retain their original orientation as they pass through the warp bubble.

Any vector field with zero curl can be represented as the gradient of a suitable scalar function. Thus, the Fell-Heisenberg drive can be completely described by just scalar field (one number at all points of space and time) rather than a vector field.

Previous examples of warp drives, such as the Alcubierre and zero-expansion Natário drives, tend toward a flat space-time as you go far enough away faster than would occur if there were any mass present in the warp drive. That is, the ADM mass of such drives is zero. However, one of the Fell-Heisenberg examples described a spacetime where the geometry approaches that of what you would get for a massive object at the warp location, giving one of the first examples of a warp drive with mass.

In the Fell-Heisenberg drive the momentum vanishes everywhere. Yet the energy occupies regions where the shift vector is varying rapidly. The lack of momentum means that the energy will not move to keep up with the differential expansion and movement of the spacetime. As a consequence, at later times the energy will have a different distribution than what is necessary to maintain the given warp configuration. The exact time evolution is not solved but this likely leads to collapse of warp bubble. Note that this is not a problem unique to the Fell-Heisenberg drive. Assuming a geometry that describes a warp produces a specific distribution of the stress-energy tensor that satisfies Einstein's equation. But there is no guarantee that this stress energy tensor satisfies the continuity equation, which describes how the distribution of matter and momentum change with time. The issue of stability of any warp drive is not well understood, but most warp drive geometries will be unstable in the sense that their matter that generates the warp fields will not be able to keep up with the motion of the warp. The Fell-Heisenberg drive is unique only in that, with vanishing momentum, there is no configuration that allows stable propagation.

(n. b. The Heisenberg here is Lavinia Heisenberg, not the Werner Heisenberg of quantum physics and uncertainty principle fame.)

=== Lentz warp drive ===

E. Lentz<ref name="Lentz">E. W. Lentz, "Breaking the warp barrier: hyper-fast solitons in Einstein–Maxwell-plasma theory", Classical and Quantum Gravity. 38 075015 (2021). arXiv:[https://arxiv.org/abs/2006.07125 2006.07125]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2021CQGra..38g5015L 2021CQGra..38g5015L]. doi:[https://doi.org/10.1088%2F1361-6382%2Fabe692 10.1088/1361-6382/abe692]. ISSN [https://search.worldcat.org/issn/0264-9381 0264-9381]. S2CID [https://api.semanticscholar.org/CorpusID:219635854 219635854].</ref> investigated a particular configuration of the Fell-Heisenberg class of warp drives, where the scalar field forms a selection of rhomboid regions, resulting in an energy density that is everywhere positive and a shift vector with vanishing tidal force at the center for a comfortable ride.

However, Santiago, Schuster, and Visser<ref name="Santiago Schuster Visser 2022"></ref> dispute claims that the Lentz drive satisfies the energy conditions, noting that everywhere positive energy density in one frame of reference is insufficient to establish that the energy density is positive in all reference frames; knowledge of the Cauchy stress tensor is also needed. They show that any generic warp drive will violate the strong energy condition, null energy condition, and weak energy condition.

== Conservation laws ==

=== General introduction ===

All the warp drives investigated so far have geometries that quickly fall off from the highly curved region near the warp bubble to a region of spacetime that is almost flat; and becomes even more flat the farther away from the warp bubble you get. There is a word for geometries of this kind – they are called asymptotically flat. Basically, you can always go far enough away in any direction to reach a place where the geometry is sufficiently flat to meet any flatness criterion you choose.

When you have an asymptotically flat geometry, then in the far away mostly flat regions gravity is well described by a linearized theory of gravity – where the perturbations to the geometry are linear in the stress energy tensor. This means that gravity is basically Newtonian, along with a few other bells and whistles like gravitational waves and frame dragging.

When linearized gravity is a good approximation, then it is known that energy, momentum, and linear momentum are conserved. Although parts of a warp geometry are highly curved and non-linear, you can measure the conserved energy and momentum and angular momentum from far away in the linear region. This ADM energy and etcetera is the quantity that is conserved when you are worrying about conserving such things, not necessarily the energy and so forth you get in the highly curved spacetime region. In fact, it is not even simple to define where the energy and such actually is withing the highly curved spacetime region; it is non-local and different observers will disagree on how it is distributed. But everyone agrees on the ADM conserved quantities from out in nearly flat spacetime (up to the usual changes due to changing speeds, which are trivial and come up with normal objects as well).

And now we run into a bit of a conundrum: most of the warp geometries so far proposed have zero ADM mass<ref>Sebastian Schuster, Jessica Santiago,and Matt Visser, "ADM mass in warp drive spacetimes" General Relativity and Gravitation (2023) 55:14 https://doi.org/10.1007/s10714-022-03061-9</ref>, meaning that things like the Alcubierre drive technically have no energy or momentum or angular momentum at all. Despite all of the extremes of negative energy regions within the wall of the warp bubble, the actual energy that you need to spend to make one is zero. This can lead to a bit of a problem. If you have a spacecraft that has mass, that mass-energy has to be conserved. So when you turn on your warp drive and the total warp mass is zero, where does that energy go?

One resolution is just that the mass of the spacecraft is negligible and the small amount of curvature it causes can be ignored. In this case, the warp geometries discussed so far are merely an approximation that neglects this effect. In fact, some of the proposed Fell-Heisenberg drives map not onto entirely flat spacetime geometries at large distances, but onto the Schwarzschild geometry that is appropriate for systems with mass at the center. If this is the case, then the warping bubble and spacecraft system will need to worry about conservation of all their conserved properties.

=== All the science, demonstration of the problem ===

Let's look at what the conservation laws mean for a spacecraft equipped with a warp drive, assuming that no matter or radiation are emitted by activating or deactivating the warp drive.

We can start with an observer floating in space. We'll call her Alice. And, approaching Alice at 100 m/s is the starship Enterprise with a mass of a cool 200,000 tons (2×108 kg). from this we know that the Enterprise has, in Alice's frame of reference, a kinetic energy of 1 TJ, a total energy (including mass and kinetic energy) of 1.8×1025 J, a momentum of 2×1010 kg m/s, and – because it is headed straight toward Alice – an angular momentum of zero. All of this is without any warp drive activated.

On the bridge of the Enterprise, Spock picks up Alice on the sensors and tells Captain Kirk. Kirk does not want to run into Alice (he'd rather date her), so he orders helm to turn on the Enterprise's warp drive and move them 1 km up so that their course takes them 1 km from Alice instead of straight into her. Sulu warps the Enterprise according to these commands, and then turns off the warp drive.

After this maneuver, keeping the Enterprise's total energy and momentum the same means that the Enterprise still has a mass of 200,000 tons and a velocity of 100 m/s in its original direction. So we can see that the warp drive must preserve the velocity vector of the thing in it (at least in flat spacetime where outside forces are not acting on it). The problem comes from angular momentum. With a known mass of 200,000 tons, a speed of 100 m/s, and a distance of closest approach of 1 km; from Alice's frame of reference the Enterprise now has an angular momentum of 2×1013 kg m2/s after the warp maneuver, while before the maneuver it has an angular momentum of zero.

From this we can see that warp drives cannot be reactionless. Arbitrary displacements using warp without emitting propellant result in violation of the law of conservation of angular momentum. As long as we restrict ourselves to accepted science based on what we know of general relativity, to move a warp drive with mass you need to accelerate it or decelerate it with rockets or other normal means of delivering momentum (such as light sails, mag sails, or pushing off of a planet). Warp drives must be inertial to comply with physics. And all of this was analyzed only from times when the warp drive was turned off; it does not depend on any assumptions about what happens when the warp is on.

=== A zero mass warp "solution" ===

So let's start picking at our assumptions. First, what if the warp bubble actually does have zero mass? Now, while in warp, there is no momentum or angular momentum or energy to be conserved. A spacecraft in warp can bounce around like a ferret on stimulants and the universe doesn't care.

But there's a problem, and it's probably pretty obvious. Our Starship Enterprise has a lot of mass when it hasn't activated its warp bubble. So, 200,000 tons × c2 = 1.8×1025 J of energy before the warp is turned on, and 0 J of energy afterward. This seems like a flagrant violation of conservation of energy.

Well, there's a solution. Maybe turning on the warp drive somehow dumps all that energy. You could get a pulse of gravitational and electromagnetic radiation rippling away from the turn-on point at the speed of light, maybe with various kinds of particle radiation thrown in. Or maybe a 200,000 ton black hole is pooped out behind the Enterprise when the warp bubble is turned on. Anything that dumps the energy and leaves it behind.

Okay, so, the good ship Enterprise zips off to Starbase 9 with in a zero mass warp bubble. But now they're in a bit of a pickle. In order to come out of warp so they can visit the starbase, they somehow need to acquire 200,000 tons worth of mass-energy from somewhere.

One option is that the Enterprise rams its warp bubble into some random asteroid and grabs 200,000 tons of it, or skims 200,000 tons of gas from the top of the atmosphere of a nearby world, or maybe the starbase keeps around millions of tons of ballast to give to visiting starships so they can de-warp. It's a perfectly reasonable way to go about conserving all the conserved things the universe seems to care about.

But their is an even weirder option. It's pretty speculative &ndash all of this is speculative, but this is even more so – but it is fun so it's worth mentioning. (more soon ...)

(Zero mass warp drives, relaxing conservation requirements, and activation/deactivation)

(If angular momentum conservation is ignored, discuss how momentum & energy are affected by outside forces. In a gravitational field equivalent to inertial frame ... like being in an accelerated elevator; will acquire momentum buildup while staying "at rest".)

== Warp Railroads ==

Van Den Broeck<ref name="VanDenBroeck_2000"></ref> suggested a warp railroad,

You can set up devices ahead of time along the path of the bubble that produce the negative energy regions for the warp bubble at the appropriate time without needing the negative energy to ever move with the bubble at all. At least for bubbles on a predictable schedule and route.

helps with:

-stability (elaborate)

-conservation laws (elaborate)

Mention Krasnikov<ref name="Krasnikov 1998">S. V. Krasnikov, "Hyperfast travel in general relativity" Phys. Rev. D 57 4760</ref>

Hyperwave<ref name="Pieri 2024">Lorenzo Pieri, "Hyperwave: hyper-Fast Communication within General Relativity", arXiv:2311.12069 [gr-qc] https://doi.org/10.48550/arXiv.2311.12069</ref> makes use of this warp railroad idea.

== Quantum effects ==

(quantum stuff here<ref name="Hiscock1997">W. A. Hiscock, "Quantum effects in the Alcubierre warp drive spacetime", Classical and Quantum Gravity 14 L183 https://doi.org/10.1088/0264-9381/14/11/002 https://arxiv.org/abs/gr-qc/9707024</ref><ref name="Finazzi et al 2009">S. Finazzi, S. Liberati, C. Barceló, "Semiclassical instability of dynamical warp drives", Physical Review D 79, 124017 (2009)https://doi.org/10.1103/PhysRevD.79.124017 https://arxiv.org/abs/0904.0141</ref>)

From<ref name="Pieri 2024"></ref>
<blockquote>
It should be emphasised that Quantum Inequalities have not been proven for realistic interacting quantum fields, where the they are expected to be weaker if they exist [Cadamuro, 2019], and they often require Minkowskian conditions at infinity, a condition which is not appropriate for fields inside bounded regions, as in a Casimir devices.
</blockquote>

== Warp drives and black holes ==

What happens if you drive your warp drive into a black hole?

The pop-sci reason that nothing can escape a black hole is that at the event horizon the escape velocity is faster than light and nothing can go faster than light. A warp drive can go faster than light.

The more detailed reason from general relativity is that at the event horizon space and time are rotated sufficiently that "inward" becomes "forward in time." You can no more go farther away from the singularity in a black hole once you are inside the event horizon than you can go backward in time. Any super-luminal travel can go backward in time, and warp drives can engage in super-luminal travel.

The Hawking area theorem that says that black holes can only grow, never shrink relies on the condition that the energy is everywhere positive. Warp drives have negative energy density.

So in principle, there's nothing that prevents a warp drive from taking a dip into a black hole and coming back out again.

Garattini and Zatrimaylov looked into the problem of a warp drive near (and in) a black hole<ref name="Garattini and Zatrimaylov">R. Garattini and K. Zatrimaylov, "Black holes, warp drives, and energy conditions", Physics Letters B 856 138910 (2024) https://doi.org/10.1016/j.physletb.2024.138910</ref>. The found that the black hole reduces the negative energy requirement for the warp bubble to move inward toward the black hole singularity, but increases the negative energy needed to move away from the center of the black hole. In addition, a warp drive parked at the event horizon would allow light from inside the horizon to pass through the bubble and escape back outside.

== Credit ==
Author: Luke Campbell

== References ==

[[Category:Physics]][[Category:Infrastructure]][[Category:Physics & Engineering‏‎]][[Category:Physics & Math & Engineering]][[Category:Transportation & Infrastructure‏‎]][[Category:Metric Engineering]]

Warp Drives

2026-07-31T00:41:57Z

Lwcamp: /* Conservation laws */

{{PageConstructionNotice}}

Science fiction often features spacecraft that can seemingly move across space and get between the place of departure and the destination much faster than light could have done. This appears to contradict the theory of relativity, which predicts unequivocally that nothing can move through space faster than light. Because relativity has been incredibly successful at describing nature, with its many other predictions regularly being confirmed to extraordinary accuracy and within the bounds of uncertainty of all the experiments that tested them, it gives confidence that relativity is a correct description of reality. Which seems to rather throw a wet towel on our hopes for rapid travel between stars.

However, while relativity does not allow things to move through space faster than light, it places no such restrictions on how fast space-time itself can expand, contract, or move around. This leads to the idea of a warp drive – the spacecraft remains stationary within a region of highly curved space-time, and that region moves at super-luminal speeds rather than the spacecraft.

== The Alcubierre warp drive ==

The first warp drive geometry that satisfied the Einstein field equations of relativity was proposed by Miguel Alcubierre<ref>M. Alcubierre, "The warp drive: hyper-fast travel within general relativity." Classical and Quantum Gravity. 11 (5): L73–L77 (1994). [https://arxiv.org/abs/gr-qc/0009013 arXiv:gr-qc/0009013]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1994CQGra..11L..73A 1994CQGra..11L..73A]. doi:[https://doi.org/10.1088%2F0264-9381%2F11%2F5%2F001 10.1088/0264-9381/11/5/001]. S2CID [https://api.semanticscholar.org/CorpusID:4797900 4797900].</ref>. In this geometry, a sphere of space-time moves at an arbitrary speed (potentially but not necessarily a speed much faster than light). Objects within the sphere are moved along with the sphere; an object at rest within the sphere would be moved along with the sphere indefinitely. Space is expanding at the rear boundary and contracting at the front boundary in order to keep the sphere moving. In order to satisfy the Einstein field equations, the boundary of the sphere must have a negative energy density. The challenges of space-time geometries with negative energy densities are described in our page on [[Wormholes#Exotic_energy_conditions|wormholes]], for our purposes it is enough to note that negative energy densities can pose problems if not handled carefully, there may be limits on how much negative energy you can have without nearby positive energy density, and it may not be possible to get enough negative energy to support a warp drive; although none of this is ruled out by physics – yet!

<table align=center>
<tr><td width=650>[[File:Energy_magnitude_of_Alcubierre_drive.png]]
<td width=650>[[File:Expansion_of_Alcubierre_drive.png]]
<tr><td>The magnitude of the energy of the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. (The energy density is negative everywhere in the shell, here only the magnitude is shown to aid visual comprehension.)
<td>The expansion of space in the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. Negative expansion means that space is contracting in that direction, positive expansion that it is expanding.
</table>

NOTE: What the? The energy distribution is symmetric! How does the drive know which way to go? There must be significant contributions of other components of the stress-energy tensor, and those have got to be asymmetric along the forward/backward axis. Check this when I get time.

(And what's with the ring in all the artwork, anyway?)

=== So what is going on? ===

What is it that makes a warp drive a warp drive, and how does it work?

A warp drive is a concept that comes out of the science of general relativity. General relativity collects space and time together into a single "structure" called spacetime. This structure is curved, and its curvature is determined by the material that is in it. Absent forces acting between matter, particles of matter move through spacetime on the shortest possible paths in this curved environment.
<blockquote>
"Spacetime tells matter how to move; matter tells spacetime how to curve." - John Archibald Wheeler
</blockquote>
All of this is encompassed in the Einstein field equation:
<div align=center>
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 8πG
<td rowspan="2" nowrap="nowrap" align="right"> T = R -  
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right"> R g
<tr>
<td style="border-top:solid 1px black;"> c4
<td style="border-top:solid 1px black;"> 2
</table>
</div>
Unfortunately, by convention, both the symbols G and R get used for a lot of different things. Here, symbols in bold are second rank tensors – constructs of arrays of 4×4 numbers where 4 is necessary because that is the number of dimensions of spacetime. Conventionally, these numbers are indexed by a value going from 0 to 3, with 0 being the part corresponding to the coordinate of time and 1 to 3 being the Cartesian coordinates that you might know as x, y, and z.

T is the stress-energy tensor: the (0,0) component represents the energy density at a given point; the (0,i) and (i, 0) components (with i going from 1 to 3) represent the momentum density; and the (i,j) components represent the stresses present, such as pressure or tension or shear.

The other term, R - R g/2, involves a partial description of the curvature. A full description of the curvature requires a construct of 4×4×4×4 components; too much for the stress energy tensor to fully constrain. Which is okay, because this allows curvature to exist away from where matter is, allowing gravity to have long range effects across empty space as well as for phenomena like gravitational waves. But it is enough that a given distribution of matter T gives enough constraints (along with specifying how the geometry behaves far away) that you can generally figure out what the actual gravity is.

This is the usual story that we are told. But there is an alternate way of looking at the same geometry such that the math works out the same, but it is described conceptually in a very different way. This divides spacetime up into "layers", called foliations, that represent slices of space and going from one foliation to the next is increasing in time. This is useful for computation because you can assign coordinates to your system. There are two additional constructs – the lapse function and the shift vector field. Starting at an event on one foliation, the lapse function at that position tells you that after a certain "proper time" (time perceived by an observer at that event) the time coordinate of the observer will be the proper time times the lapse function at that event. And the new spatial coordinates of the observer will be the proper time times the shift vector at that event. (Strictly speaking, this is only exactly true in the limit of the proper time approaching zero or when the lapse function and shift vectors are constant. But hopefully this gives a good enough insight to what these entities are so that we can go on to describe warp drives.)

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Lapse_shift.png]]
<tr><td>A folliation of space-time into slices of constant time coordinates (grey). At an event (black circle), the lapse function (yellow) shows how far into the future the point at that event will be projected in a unit amount of proper time experienced by an observer initially at the event. The shift vector (red) shows how that point changes coordinates after the unit amount of proper time. The cyan vector shows the new coordinates the point is projected onto. The lower black surface indicates a foliation of spacetime at the same time coordinate; after unit proper time, the events on the lower foliation get projected to the curved upper black foliation according to the shift vectors and lapse functions at each of their respective points.
</table>
</table>

Warp drives are generally restricted to spacetime geometries with a constant lapse function - all observers go forward by the same time coordinate after experiencing the same amount of proper time. Thus, all warp drives have no time dilation.
Far from the warp bubble, the shift vector vanishes – each spatial coordinate is mapped to the same spatial coordinate after any amount of proper time so long as the warp drive is not near. But the inside of the warp bubble is an extended region where the shift vector field is non-zero and constant. In this bubble, all spatial coordinates are mapped to a coordinate that shifts linearly in time with the proper time of the observer. Astute readers will see that this linear change in position with time is very similar to a velocity. This region of constant shift vector then moves with the shift vector velocity to keep stuff originally in the bubble so that it stays in the bubble. In between the inside of the bubble and the far away regions with zero shift vector, the shift vector field falls away to match one region to the other.

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Warp_shift.png]]
<tr><td>The shift vector of a one-dimensional slice through warp geometry, in red. At the edges there is no shift. At the center (inside the warp bubble) there is a region of uniform shift so that at a later time the inside of the warp bubble is rigidly translated to different spatial coordinates (in blue). The two foliations remain the same distance apart, because warp drives maintain a unit lapse function.
</table>
</table>

Traditionally, scientists would start with an interesting distribution of matter given by T and then find out what kind of curvature it created. This works well for planets and stars and black holes and cosmic strings and other things which are observed in the universe and you want to see what sort of effect they had. But starting in the 1980's, some physicists started looking at Einstein's field equations the other way around. They start with a desired geometry, find the curvature part of the field equation R - R g/2, and use that to find out what T has to be in order to get that curvature. This was originally done for [[Wormholes|wormholes]], but in 1994 Miguel Alcubierre used this same trick to figure out what could create a region of constant shift vector that moved with the shift vector - the warp drive. With this demand for a warp geometry, the required distribution of matter and energy and momentum and stress can be determined.

The original Alcubierre shift vector N→ produces a spherical warp bubble with an axi-symmetric distribution of, well, everything along the direction of motion with warp velocity vector v→.
<div align="center">
N→ = v→ f(rs)
</div>
Here, rs is the distance from the center of the warp bubble, and f(rs) can be any function that is 1 near rs = 0 and is 0 for rs well beyond the radius of the warp bubble.

=== Challenges and possible resolutions ===

Almost as soon as Alcubierre proposed his warp drive, others began picking it apart.
<ol>
<li>Pfenning and Ford<ref name="PfenningFord_2001">M. J. Pfenning and L. H. Ford, "The unphysical nature of 'Warp Drive'", Classical and Quantum Gravity. 14 (7): 1743–1751 (1997). arXiv:[https://arxiv.org/abs/gr-qc/9702026 gr-qc/9702026]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1997CQGra..14.1743P 1997CQGra..14.1743P]. doi:[https://doi.org/10.1088%2F0264-9381%2F14%2F7%2F011 10.1088/0264-9381/14/7/011]. S2CID [https://api.semanticscholar.org/CorpusID:15279207 15279207].</ref> found that if the negative energy density wall around the bubble obeys [[Wormholes#Quantum_energy_inequalities|quantum energy inequalities]], then for warp speeds of around light speed the shell thickness would be on the order of a hundred Planck lengths; a distance far smaller than any other known physical phenomenon or object.
<li>In the same work, Pfenning and Ford also found that for a bubble a hundred meters in radius, a warp speed of around the speed of light, and the required thickness of about 100 Planck lengths the energy in the bubble shell would be E ≈ -1063 kg c2. The magnitude of this latter value is ten orders of magnitude larger than the energy of the entire visible universe. They do note, however, that if quantum energy inequalities can be ignored then a warp drive with a hundred meters radius but a shell thickness of one meter would "only" have an energy magnitude of about a quarter solar mass.
<li>As with any method of faster than light travel, the warp drive could be used to make a time machine.
<li>Perhaps most seriously, if the warp drive is going faster than light speed, the negative energy regions on the outside of the shell won't be in the warp parts of the bubble. They will have to be moving through space at faster than light speed if the bubble is to maintain its integrity, which is the very problem that the warp drive was designed to avoid. The outside of the warp bubble shell that maintains the warp bubble would fall away and the warp bubble would quickly erode to sub-luminal speeds.<ref name="Van Den Broeck 2000">Chris Van Den Broeck, "On the (im)possibility of warp bubbles", arXiv:gr-qc/9906050 https://doi.org/10.48550/arXiv.gr-qc/9906050</ref>
<li>A more general problem that includes the previous one is that the geometry of the warp drive determines the distribution of energy, momentum, and stress (pressure, shear, and tension) by Einstein's field equation, which states that the stress energy tensor is directly proportional to a reduced order curvature tensor of space-time. However, the stress-energy tensor must also obey the continuity equation – basically, for the energy to change in a given volume of space that energy must come in or go out of the boundary of that volume and similarly for the momentum. So a warp drive can have the warp condition for one moment in time. But after that, the stress-energy that maintains the warp will start to move according to the forces on it and according to the natural paths that it takes through the warped space-time. There is no guarantee that the configuration of the stress-energy at later times will be sufficient to maintain the warp.
<blockquote>
From the perspective of simulating the warp drive dynamically, the key challenge is stability. One can straightforwardly generate the initial metric that describes the Alcubierre metric and infer the instantaneous matter configuration that must support it using the Einstein Equation, but a time-domain evolution of the coupled matter-gravity system necessitates that we specify the equation of state of the matter, which in turn determines how it evolves. In a perfect fluid, this involves specifying the relation between the fluid pressure and density, but in a more general case, the full relation between the
stress-energy components must be given. There is (to our knowledge) no known equation of state that would maintain the warp drive metric in a stable configuration over time - therefore, whilst one can require that initially, the warp bubble is constant, it will quickly evolve away from that state and, in most cases, the warp fluid and spacetime deformations will disperse or collapse into a central point.
 
– Clough, Dietrich, and Khan<ref>Katy Clough, Tim Dietrich, and Sebastian Khan, "What no one has seen before: gravitational waveforms from warp drive collapse", arXiv:2406.02466 [gr-qc] https://doi.org/10.48550/arXiv.2406.02466</ref>
</blockquote>
<li>In order to get the matter (or more generally, stress-energy) that forms the warp shell up to speeds that could allow the shell to exist, you need to be able to, well, accelerate that matter to warp speeds. Which gives a chicken-and-egg problem - in order to achieve superluminal travel you need to be able to get matter or energy moving superluminally in the first place<ref name="Bobrick and Martire">Alexey Bobrick and Gianni Martire, "Introducing Physical Warp Drives", arXiv:2102.06824 [gr-qc] https://doi.org/10.48550/arXiv.2102.06824</ref><ref>D. H. Coule, "No warp drive", Class. Quantum Grav. 15 (1998) 2523–2527. https://doi.org/10.1088/0264-9381/15/8/026</ref>, which rather makes warp drives unnecessary from the get-go.
<li>A warp bubble moving faster than light will not be able to get any signals in front of it as it will overtake its own signal. In fact, anyone in the interior of the bubble would not be able to send signals to the front surface of the bubble. In science-speak, a horizon forms at the front of the bubble<ref name="Natario">José Natário, "Warp drive with zero expansion", Classical and Quantum Gravity. 19 (6): 1157–1166 (2002). arXiv:[https://arxiv.org/abs/gr-qc/0110086 gr-qc/0110086]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2002CQGra..19.1157N 2002CQGra..19.1157N]. doi:[https://doi.org/10.1088%2F0264-9381%2F19%2F6%2F308 10.1088/0264-9381/19/6/308]. S2CID [https://api.semanticscholar.org/CorpusID:15859984 15859984].</ref>. As a consequence, the bubble cannot be controlled, steered, nor stopped from within the bubble itself.
<li>A Horizon will naturally emit Hawking radiation. It is estimated that the radiation will raise the temperature of the interior of the bubble to approximately 1032 K<ref name="Finazzi 2009">Stefano Finazzi, Stefano Liberati, and Carlos Barceló, "Semiclassical instability of dynamical warp drives", Phys. Rev. D 79, 124017 (12 June, 2009) https://doi.org/10.1103/PhysRevD.79.124017 , arXiv:0904.0141 [gr-qc] https://doi.org/10.48550/arXiv.0904.0141</ref> (at this level of accuracy, the result is the same if you replace temperatures in K by the same number in °C or °F). This temperature, however, may be lower if quantum inequalities do not have to be fulfilled.
<li>Semi-classical instabilities at the horizon are expected to lead to a divergence of the stress-energy tensor at the boundary in a short period of time, thus leading to the destruction of the warp bubble<ref name="Finazzi 2009"></ref>.
</ol>
Helpfully, Van Den Broeck<ref name="VanDenBroeck_2000">C. Van Den Broeck, "Alcubierre’s Warp Drive: Problems and Prospects" AIP Conference Proceedings. 504: 1105–1110 (2000). Bibcode:[https://ui.adsabs.harvard.edu/abs/2000AIPC..504.1105V 2000AIPC..504.1105V]. doi:[https://doi.org/10.1063%2F1.1290913 10.1063/1.1290913].</ref> suggested several solutions – or at least mitigations – for some of these problems
<ol>
<li>Quantum inequalities had not been shown to be true in general for highly curved space-times.
<li>A proposed warp drive geometry<ref name="VanDenBroeck_1999">C. Van Den Broeck, "A 'warp drive' with more reasonable total energy requirements". Classical and Quantum Gravity. 16 (12): 3973–3979 (1999). arXiv:[https://arxiv.org/abs/gr-qc/9905084 gr-qc/9905084]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1999CQGra..16.3973V 1999CQGra..16.3973V]. doi:[https://doi.org/10.1088%2F0264-9381%2F16%2F12%2F314 10.1088/0264-9381/16/12/314]. S2CID [https://api.semanticscholar.org/CorpusID:15466313 15466313]. </ref> (see the van Den Broeck drive, below) would be able to minimize the magnitude of the energy required to about that of the mass of our sun. While still large, it is not unphysically large.
<li>Time travel is always going to be a worry with faster than light travel. There's no neat solution to this one.
<li>It may be possible for the negative energy outside of the superluminal region to compress into the superluminal region to form a shock in space-time. The front surface of the bubble would then be a singularity. It is not clear if such a sharp jump in physical properties is possible, but neither is sure that it is impossible, either. However, while this might help with the negative energy in front of the bubble getting swept up because it cannot move faster than light, it does little to help with the material at the back of the bubble on the outside getting left behind because it cannot keep up.
</ol>
Other mitigating research has also shown
<ol start=9>
<li>Semi-classical divergence only occurs in warp drives with only one spatial dimensions. Those with two or more spatial dimensions (as in our universe) only contain points where divergences occur, and these divergences can be managed. In fact, their effects can be reduced by making the warp shape more elongated<ref>Carlos Barceló, Valentin Boyanov, Luis J. Garay, Eduardo Martín-Martínez,
and Jose M. Sánchez Velázquez, "Warp drive aerodynamics", JHEP08 (2022) 288 https://doi.org/10.1007/JHEP08(2022)288 , arXiv:2207.06458 [gr-qc] https://doi.org/10.48550/arXiv.2207.06458</ref>.
</ol>

=== Alcubierre warp interactions with light and matter ===

Space is not entirely empty. It is filled with a diffuse plasma in the form of the interstellar medium, as well as cosmic radiation, light from stars, and cosmic microwave background radiation. A warp drive propagating through space will encounter this stuff. What happens when this matter and radiation have a warp bubble pass across them?

The first analysis of matter encountering a warp bubble was performed by Pfenning and Ford<ref name="PfenningFord_2001"></ref>. They looked at the warp bubble interaction with a massive object at rest with the frame of reference of the warp drive (keep in mind that the rest frame is the same inside and outside the warp bubble; in this rest frame objects inside the bubble have no momentum even though they are, in some sense, changing location rapidly with time). When the warp bubble passes, the object experiences an acceleration in the direction of the warp bubble motion. When the warp shell passes and the object is inside the bubble, it will be moving with approximately the speed of the bubble. Pfenning and Ford analyzed this problem with a continuous distribution of shell energy that, strictly speaking, never falls to zero except at the bubble center and at spatial infinity, so unless the object passes through the center in Pfenning and Ford's description it will never quite get up to the bubble's speed. In this case, the object will move almost as fast as the object but will pass through the bubble in a finite time, after which it will again be at rest with respect to the reference frame but displaced along the direction of the warp bubble motion by some distance. With this description of the warp bubble, a spacecraft of finite size will always be moving a little slower than the warp bubble and would have to use rockets to keep up with it. in addition, the spacecraft would experience tidal forces that would cause stress on the spacecraft's structure.

Pfenning and Ford also examined cases where the shell is of a finite (possibly infinitesimal) width and falls to zero both inside and outside the bubble. In this case any matter encountering the bubble would be collected at the bow of the bubble and thereafter move along with it.

McMonigal et al. analyzed the situation for both massive particles and light moving along the axis of travel of the bubble<ref name="McMonigalEtAl_2012">B. McMonigal, G. F. Lewis, and P. O'Byrne, "Alcubierre warp drive: On the matter of matter". Physical Review D. 85 (6) 064024 (20 March 2012). arXiv:[https://arxiv.org/abs/1202.5708 1202.5708]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2012PhRvD..85f4024M 2012PhRvD..85f4024M]. doi:[https://doi.org/10.1103%2FPhysRevD.85.064024 10.1103/PhysRevD.85.064024]. S2CID [https://api.semanticscholar.org/CorpusID:3993148 3993148].</ref>. They found that light moving opposite the warp direction passed through the bubble without incident, being only somewhat delayed by passing through the bubble. A warp bubble that is warping at sub-luminal speeds can have light catch up from behind it. This light is able to pass through the bubble, and is somewhat advanced in its path by the speed of the bubble. For super-luminal warp bubbles, however, the situation is different. The bubble will catch up to light moving it its own direction that is originally in front of it. This light cannot escape forward, the bubble being too fast. Nor can it escape backward, as the light is propagating forward and the interior of the bubble is at rest. Thus, the light gets caught at the bow of the warp bubble, unable to escape for so long as the warp bubble is active. This light is strongly blue shifted to extremely energetic x-rays and gamma rays. The space behind the bubble is swept clear of forward-moving light.

The situation for matter over-run by a super-luminal warp bubble is similar. Matter moving backward passes through the bubble, being only swept a ways along the bubble's path as sign of its passing but otherwise continuing on their way. Matter at rest behaves the same way as Pfenning and Ford discovered. Meanwhile, matter moving in the same direction of the warp is overtaken and collects at the bow of the bubble, unable to leave for so long as the bubble is warping. This matter "surfing" on the bubble bow is highly accelerated to relativistic speeds, experiencing extreme time dilation.

For a sub-luminal bubble, there is a speed faster than the warp speed where particles coming up from behind overtake the bubble and pass through, out the front. Particles below this speed but still faster than the warp speed still overtake the bubble, but then bounce backwards out of the bubble from behind. They are still moving forward, but are now moving more slowly than the warp speed. Particles moving forward but slower than the bubble will be overtaken and then bounced out the front with a speed higher than the warp speed. Those particles moving backward will pass through the bubble and exit with their initial speed.

The spacecraft inside the bubble will observe that light moving backward compared to the warp direction is blue-shifted, and matter moving backward passes through with increased energy. Meanwhile, a sub-luminal bubble will see light that was moving forward as red-shifted and matter catching up to the bubble from behind will pass by with reduced energy (superluminal bubbles, of course, do not have any matter or radiation catching up to them from behind).

When a superluminal warp bubble that has been collecting matter and radiation for the duration of its journey and blue-shifting it to much higher energies is turned off, all that matter and energy is released as a blast of radiation in the direction the bubble was warping.

McMonigal et al. conclude by noting that any spacecraft in the warp bubble would need shielding to protect against the blue-shifted radiation and matter of increased energy. In addition the destination would be "blasted into oblivion" by the release of matter and radiation that had been caught in the bubble during the trip.

Clark et al.<ref>Chad Clark, William A. Hiscock, and Shane L. Larson, "Null geodesics in the Alcubierre warp drive spacetime: the view from the bridge", arXiv:gr-qc/9907019 (2018) https://doi.org/10.48550/arXiv.gr-qc/9907019</ref> examine what it would look like from within an operating warp drive. They find that light moving at 90 degrees to the direction of motion will look like it is still coming from 90 degrees. However, light from the front hemisphere is squished progressively toward the forward point the faster you go. For a warp bubble moving faster than the speed of light, there is a cone behind the warp bubble where no light can reach it. However, you don't see a black spot behind you – light from the regions in the backward hemisphere from outside that cone that can reach you is distorted toward the direction behind you to fill in that spot. Light from the forward direction is blue-shifted and light from behind is red-shifted. At speeds of around 200 times light speed, the cosmic microwave background would be blue shifted to energies similar to that of the solar photosphere, heating anything in the warp bubble as if they were close to the sun.

== Other warp geometries ==

The Alcubierre warp drive is the first and best known of the warp drives. It has been around long enough that there have been many studies done on it. The warp geometry is particularly simple, making analysis easier. But in the time that the Alcubierre warp was proposed, many other researchers have come up with other warp designs, and even generalized the idea to include a wide array of additional possibilities.

Most of the general statements about the challenges of the Alcubierre drive hold for these other drives. Many of the details about interactions with matter are also likely to hold, although these other styles of warp drive have not yet been subject to such rigorous investigation.

=== Van Den Broeck warp drive ===

If an Alcubierre warp bubble a hundred meters across requires a magnitude of energy greater than the entire energy in the observable universe, one option to reduce the magnitude of energy used is to make the warp bubble smaller. Much smaller. Pfenning and Ford<ref name="PfenningFord_2001"></ref> suggested making the warp bubble smaller than an atom. Of course, that brings up the problem of how to stuff a spacecraft in there. Van Den Broeck proposed a solution<ref name="VanDenBroeck_1999"></ref>: make the warp bubble only about the size of an atomic nucleus but expand the space inside the bubble enormously so that a spacecraft could fit in. This is somewhat like a nuclear sized wormhole that leads to a pocket universe that holds the spacecraft. Unfortunately, the metric doesn't fit neatly into any [[Wormholes#Three_dimensions|embedding diagram]] that I can figure out, but the math works out so that spatial coordinates inside the bubble are expanded by a factor of 1017 and a spacecraft would have a few hundred meters of bubble interior to putz around in. This trick manages to reduce the magnitude of energy to only about the mass energy of a few stars similar to our own. Other than that, it is otherwise a normal Alcubierre drive.

Further work<ref>S. Krasnikov, "The quantum inequalities do not forbid spacetime shortcuts", Physical Review D. 67 (10) 104013. https://doi.org/10.1103%2FPhysRevD.67.104013, arXiv:gr-qc/0207057 (2003) https://doi.org/10.48550/arXiv.gr-qc/0207057 </ref> has managed to use Van Den Broeck's trick to reduce the magnitude of the negative energy to only about a milligram.

=== Natário warp drive ===

The Alcubierre drive is not the only way to construct a warp drive. José Natário<ref name="Natario"></ref> showed that it was just one example of an entire class of warp drives. These are space-time geometries with a smooth and localized shift vector field and a unit lapse function.

Natário then went on to find in that class a set of warp drives that do not have any expansion or contraction of space at all. Instead, space encountering the front boundary of the warp bubble instead "slides" around the outside of the bubble until it gets to the corresponding place on the back and is left behind there. Another way to think of it is that space entering the bubble shell is compressed in the radial direction of the shell but is simultaneously expanded in the tangential direction so that there is no net change in volume. When it gets to the back, the opposite occurs. In the language of mathematics, the shift vector field is divergenceless.

<table align=center>
<tr><td width=360>[[File:Natario_drive_flow_of_space.png]]
<td width=445>[[File:Natario_warp_drive_horizons.png]]
<tr><td style="vertical-align: top;">A representation of the motion of space in a zero-expansion Natário drive with a vanishingly thin warp shell around the bubble.
<td>The horizons of the generalized Natário class of warp drives (including the Alcubierre warp drive and all fell-Heisenberg drives) at superluminal speed in the direction of the green arrow. No event inside the bubble can affect events outside the bubble in front of the blue line. No events outside the bubble behind the red line can affect events inside the bubble. The lines have an angle α with respect to the direction of motion with sin(α) = 1/vs for vs the speed of the bubble relative to light speed. This is analogous to the Mach cone of supersonic objects.
</table>

NOTE: There are two different things that can be meant by the term "Natário warp drive". The generalized Natário warp drive is the general theory of all warp drives that includes within it the Alcubierre, Fell-Heisenberg, and Lentz warp drives, among infinitely many others. The zero-expansion Natário warp drive is the specific implementation of the generalized Natário warp drive that has a divergenceless shift vector field.

The generalized class of warp drives developed by Natário (including both the Alcubierre and this zero-expansion warp drive) are shown to always have regions in the warp shell where the energy density is negative to at least some observers. You can't get away from it – to warp, you need negative energy density.

The Natário class of warp drives blueshift light coming in from the front and redshift light catching up from the back. For the case of a super-luminal warp bubble, there will of course be a horizon at the back of the bubble and light will not be able to get through from behind. For a bubble speed vs relative to the speed of light, light coming from straight ahead will be blueshifted up in frequency and photon energy by a factor of 1 + vs. For sub-luminal travel, light coming from behind will be redshifted by a factor 1 - vs. In general for n as the unit direction along which the light is propagating, the blueshift factor will be 1 + vs ⋅ n. If the warp shell is infinitesimally thin, light from outside that reaches the center of the bubble will not be distorted in direction although it will be frequency shifted, but observers still inside the bubble but displaced from the center will see the field of view distorted as well as frequency shifted.

Most of the difficulties that apply to the Alcubierre warp drive also apply to the generalized Natário warp drive (and thus to the zero-expansion Natário drive and Fell-Heisenberg drive as well). In particular, quantum energy inequalities, high magnitudes of the energy required, formation of time machines, instabilities related to the stress-energy making up the warp shell being unable to keep up with the bubble, the chicken-and-egg problem of needing FTL to achieve FTL, the formation of horizons making warp bubbles uncontrollable, and Hawking radiation at the horizons, are all still concerns for generic warp drives.

=== Fell-Heisenberg warp drives ===

S. D. B. Fell and L. Heisenberg investigated what sort of warp structure is necessary to produce a warp drive with positive energy density everywhere<ref name="Fell-Heisenberg">S. D. B. Fell and L> Heisenberg, "Positive energy warp drive from hidden geometric structures", Classical and Quantum Gravity 38 155020 (2021) https://doi.org/10.1088/1361-6382/ac0e47 https://arxiv.org/abs/2104.06488</ref>. They found that warp drives of the generic Natário form with irrotational shift vectors had a simple expression for the energy density and vanishing momentum (irrotational means that the shift vector can be described as the gradient of a scalar potential field or, equivalently, that the curl of the shift vector field vanishes everywhere). Fell and Heisenberg proposed several potential fields that either had positive energy density everywhere except at discontinuities, or that were continuous but had regions that violated the weak energy condition (i. e. negative energy densities in some frames of reference). One specific example they provided of a warp field had an energy that is about 10,000 times less than the mass-energy of our sun. Or only about half the mass-energy of Jupiter. This is a significant improvement over many other proposed drives.

Despite the introduction discussing warp drive configurations that satisfy the various energy conditions, the configurations described in the paper are shown to locally violate the weak and strong energy conditions. Nonetheless, the energy density is still mostly positive. Santiago, Schuster, and Visser's<ref name="Santiago Schuster Visser 2022">J. Santiago, S. Schuster, and M. Visser, "Generic warp drives violate the null energy condition", Physical Review D 105, 064038 (2022) https://doi.org/10.1103/PhysRevD.105.064038</ref> work shows that the various energy conditions must still be violated by this warp drive; to not violate these, energy density must be positive in all reference frames not just those of the co-moving observer.

Any vector field can be described by its divergence, its curl, and a constant. The constant part of a warp drive's shift vector can be discarded because then the warp drive would not be localized. It is interesting, therefore, that the Natário zero-expansion drive has zero divergence and can be described only by its curl, while the Fell-Heisenberg drive has zero curl and can be described only by its divergence. These two classes of drives thus rely on opposite choices of setting one of the two different parts describing a vector field to zero. As might be expected, the Fell-Heisenberg drive is not zero expansion – space-time will change in volume as it passes through the warp bubble (although after going all the way through it will return to its original volume). However, objects in space-time will retain their original orientation as they pass through the warp bubble.

Any vector field with zero curl can be represented as the gradient of a suitable scalar function. Thus, the Fell-Heisenberg drive can be completely described by just scalar field (one number at all points of space and time) rather than a vector field.

Previous examples of warp drives, such as the Alcubierre and zero-expansion Natário drives, tend toward a flat space-time as you go far enough away faster than would occur if there were any mass present in the warp drive. That is, the ADM mass of such drives is zero. However, one of the Fell-Heisenberg examples described a spacetime where the geometry approaches that of what you would get for a massive object at the warp location, giving one of the first examples of a warp drive with mass.

In the Fell-Heisenberg drive the momentum vanishes everywhere. Yet the energy occupies regions where the shift vector is varying rapidly. The lack of momentum means that the energy will not move to keep up with the differential expansion and movement of the spacetime. As a consequence, at later times the energy will have a different distribution than what is necessary to maintain the given warp configuration. The exact time evolution is not solved but this likely leads to collapse of warp bubble. Note that this is not a problem unique to the Fell-Heisenberg drive. Assuming a geometry that describes a warp produces a specific distribution of the stress-energy tensor that satisfies Einstein's equation. But there is no guarantee that this stress energy tensor satisfies the continuity equation, which describes how the distribution of matter and momentum change with time. The issue of stability of any warp drive is not well understood, but most warp drive geometries will be unstable in the sense that their matter that generates the warp fields will not be able to keep up with the motion of the warp. The Fell-Heisenberg drive is unique only in that, with vanishing momentum, there is no configuration that allows stable propagation.

(n. b. The Heisenberg here is Lavinia Heisenberg, not the Werner Heisenberg of quantum physics and uncertainty principle fame.)

=== Lentz warp drive ===

E. Lentz<ref name="Lentz">E. W. Lentz, "Breaking the warp barrier: hyper-fast solitons in Einstein–Maxwell-plasma theory", Classical and Quantum Gravity. 38 075015 (2021). arXiv:[https://arxiv.org/abs/2006.07125 2006.07125]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2021CQGra..38g5015L 2021CQGra..38g5015L]. doi:[https://doi.org/10.1088%2F1361-6382%2Fabe692 10.1088/1361-6382/abe692]. ISSN [https://search.worldcat.org/issn/0264-9381 0264-9381]. S2CID [https://api.semanticscholar.org/CorpusID:219635854 219635854].</ref> investigated a particular configuration of the Fell-Heisenberg class of warp drives, where the scalar field forms a selection of rhomboid regions, resulting in an energy density that is everywhere positive and a shift vector with vanishing tidal force at the center for a comfortable ride.

However, Santiago, Schuster, and Visser<ref name="Santiago Schuster Visser 2022"></ref> dispute claims that the Lentz drive satisfies the energy conditions, noting that everywhere positive energy density in one frame of reference is insufficient to establish that the energy density is positive in all reference frames; knowledge of the Cauchy stress tensor is also needed. They show that any generic warp drive will violate the strong energy condition, null energy condition, and weak energy condition.

== Conservation laws ==

=== General introduction ===

All the warp drives investigated so far have geometries that quickly fall off from the highly curved region near the warp bubble to a region of spacetime that is almost flat; and becomes even more flat the farther away from the warp bubble you get. There is a word for geometries of this kind – they are called asymptotically flat. Basically, you can always go far enough away in any direction to reach a place where the geometry is sufficiently flat to meet any flatness criterion you choose.

When you have an asymptotically flat geometry, then in the far away mostly flat regions gravity is well described by a linearized theory of gravity – where the perturbations to the geometry are linear in the stress energy tensor. This means that gravity is basically Newtonian, along with a few other bells and whistles like gravitational waves and frame dragging.

When linearized gravity is a good approximation, then it is known that energy, momentum, and linear momentum are conserved. Although parts of a warp geometry are highly curved and non-linear, you can measure the conserved energy and momentum and angular momentum from far away in the linear region. This ADM energy and etcetera is the quantity that is conserved when you are worrying about conserving such things, not necessarily the energy and so forth you get in the highly curved spacetime region. In fact, it is not even simple to define where the energy and such actually is withing the highly curved spacetime region; it is non-local and different observers will disagree on how it is distributed. But everyone agrees on the ADM conserved quantities from out in nearly flat spacetime (up to the usual changes due to changing speeds, which are trivial and come up with normal objects as well).

And now we run into a bit of a conundrum: most of the warp geometries so far proposed have zero ADM mass<ref>Sebastian Schuster, Jessica Santiago,and Matt Visser, "ADM mass in warp drive spacetimes" General Relativity and Gravitation (2023) 55:14 https://doi.org/10.1007/s10714-022-03061-9</ref>, meaning that things like the Alcubierre drive technically have no energy or momentum or angular momentum at all. Despite all of the extremes of negative energy regions within the wall of the warp bubble, the actual energy that you need to spend to make one is zero. This can lead to a bit of a problem. If you have a spacecraft that has mass, that mass-energy has to be conserved. So when you turn on your warp drive and the total warp mass is zero, where does that energy go?

One resolution is just that the mass of the spacecraft is negligible and the small amount of curvature it causes can be ignored. In this case, the warp geometries discussed so far are merely an approximation that neglects this effect. In fact, some of the proposed Fell-Heisenberg drives map not onto entirely flat spacetime geometries at large distances, but onto the Schwarzschild geometry that is appropriate for systems with mass at the center. If this is the case, then the warping bubble and spacecraft system will need to worry about conservation of all their conserved properties.

=== All the science, demonstration of the problem ===

Let's look at what the conservation laws mean for a spacecraft equipped with a warp drive, assuming that no matter or radiation are emitted by activating or deactivating the warp drive.

We can start with an observer floating in space. We'll call her Alice. And, approaching Alice at 100 m/s is the starship Enterprise with a mass of a cool 200,000 tons (2×108 kg). from this we know that the Enterprise has, in Alice's frame of reference, a kinetic energy of 1 TJ, a total energy (including mass and kinetic energy) of 1.8×1025 J, a momentum of 2×1010 kg m/s, and – because it is headed straight toward Alice – an angular momentum of zero. All of this is without any warp drive activated.

On the bridge of the Enterprise, Spock picks up Alice on the sensors and tells Captain Kirk. Kirk does not want to run into Alice (he'd rather date her), so he orders helm to turn on the Enterprise's warp drive and move them 1 km up so that their course takes them 1 km from Alice instead of straight into her. Sulu warps the Enterprise according to these commands, and then turns off the warp drive.

After this maneuver, keeping the Enterprise's total energy and momentum the same means that the Enterprise still has a mass of 200,000 tons and a velocity of 100 m/s in its original direction. So we can see that the warp drive must preserve the velocity vector of the thing in it (at least in flat spacetime where outside forces are not acting on it). The problem comes from angular momentum. With a known mass of 200,000 tons, a speed of 100 m/s, and a distance of closest approach of 1 km; from Alice's frame of reference the Enterprise now has an angular momentum of 2×1013 kg m2/s after the warp maneuver, while before the maneuver it has an angular momentum of zero.

From this we can see that warp drives cannot be reactionless. Arbitrary displacements using warp without emitting propellant result in violation of the law of conservation of angular momentum. As long as we restrict ourselves to accepted science based on what we know of general relativity, to move a warp drive with mass you need to accelerate it or decelerate it with rockets or other normal means of delivering momentum (such as light sails, mag sails, or pushing off of a planet). Warp drives must be inertial to comply with physics. And all of this was analyzed only from times when the warp drive was turned off; it does not depend on any assumptions about what happens when the warp is on.

(Zero mass warp drives, relaxing conservation requirements, and activation/deactivation)

(If angular momentum conservation is ignored, discuss how momentum & energy are affected by outside forces. In a gravitational field equivalent to inertial frame ... like being in an accelerated elevator; will acquire momentum buildup while staying "at rest".)

== Warp Railroads ==

Van Den Broeck<ref name="VanDenBroeck_2000"></ref> suggested a warp railroad,

You can set up devices ahead of time along the path of the bubble that produce the negative energy regions for the warp bubble at the appropriate time without needing the negative energy to ever move with the bubble at all. At least for bubbles on a predictable schedule and route.

helps with:

-stability (elaborate)

-conservation laws (elaborate)

Mention Krasnikov<ref name="Krasnikov 1998">S. V. Krasnikov, "Hyperfast travel in general relativity" Phys. Rev. D 57 4760</ref>

Hyperwave<ref name="Pieri 2024">Lorenzo Pieri, "Hyperwave: hyper-Fast Communication within General Relativity", arXiv:2311.12069 [gr-qc] https://doi.org/10.48550/arXiv.2311.12069</ref> makes use of this warp railroad idea.

== Quantum effects ==

(quantum stuff here<ref name="Hiscock1997">W. A. Hiscock, "Quantum effects in the Alcubierre warp drive spacetime", Classical and Quantum Gravity 14 L183 https://doi.org/10.1088/0264-9381/14/11/002 https://arxiv.org/abs/gr-qc/9707024</ref><ref name="Finazzi et al 2009">S. Finazzi, S. Liberati, C. Barceló, "Semiclassical instability of dynamical warp drives", Physical Review D 79, 124017 (2009)https://doi.org/10.1103/PhysRevD.79.124017 https://arxiv.org/abs/0904.0141</ref>)

From<ref name="Pieri 2024"></ref>
<blockquote>
It should be emphasised that Quantum Inequalities have not been proven for realistic interacting quantum fields, where the they are expected to be weaker if they exist [Cadamuro, 2019], and they often require Minkowskian conditions at infinity, a condition which is not appropriate for fields inside bounded regions, as in a Casimir devices.
</blockquote>

== Warp drives and black holes ==

What happens if you drive your warp drive into a black hole?

The pop-sci reason that nothing can escape a black hole is that at the event horizon the escape velocity is faster than light and nothing can go faster than light. A warp drive can go faster than light.

The more detailed reason from general relativity is that at the event horizon space and time are rotated sufficiently that "inward" becomes "forward in time." You can no more go farther away from the singularity in a black hole once you are inside the event horizon than you can go backward in time. Any super-luminal travel can go backward in time, and warp drives can engage in super-luminal travel.

The Hawking area theorem that says that black holes can only grow, never shrink relies on the condition that the energy is everywhere positive. Warp drives have negative energy density.

So in principle, there's nothing that prevents a warp drive from taking a dip into a black hole and coming back out again.

Garattini and Zatrimaylov looked into the problem of a warp drive near (and in) a black hole<ref name="Garattini and Zatrimaylov">R. Garattini and K. Zatrimaylov, "Black holes, warp drives, and energy conditions", Physics Letters B 856 138910 (2024) https://doi.org/10.1016/j.physletb.2024.138910</ref>. The found that the black hole reduces the negative energy requirement for the warp bubble to move inward toward the black hole singularity, but increases the negative energy needed to move away from the center of the black hole. In addition, a warp drive parked at the event horizon would allow light from inside the horizon to pass through the bubble and escape back outside.

== Credit ==
Author: Luke Campbell

== References ==

[[Category:Physics]][[Category:Infrastructure]][[Category:Physics & Engineering‏‎]][[Category:Physics & Math & Engineering]][[Category:Transportation & Infrastructure‏‎]][[Category:Metric Engineering]]

Warp Drives

2026-07-30T04:25:59Z

Lwcamp: /* Conservation laws */

{{PageConstructionNotice}}

Science fiction often features spacecraft that can seemingly move across space and get between the place of departure and the destination much faster than light could have done. This appears to contradict the theory of relativity, which predicts unequivocally that nothing can move through space faster than light. Because relativity has been incredibly successful at describing nature, with its many other predictions regularly being confirmed to extraordinary accuracy and within the bounds of uncertainty of all the experiments that tested them, it gives confidence that relativity is a correct description of reality. Which seems to rather throw a wet towel on our hopes for rapid travel between stars.

However, while relativity does not allow things to move through space faster than light, it places no such restrictions on how fast space-time itself can expand, contract, or move around. This leads to the idea of a warp drive – the spacecraft remains stationary within a region of highly curved space-time, and that region moves at super-luminal speeds rather than the spacecraft.

== The Alcubierre warp drive ==

The first warp drive geometry that satisfied the Einstein field equations of relativity was proposed by Miguel Alcubierre<ref>M. Alcubierre, "The warp drive: hyper-fast travel within general relativity." Classical and Quantum Gravity. 11 (5): L73–L77 (1994). [https://arxiv.org/abs/gr-qc/0009013 arXiv:gr-qc/0009013]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1994CQGra..11L..73A 1994CQGra..11L..73A]. doi:[https://doi.org/10.1088%2F0264-9381%2F11%2F5%2F001 10.1088/0264-9381/11/5/001]. S2CID [https://api.semanticscholar.org/CorpusID:4797900 4797900].</ref>. In this geometry, a sphere of space-time moves at an arbitrary speed (potentially but not necessarily a speed much faster than light). Objects within the sphere are moved along with the sphere; an object at rest within the sphere would be moved along with the sphere indefinitely. Space is expanding at the rear boundary and contracting at the front boundary in order to keep the sphere moving. In order to satisfy the Einstein field equations, the boundary of the sphere must have a negative energy density. The challenges of space-time geometries with negative energy densities are described in our page on [[Wormholes#Exotic_energy_conditions|wormholes]], for our purposes it is enough to note that negative energy densities can pose problems if not handled carefully, there may be limits on how much negative energy you can have without nearby positive energy density, and it may not be possible to get enough negative energy to support a warp drive; although none of this is ruled out by physics – yet!

<table align=center>
<tr><td width=650>[[File:Energy_magnitude_of_Alcubierre_drive.png]]
<td width=650>[[File:Expansion_of_Alcubierre_drive.png]]
<tr><td>The magnitude of the energy of the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. (The energy density is negative everywhere in the shell, here only the magnitude is shown to aid visual comprehension.)
<td>The expansion of space in the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. Negative expansion means that space is contracting in that direction, positive expansion that it is expanding.
</table>

NOTE: What the? The energy distribution is symmetric! How does the drive know which way to go? There must be significant contributions of other components of the stress-energy tensor, and those have got to be asymmetric along the forward/backward axis. Check this when I get time.

(And what's with the ring in all the artwork, anyway?)

=== So what is going on? ===

What is it that makes a warp drive a warp drive, and how does it work?

A warp drive is a concept that comes out of the science of general relativity. General relativity collects space and time together into a single "structure" called spacetime. This structure is curved, and its curvature is determined by the material that is in it. Absent forces acting between matter, particles of matter move through spacetime on the shortest possible paths in this curved environment.
<blockquote>
"Spacetime tells matter how to move; matter tells spacetime how to curve." - John Archibald Wheeler
</blockquote>
All of this is encompassed in the Einstein field equation:
<div align=center>
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 8πG
<td rowspan="2" nowrap="nowrap" align="right"> T = R -  
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right"> R g
<tr>
<td style="border-top:solid 1px black;"> c4
<td style="border-top:solid 1px black;"> 2
</table>
</div>
Unfortunately, by convention, both the symbols G and R get used for a lot of different things. Here, symbols in bold are second rank tensors – constructs of arrays of 4×4 numbers where 4 is necessary because that is the number of dimensions of spacetime. Conventionally, these numbers are indexed by a value going from 0 to 3, with 0 being the part corresponding to the coordinate of time and 1 to 3 being the Cartesian coordinates that you might know as x, y, and z.

T is the stress-energy tensor: the (0,0) component represents the energy density at a given point; the (0,i) and (i, 0) components (with i going from 1 to 3) represent the momentum density; and the (i,j) components represent the stresses present, such as pressure or tension or shear.

The other term, R - R g/2, involves a partial description of the curvature. A full description of the curvature requires a construct of 4×4×4×4 components; too much for the stress energy tensor to fully constrain. Which is okay, because this allows curvature to exist away from where matter is, allowing gravity to have long range effects across empty space as well as for phenomena like gravitational waves. But it is enough that a given distribution of matter T gives enough constraints (along with specifying how the geometry behaves far away) that you can generally figure out what the actual gravity is.

This is the usual story that we are told. But there is an alternate way of looking at the same geometry such that the math works out the same, but it is described conceptually in a very different way. This divides spacetime up into "layers", called foliations, that represent slices of space and going from one foliation to the next is increasing in time. This is useful for computation because you can assign coordinates to your system. There are two additional constructs – the lapse function and the shift vector field. Starting at an event on one foliation, the lapse function at that position tells you that after a certain "proper time" (time perceived by an observer at that event) the time coordinate of the observer will be the proper time times the lapse function at that event. And the new spatial coordinates of the observer will be the proper time times the shift vector at that event. (Strictly speaking, this is only exactly true in the limit of the proper time approaching zero or when the lapse function and shift vectors are constant. But hopefully this gives a good enough insight to what these entities are so that we can go on to describe warp drives.)

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Lapse_shift.png]]
<tr><td>A folliation of space-time into slices of constant time coordinates (grey). At an event (black circle), the lapse function (yellow) shows how far into the future the point at that event will be projected in a unit amount of proper time experienced by an observer initially at the event. The shift vector (red) shows how that point changes coordinates after the unit amount of proper time. The cyan vector shows the new coordinates the point is projected onto. The lower black surface indicates a foliation of spacetime at the same time coordinate; after unit proper time, the events on the lower foliation get projected to the curved upper black foliation according to the shift vectors and lapse functions at each of their respective points.
</table>
</table>

Warp drives are generally restricted to spacetime geometries with a constant lapse function - all observers go forward by the same time coordinate after experiencing the same amount of proper time. Thus, all warp drives have no time dilation.
Far from the warp bubble, the shift vector vanishes – each spatial coordinate is mapped to the same spatial coordinate after any amount of proper time so long as the warp drive is not near. But the inside of the warp bubble is an extended region where the shift vector field is non-zero and constant. In this bubble, all spatial coordinates are mapped to a coordinate that shifts linearly in time with the proper time of the observer. Astute readers will see that this linear change in position with time is very similar to a velocity. This region of constant shift vector then moves with the shift vector velocity to keep stuff originally in the bubble so that it stays in the bubble. In between the inside of the bubble and the far away regions with zero shift vector, the shift vector field falls away to match one region to the other.

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Warp_shift.png]]
<tr><td>The shift vector of a one-dimensional slice through warp geometry, in red. At the edges there is no shift. At the center (inside the warp bubble) there is a region of uniform shift so that at a later time the inside of the warp bubble is rigidly translated to different spatial coordinates (in blue). The two foliations remain the same distance apart, because warp drives maintain a unit lapse function.
</table>
</table>

Traditionally, scientists would start with an interesting distribution of matter given by T and then find out what kind of curvature it created. This works well for planets and stars and black holes and cosmic strings and other things which are observed in the universe and you want to see what sort of effect they had. But starting in the 1980's, some physicists started looking at Einstein's field equations the other way around. They start with a desired geometry, find the curvature part of the field equation R - R g/2, and use that to find out what T has to be in order to get that curvature. This was originally done for [[Wormholes|wormholes]], but in 1994 Miguel Alcubierre used this same trick to figure out what could create a region of constant shift vector that moved with the shift vector - the warp drive. With this demand for a warp geometry, the required distribution of matter and energy and momentum and stress can be determined.

The original Alcubierre shift vector N→ produces a spherical warp bubble with an axi-symmetric distribution of, well, everything along the direction of motion with warp velocity vector v→.
<div align="center">
N→ = v→ f(rs)
</div>
Here, rs is the distance from the center of the warp bubble, and f(rs) can be any function that is 1 near rs = 0 and is 0 for rs well beyond the radius of the warp bubble.

=== Challenges and possible resolutions ===

Almost as soon as Alcubierre proposed his warp drive, others began picking it apart.
<ol>
<li>Pfenning and Ford<ref name="PfenningFord_2001">M. J. Pfenning and L. H. Ford, "The unphysical nature of 'Warp Drive'", Classical and Quantum Gravity. 14 (7): 1743–1751 (1997). arXiv:[https://arxiv.org/abs/gr-qc/9702026 gr-qc/9702026]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1997CQGra..14.1743P 1997CQGra..14.1743P]. doi:[https://doi.org/10.1088%2F0264-9381%2F14%2F7%2F011 10.1088/0264-9381/14/7/011]. S2CID [https://api.semanticscholar.org/CorpusID:15279207 15279207].</ref> found that if the negative energy density wall around the bubble obeys [[Wormholes#Quantum_energy_inequalities|quantum energy inequalities]], then for warp speeds of around light speed the shell thickness would be on the order of a hundred Planck lengths; a distance far smaller than any other known physical phenomenon or object.
<li>In the same work, Pfenning and Ford also found that for a bubble a hundred meters in radius, a warp speed of around the speed of light, and the required thickness of about 100 Planck lengths the energy in the bubble shell would be E ≈ -1063 kg c2. The magnitude of this latter value is ten orders of magnitude larger than the energy of the entire visible universe. They do note, however, that if quantum energy inequalities can be ignored then a warp drive with a hundred meters radius but a shell thickness of one meter would "only" have an energy magnitude of about a quarter solar mass.
<li>As with any method of faster than light travel, the warp drive could be used to make a time machine.
<li>Perhaps most seriously, if the warp drive is going faster than light speed, the negative energy regions on the outside of the shell won't be in the warp parts of the bubble. They will have to be moving through space at faster than light speed if the bubble is to maintain its integrity, which is the very problem that the warp drive was designed to avoid. The outside of the warp bubble shell that maintains the warp bubble would fall away and the warp bubble would quickly erode to sub-luminal speeds.<ref name="Van Den Broeck 2000">Chris Van Den Broeck, "On the (im)possibility of warp bubbles", arXiv:gr-qc/9906050 https://doi.org/10.48550/arXiv.gr-qc/9906050</ref>
<li>A more general problem that includes the previous one is that the geometry of the warp drive determines the distribution of energy, momentum, and stress (pressure, shear, and tension) by Einstein's field equation, which states that the stress energy tensor is directly proportional to a reduced order curvature tensor of space-time. However, the stress-energy tensor must also obey the continuity equation – basically, for the energy to change in a given volume of space that energy must come in or go out of the boundary of that volume and similarly for the momentum. So a warp drive can have the warp condition for one moment in time. But after that, the stress-energy that maintains the warp will start to move according to the forces on it and according to the natural paths that it takes through the warped space-time. There is no guarantee that the configuration of the stress-energy at later times will be sufficient to maintain the warp.
<blockquote>
From the perspective of simulating the warp drive dynamically, the key challenge is stability. One can straightforwardly generate the initial metric that describes the Alcubierre metric and infer the instantaneous matter configuration that must support it using the Einstein Equation, but a time-domain evolution of the coupled matter-gravity system necessitates that we specify the equation of state of the matter, which in turn determines how it evolves. In a perfect fluid, this involves specifying the relation between the fluid pressure and density, but in a more general case, the full relation between the
stress-energy components must be given. There is (to our knowledge) no known equation of state that would maintain the warp drive metric in a stable configuration over time - therefore, whilst one can require that initially, the warp bubble is constant, it will quickly evolve away from that state and, in most cases, the warp fluid and spacetime deformations will disperse or collapse into a central point.
 
– Clough, Dietrich, and Khan<ref>Katy Clough, Tim Dietrich, and Sebastian Khan, "What no one has seen before: gravitational waveforms from warp drive collapse", arXiv:2406.02466 [gr-qc] https://doi.org/10.48550/arXiv.2406.02466</ref>
</blockquote>
<li>In order to get the matter (or more generally, stress-energy) that forms the warp shell up to speeds that could allow the shell to exist, you need to be able to, well, accelerate that matter to warp speeds. Which gives a chicken-and-egg problem - in order to achieve superluminal travel you need to be able to get matter or energy moving superluminally in the first place<ref name="Bobrick and Martire">Alexey Bobrick and Gianni Martire, "Introducing Physical Warp Drives", arXiv:2102.06824 [gr-qc] https://doi.org/10.48550/arXiv.2102.06824</ref><ref>D. H. Coule, "No warp drive", Class. Quantum Grav. 15 (1998) 2523–2527. https://doi.org/10.1088/0264-9381/15/8/026</ref>, which rather makes warp drives unnecessary from the get-go.
<li>A warp bubble moving faster than light will not be able to get any signals in front of it as it will overtake its own signal. In fact, anyone in the interior of the bubble would not be able to send signals to the front surface of the bubble. In science-speak, a horizon forms at the front of the bubble<ref name="Natario">José Natário, "Warp drive with zero expansion", Classical and Quantum Gravity. 19 (6): 1157–1166 (2002). arXiv:[https://arxiv.org/abs/gr-qc/0110086 gr-qc/0110086]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2002CQGra..19.1157N 2002CQGra..19.1157N]. doi:[https://doi.org/10.1088%2F0264-9381%2F19%2F6%2F308 10.1088/0264-9381/19/6/308]. S2CID [https://api.semanticscholar.org/CorpusID:15859984 15859984].</ref>. As a consequence, the bubble cannot be controlled, steered, nor stopped from within the bubble itself.
<li>A Horizon will naturally emit Hawking radiation. It is estimated that the radiation will raise the temperature of the interior of the bubble to approximately 1032 K<ref name="Finazzi 2009">Stefano Finazzi, Stefano Liberati, and Carlos Barceló, "Semiclassical instability of dynamical warp drives", Phys. Rev. D 79, 124017 (12 June, 2009) https://doi.org/10.1103/PhysRevD.79.124017 , arXiv:0904.0141 [gr-qc] https://doi.org/10.48550/arXiv.0904.0141</ref> (at this level of accuracy, the result is the same if you replace temperatures in K by the same number in °C or °F). This temperature, however, may be lower if quantum inequalities do not have to be fulfilled.
<li>Semi-classical instabilities at the horizon are expected to lead to a divergence of the stress-energy tensor at the boundary in a short period of time, thus leading to the destruction of the warp bubble<ref name="Finazzi 2009"></ref>.
</ol>
Helpfully, Van Den Broeck<ref name="VanDenBroeck_2000">C. Van Den Broeck, "Alcubierre’s Warp Drive: Problems and Prospects" AIP Conference Proceedings. 504: 1105–1110 (2000). Bibcode:[https://ui.adsabs.harvard.edu/abs/2000AIPC..504.1105V 2000AIPC..504.1105V]. doi:[https://doi.org/10.1063%2F1.1290913 10.1063/1.1290913].</ref> suggested several solutions – or at least mitigations – for some of these problems
<ol>
<li>Quantum inequalities had not been shown to be true in general for highly curved space-times.
<li>A proposed warp drive geometry<ref name="VanDenBroeck_1999">C. Van Den Broeck, "A 'warp drive' with more reasonable total energy requirements". Classical and Quantum Gravity. 16 (12): 3973–3979 (1999). arXiv:[https://arxiv.org/abs/gr-qc/9905084 gr-qc/9905084]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1999CQGra..16.3973V 1999CQGra..16.3973V]. doi:[https://doi.org/10.1088%2F0264-9381%2F16%2F12%2F314 10.1088/0264-9381/16/12/314]. S2CID [https://api.semanticscholar.org/CorpusID:15466313 15466313]. </ref> (see the van Den Broeck drive, below) would be able to minimize the magnitude of the energy required to about that of the mass of our sun. While still large, it is not unphysically large.
<li>Time travel is always going to be a worry with faster than light travel. There's no neat solution to this one.
<li>It may be possible for the negative energy outside of the superluminal region to compress into the superluminal region to form a shock in space-time. The front surface of the bubble would then be a singularity. It is not clear if such a sharp jump in physical properties is possible, but neither is sure that it is impossible, either. However, while this might help with the negative energy in front of the bubble getting swept up because it cannot move faster than light, it does little to help with the material at the back of the bubble on the outside getting left behind because it cannot keep up.
</ol>
Other mitigating research has also shown
<ol start=9>
<li>Semi-classical divergence only occurs in warp drives with only one spatial dimensions. Those with two or more spatial dimensions (as in our universe) only contain points where divergences occur, and these divergences can be managed. In fact, their effects can be reduced by making the warp shape more elongated<ref>Carlos Barceló, Valentin Boyanov, Luis J. Garay, Eduardo Martín-Martínez,
and Jose M. Sánchez Velázquez, "Warp drive aerodynamics", JHEP08 (2022) 288 https://doi.org/10.1007/JHEP08(2022)288 , arXiv:2207.06458 [gr-qc] https://doi.org/10.48550/arXiv.2207.06458</ref>.
</ol>

=== Alcubierre warp interactions with light and matter ===

Space is not entirely empty. It is filled with a diffuse plasma in the form of the interstellar medium, as well as cosmic radiation, light from stars, and cosmic microwave background radiation. A warp drive propagating through space will encounter this stuff. What happens when this matter and radiation have a warp bubble pass across them?

The first analysis of matter encountering a warp bubble was performed by Pfenning and Ford<ref name="PfenningFord_2001"></ref>. They looked at the warp bubble interaction with a massive object at rest with the frame of reference of the warp drive (keep in mind that the rest frame is the same inside and outside the warp bubble; in this rest frame objects inside the bubble have no momentum even though they are, in some sense, changing location rapidly with time). When the warp bubble passes, the object experiences an acceleration in the direction of the warp bubble motion. When the warp shell passes and the object is inside the bubble, it will be moving with approximately the speed of the bubble. Pfenning and Ford analyzed this problem with a continuous distribution of shell energy that, strictly speaking, never falls to zero except at the bubble center and at spatial infinity, so unless the object passes through the center in Pfenning and Ford's description it will never quite get up to the bubble's speed. In this case, the object will move almost as fast as the object but will pass through the bubble in a finite time, after which it will again be at rest with respect to the reference frame but displaced along the direction of the warp bubble motion by some distance. With this description of the warp bubble, a spacecraft of finite size will always be moving a little slower than the warp bubble and would have to use rockets to keep up with it. in addition, the spacecraft would experience tidal forces that would cause stress on the spacecraft's structure.

Pfenning and Ford also examined cases where the shell is of a finite (possibly infinitesimal) width and falls to zero both inside and outside the bubble. In this case any matter encountering the bubble would be collected at the bow of the bubble and thereafter move along with it.

McMonigal et al. analyzed the situation for both massive particles and light moving along the axis of travel of the bubble<ref name="McMonigalEtAl_2012">B. McMonigal, G. F. Lewis, and P. O'Byrne, "Alcubierre warp drive: On the matter of matter". Physical Review D. 85 (6) 064024 (20 March 2012). arXiv:[https://arxiv.org/abs/1202.5708 1202.5708]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2012PhRvD..85f4024M 2012PhRvD..85f4024M]. doi:[https://doi.org/10.1103%2FPhysRevD.85.064024 10.1103/PhysRevD.85.064024]. S2CID [https://api.semanticscholar.org/CorpusID:3993148 3993148].</ref>. They found that light moving opposite the warp direction passed through the bubble without incident, being only somewhat delayed by passing through the bubble. A warp bubble that is warping at sub-luminal speeds can have light catch up from behind it. This light is able to pass through the bubble, and is somewhat advanced in its path by the speed of the bubble. For super-luminal warp bubbles, however, the situation is different. The bubble will catch up to light moving it its own direction that is originally in front of it. This light cannot escape forward, the bubble being too fast. Nor can it escape backward, as the light is propagating forward and the interior of the bubble is at rest. Thus, the light gets caught at the bow of the warp bubble, unable to escape for so long as the warp bubble is active. This light is strongly blue shifted to extremely energetic x-rays and gamma rays. The space behind the bubble is swept clear of forward-moving light.

The situation for matter over-run by a super-luminal warp bubble is similar. Matter moving backward passes through the bubble, being only swept a ways along the bubble's path as sign of its passing but otherwise continuing on their way. Matter at rest behaves the same way as Pfenning and Ford discovered. Meanwhile, matter moving in the same direction of the warp is overtaken and collects at the bow of the bubble, unable to leave for so long as the bubble is warping. This matter "surfing" on the bubble bow is highly accelerated to relativistic speeds, experiencing extreme time dilation.

For a sub-luminal bubble, there is a speed faster than the warp speed where particles coming up from behind overtake the bubble and pass through, out the front. Particles below this speed but still faster than the warp speed still overtake the bubble, but then bounce backwards out of the bubble from behind. They are still moving forward, but are now moving more slowly than the warp speed. Particles moving forward but slower than the bubble will be overtaken and then bounced out the front with a speed higher than the warp speed. Those particles moving backward will pass through the bubble and exit with their initial speed.

The spacecraft inside the bubble will observe that light moving backward compared to the warp direction is blue-shifted, and matter moving backward passes through with increased energy. Meanwhile, a sub-luminal bubble will see light that was moving forward as red-shifted and matter catching up to the bubble from behind will pass by with reduced energy (superluminal bubbles, of course, do not have any matter or radiation catching up to them from behind).

When a superluminal warp bubble that has been collecting matter and radiation for the duration of its journey and blue-shifting it to much higher energies is turned off, all that matter and energy is released as a blast of radiation in the direction the bubble was warping.

McMonigal et al. conclude by noting that any spacecraft in the warp bubble would need shielding to protect against the blue-shifted radiation and matter of increased energy. In addition the destination would be "blasted into oblivion" by the release of matter and radiation that had been caught in the bubble during the trip.

Clark et al.<ref>Chad Clark, William A. Hiscock, and Shane L. Larson, "Null geodesics in the Alcubierre warp drive spacetime: the view from the bridge", arXiv:gr-qc/9907019 (2018) https://doi.org/10.48550/arXiv.gr-qc/9907019</ref> examine what it would look like from within an operating warp drive. They find that light moving at 90 degrees to the direction of motion will look like it is still coming from 90 degrees. However, light from the front hemisphere is squished progressively toward the forward point the faster you go. For a warp bubble moving faster than the speed of light, there is a cone behind the warp bubble where no light can reach it. However, you don't see a black spot behind you – light from the regions in the backward hemisphere from outside that cone that can reach you is distorted toward the direction behind you to fill in that spot. Light from the forward direction is blue-shifted and light from behind is red-shifted. At speeds of around 200 times light speed, the cosmic microwave background would be blue shifted to energies similar to that of the solar photosphere, heating anything in the warp bubble as if they were close to the sun.

== Other warp geometries ==

The Alcubierre warp drive is the first and best known of the warp drives. It has been around long enough that there have been many studies done on it. The warp geometry is particularly simple, making analysis easier. But in the time that the Alcubierre warp was proposed, many other researchers have come up with other warp designs, and even generalized the idea to include a wide array of additional possibilities.

Most of the general statements about the challenges of the Alcubierre drive hold for these other drives. Many of the details about interactions with matter are also likely to hold, although these other styles of warp drive have not yet been subject to such rigorous investigation.

=== Van Den Broeck warp drive ===

If an Alcubierre warp bubble a hundred meters across requires a magnitude of energy greater than the entire energy in the observable universe, one option to reduce the magnitude of energy used is to make the warp bubble smaller. Much smaller. Pfenning and Ford<ref name="PfenningFord_2001"></ref> suggested making the warp bubble smaller than an atom. Of course, that brings up the problem of how to stuff a spacecraft in there. Van Den Broeck proposed a solution<ref name="VanDenBroeck_1999"></ref>: make the warp bubble only about the size of an atomic nucleus but expand the space inside the bubble enormously so that a spacecraft could fit in. This is somewhat like a nuclear sized wormhole that leads to a pocket universe that holds the spacecraft. Unfortunately, the metric doesn't fit neatly into any [[Wormholes#Three_dimensions|embedding diagram]] that I can figure out, but the math works out so that spatial coordinates inside the bubble are expanded by a factor of 1017 and a spacecraft would have a few hundred meters of bubble interior to putz around in. This trick manages to reduce the magnitude of energy to only about the mass energy of a few stars similar to our own. Other than that, it is otherwise a normal Alcubierre drive.

Further work<ref>S. Krasnikov, "The quantum inequalities do not forbid spacetime shortcuts", Physical Review D. 67 (10) 104013. https://doi.org/10.1103%2FPhysRevD.67.104013, arXiv:gr-qc/0207057 (2003) https://doi.org/10.48550/arXiv.gr-qc/0207057 </ref> has managed to use Van Den Broeck's trick to reduce the magnitude of the negative energy to only about a milligram.

=== Natário warp drive ===

The Alcubierre drive is not the only way to construct a warp drive. José Natário<ref name="Natario"></ref> showed that it was just one example of an entire class of warp drives. These are space-time geometries with a smooth and localized shift vector field and a unit lapse function.

Natário then went on to find in that class a set of warp drives that do not have any expansion or contraction of space at all. Instead, space encountering the front boundary of the warp bubble instead "slides" around the outside of the bubble until it gets to the corresponding place on the back and is left behind there. Another way to think of it is that space entering the bubble shell is compressed in the radial direction of the shell but is simultaneously expanded in the tangential direction so that there is no net change in volume. When it gets to the back, the opposite occurs. In the language of mathematics, the shift vector field is divergenceless.

<table align=center>
<tr><td width=360>[[File:Natario_drive_flow_of_space.png]]
<td width=445>[[File:Natario_warp_drive_horizons.png]]
<tr><td style="vertical-align: top;">A representation of the motion of space in a zero-expansion Natário drive with a vanishingly thin warp shell around the bubble.
<td>The horizons of the generalized Natário class of warp drives (including the Alcubierre warp drive and all fell-Heisenberg drives) at superluminal speed in the direction of the green arrow. No event inside the bubble can affect events outside the bubble in front of the blue line. No events outside the bubble behind the red line can affect events inside the bubble. The lines have an angle α with respect to the direction of motion with sin(α) = 1/vs for vs the speed of the bubble relative to light speed. This is analogous to the Mach cone of supersonic objects.
</table>

NOTE: There are two different things that can be meant by the term "Natário warp drive". The generalized Natário warp drive is the general theory of all warp drives that includes within it the Alcubierre, Fell-Heisenberg, and Lentz warp drives, among infinitely many others. The zero-expansion Natário warp drive is the specific implementation of the generalized Natário warp drive that has a divergenceless shift vector field.

The generalized class of warp drives developed by Natário (including both the Alcubierre and this zero-expansion warp drive) are shown to always have regions in the warp shell where the energy density is negative to at least some observers. You can't get away from it – to warp, you need negative energy density.

The Natário class of warp drives blueshift light coming in from the front and redshift light catching up from the back. For the case of a super-luminal warp bubble, there will of course be a horizon at the back of the bubble and light will not be able to get through from behind. For a bubble speed vs relative to the speed of light, light coming from straight ahead will be blueshifted up in frequency and photon energy by a factor of 1 + vs. For sub-luminal travel, light coming from behind will be redshifted by a factor 1 - vs. In general for n as the unit direction along which the light is propagating, the blueshift factor will be 1 + vs ⋅ n. If the warp shell is infinitesimally thin, light from outside that reaches the center of the bubble will not be distorted in direction although it will be frequency shifted, but observers still inside the bubble but displaced from the center will see the field of view distorted as well as frequency shifted.

Most of the difficulties that apply to the Alcubierre warp drive also apply to the generalized Natário warp drive (and thus to the zero-expansion Natário drive and Fell-Heisenberg drive as well). In particular, quantum energy inequalities, high magnitudes of the energy required, formation of time machines, instabilities related to the stress-energy making up the warp shell being unable to keep up with the bubble, the chicken-and-egg problem of needing FTL to achieve FTL, the formation of horizons making warp bubbles uncontrollable, and Hawking radiation at the horizons, are all still concerns for generic warp drives.

=== Fell-Heisenberg warp drives ===

S. D. B. Fell and L. Heisenberg investigated what sort of warp structure is necessary to produce a warp drive with positive energy density everywhere<ref name="Fell-Heisenberg">S. D. B. Fell and L> Heisenberg, "Positive energy warp drive from hidden geometric structures", Classical and Quantum Gravity 38 155020 (2021) https://doi.org/10.1088/1361-6382/ac0e47 https://arxiv.org/abs/2104.06488</ref>. They found that warp drives of the generic Natário form with irrotational shift vectors had a simple expression for the energy density and vanishing momentum (irrotational means that the shift vector can be described as the gradient of a scalar potential field or, equivalently, that the curl of the shift vector field vanishes everywhere). Fell and Heisenberg proposed several potential fields that either had positive energy density everywhere except at discontinuities, or that were continuous but had regions that violated the weak energy condition (i. e. negative energy densities in some frames of reference). One specific example they provided of a warp field had an energy that is about 10,000 times less than the mass-energy of our sun. Or only about half the mass-energy of Jupiter. This is a significant improvement over many other proposed drives.

Despite the introduction discussing warp drive configurations that satisfy the various energy conditions, the configurations described in the paper are shown to locally violate the weak and strong energy conditions. Nonetheless, the energy density is still mostly positive. Santiago, Schuster, and Visser's<ref name="Santiago Schuster Visser 2022">J. Santiago, S. Schuster, and M. Visser, "Generic warp drives violate the null energy condition", Physical Review D 105, 064038 (2022) https://doi.org/10.1103/PhysRevD.105.064038</ref> work shows that the various energy conditions must still be violated by this warp drive; to not violate these, energy density must be positive in all reference frames not just those of the co-moving observer.

Any vector field can be described by its divergence, its curl, and a constant. The constant part of a warp drive's shift vector can be discarded because then the warp drive would not be localized. It is interesting, therefore, that the Natário zero-expansion drive has zero divergence and can be described only by its curl, while the Fell-Heisenberg drive has zero curl and can be described only by its divergence. These two classes of drives thus rely on opposite choices of setting one of the two different parts describing a vector field to zero. As might be expected, the Fell-Heisenberg drive is not zero expansion – space-time will change in volume as it passes through the warp bubble (although after going all the way through it will return to its original volume). However, objects in space-time will retain their original orientation as they pass through the warp bubble.

Any vector field with zero curl can be represented as the gradient of a suitable scalar function. Thus, the Fell-Heisenberg drive can be completely described by just scalar field (one number at all points of space and time) rather than a vector field.

Previous examples of warp drives, such as the Alcubierre and zero-expansion Natário drives, tend toward a flat space-time as you go far enough away faster than would occur if there were any mass present in the warp drive. That is, the ADM mass of such drives is zero. However, one of the Fell-Heisenberg examples described a spacetime where the geometry approaches that of what you would get for a massive object at the warp location, giving one of the first examples of a warp drive with mass.

In the Fell-Heisenberg drive the momentum vanishes everywhere. Yet the energy occupies regions where the shift vector is varying rapidly. The lack of momentum means that the energy will not move to keep up with the differential expansion and movement of the spacetime. As a consequence, at later times the energy will have a different distribution than what is necessary to maintain the given warp configuration. The exact time evolution is not solved but this likely leads to collapse of warp bubble. Note that this is not a problem unique to the Fell-Heisenberg drive. Assuming a geometry that describes a warp produces a specific distribution of the stress-energy tensor that satisfies Einstein's equation. But there is no guarantee that this stress energy tensor satisfies the continuity equation, which describes how the distribution of matter and momentum change with time. The issue of stability of any warp drive is not well understood, but most warp drive geometries will be unstable in the sense that their matter that generates the warp fields will not be able to keep up with the motion of the warp. The Fell-Heisenberg drive is unique only in that, with vanishing momentum, there is no configuration that allows stable propagation.

(n. b. The Heisenberg here is Lavinia Heisenberg, not the Werner Heisenberg of quantum physics and uncertainty principle fame.)

=== Lentz warp drive ===

E. Lentz<ref name="Lentz">E. W. Lentz, "Breaking the warp barrier: hyper-fast solitons in Einstein–Maxwell-plasma theory", Classical and Quantum Gravity. 38 075015 (2021). arXiv:[https://arxiv.org/abs/2006.07125 2006.07125]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2021CQGra..38g5015L 2021CQGra..38g5015L]. doi:[https://doi.org/10.1088%2F1361-6382%2Fabe692 10.1088/1361-6382/abe692]. ISSN [https://search.worldcat.org/issn/0264-9381 0264-9381]. S2CID [https://api.semanticscholar.org/CorpusID:219635854 219635854].</ref> investigated a particular configuration of the Fell-Heisenberg class of warp drives, where the scalar field forms a selection of rhomboid regions, resulting in an energy density that is everywhere positive and a shift vector with vanishing tidal force at the center for a comfortable ride.

However, Santiago, Schuster, and Visser<ref name="Santiago Schuster Visser 2022"></ref> dispute claims that the Lentz drive satisfies the energy conditions, noting that everywhere positive energy density in one frame of reference is insufficient to establish that the energy density is positive in all reference frames; knowledge of the Cauchy stress tensor is also needed. They show that any generic warp drive will violate the strong energy condition, null energy condition, and weak energy condition.

== Conservation laws ==

All the warp drives investigated so far have geometries that quickly fall off from the highly curved region near the warp bubble to a region of spacetime that is almost flat; and becomes even more flat the farther away from the warp bubble you get. There is a word for geometries of this kind – they are called asymptotically flat. Basically, you can always go far enough away in any direction to reach a place where the geometry is sufficiently flat to meet any flatness criterion you choose.

When you have an asymptotically flat geometry, then in the far away mostly flat regions gravity is well described by a linearized theory of gravity – where the perturbations to the geometry are linear in the stress energy tensor. This means that gravity is basically Newtonian, along with a few other bells and whistles like gravitational waves and frame dragging.

When linearized gravity is a good approximation, then it is known that energy, momentum, and linear momentum are conserved. Although parts of a warp geometry are highly curved and non-linear, you can measure the conserved energy and momentum and angular momentum from far away in the linear region. This ADM energy and etcetera is the quantity that is conserved when you are worrying about conserving such things, not necessarily the energy and so forth you get in the highly curved spacetime region. In fact, it is not even simple to define where the energy and such actually is withing the highly curved spacetime region; it is non-local and different observers will disagree on how it is distributed. But everyone agrees on the ADM conserved quantities from out in nearly flat spacetime (up to the usual changes due to changing speeds, which are trivial and come up with normal objects as well).

And now we run into a bit of a conundrum: most of the warp geometries so far proposed have zero ADM mass<ref>Sebastian Schuster, Jessica Santiago,and Matt Visser, "ADM mass in warp drive spacetimes" General Relativity and Gravitation (2023) 55:14 https://doi.org/10.1007/s10714-022-03061-9</ref>, meaning that things like the Alcubierre drive technically have no energy or momentum or angular momentum at all. Despite all of the extremes of negative energy regions within the wall of the warp bubble, the actual energy that you need to spend to make one is zero. This can lead to a bit of a problem. If you have a spacecraft that has mass, that mass-energy has to be conserved. So when you turn on your warp drive and the total warp mass is zero, where does that energy go?

One resolution is just that the mass of the spacecraft is negligible and the small amount of curvature it causes can be ignored. In this case, the warp geometries discussed so far are merely an approximation that neglects this effect. In fact, some of the proposed Fell-Heisenberg drives map not onto entirely flat spacetime geometries at large distances, but onto the Schwarzschild geometry that is appropriate for systems with mass at the center. If this is the case, then the warping bubble and spacecraft system will need to worry about conservation of all their conserved properties.

Let's look at what the conservation laws mean for a spacecraft equipped with a warp drive, assuming that no matter or radiation are emitted by activating or deactivating the warp drive.

We can start with an observer floating in space. We'll call her Alice. And, approaching Alice at 100 m/s is the starship Enterprise with a mass of a cool 200,000 tons (2×108 kg). from this we know that the Enterprise has, in Alice's frame of reference, a kinetic energy of 1 TJ, a total energy (including mass and kinetic energy) of 1.8×1025 J, a momentum of 2×1010 kg m/s, and – because it is headed straight toward Alice – an angular momentum of zero. All of this is without any warp drive activated.

On the bridge of the Enterprise, Spock picks up Alice on the sensors and tells Captain Kirk. Kirk does not want to run into Alice (he'd rather date her), so he orders helm to turn on the Enterprise's warp drive and move them 1 km up so that their course takes them 1 km from Alice instead of straight into her. Sulu warps the Enterprise according to these commands, and then turns off the warp drive.

After this maneuver, keeping the Enterprise's total energy and momentum the same means that the Enterprise still has a mass of 200,000 tons and a velocity of 100 m/s in its original direction. So we can see that the warp drive must preserve the velocity vector of the thing in it (at least in flat spacetime where outside forces are not acting on it). The problem comes from angular momentum. With a known mass of 200,000 tons, a speed of 100 m/s, and a distance of closest approach of 1 km; from Alice's frame of reference the Enterprise now has an angular momentum of 2×1013 kg m2/s after the warp maneuver, while before the maneuver it has an angular momentum of zero.

From this we can see that warp drives cannot be reactionless. Arbitrary displacements using warp without emitting propellant result in violation of the law of conservation of angular momentum. As long as we restrict ourselves to accepted science based on what we know of general relativity, to move a warp drive with mass you need to accelerate it or decelerate it with rockets or other normal means of delivering momentum (such as light sails, mag sails, or pushing off of a planet). Warp drives must be inertial to comply with physics. And all of this was analyzed only from times when the warp drive was turned off; it does not depend on any assumptions about what happens when the warp is on.

(Zero mass warp drives, relaxing conservation requirements, and activation/deactivation)

(If angular momentum conservation is ignored, discuss how momentum & energy are affected by outside forces. In a gravitational field equivalent to inertial frame ... like being in an accelerated elevator; will acquire momentum buildup while staying "at rest".)

== Warp Railroads ==

Van Den Broeck<ref name="VanDenBroeck_2000"></ref> suggested a warp railroad,

You can set up devices ahead of time along the path of the bubble that produce the negative energy regions for the warp bubble at the appropriate time without needing the negative energy to ever move with the bubble at all. At least for bubbles on a predictable schedule and route.

helps with:

-stability (elaborate)

-conservation laws (elaborate)

Mention Krasnikov<ref name="Krasnikov 1998">S. V. Krasnikov, "Hyperfast travel in general relativity" Phys. Rev. D 57 4760</ref>

Hyperwave<ref name="Pieri 2024">Lorenzo Pieri, "Hyperwave: hyper-Fast Communication within General Relativity", arXiv:2311.12069 [gr-qc] https://doi.org/10.48550/arXiv.2311.12069</ref> makes use of this warp railroad idea.

== Quantum effects ==

(quantum stuff here<ref name="Hiscock1997">W. A. Hiscock, "Quantum effects in the Alcubierre warp drive spacetime", Classical and Quantum Gravity 14 L183 https://doi.org/10.1088/0264-9381/14/11/002 https://arxiv.org/abs/gr-qc/9707024</ref><ref name="Finazzi et al 2009">S. Finazzi, S. Liberati, C. Barceló, "Semiclassical instability of dynamical warp drives", Physical Review D 79, 124017 (2009)https://doi.org/10.1103/PhysRevD.79.124017 https://arxiv.org/abs/0904.0141</ref>)

From<ref name="Pieri 2024"></ref>
<blockquote>
It should be emphasised that Quantum Inequalities have not been proven for realistic interacting quantum fields, where the they are expected to be weaker if they exist [Cadamuro, 2019], and they often require Minkowskian conditions at infinity, a condition which is not appropriate for fields inside bounded regions, as in a Casimir devices.
</blockquote>

== Warp drives and black holes ==

What happens if you drive your warp drive into a black hole?

The pop-sci reason that nothing can escape a black hole is that at the event horizon the escape velocity is faster than light and nothing can go faster than light. A warp drive can go faster than light.

The more detailed reason from general relativity is that at the event horizon space and time are rotated sufficiently that "inward" becomes "forward in time." You can no more go farther away from the singularity in a black hole once you are inside the event horizon than you can go backward in time. Any super-luminal travel can go backward in time, and warp drives can engage in super-luminal travel.

The Hawking area theorem that says that black holes can only grow, never shrink relies on the condition that the energy is everywhere positive. Warp drives have negative energy density.

So in principle, there's nothing that prevents a warp drive from taking a dip into a black hole and coming back out again.

Garattini and Zatrimaylov looked into the problem of a warp drive near (and in) a black hole<ref name="Garattini and Zatrimaylov">R. Garattini and K. Zatrimaylov, "Black holes, warp drives, and energy conditions", Physics Letters B 856 138910 (2024) https://doi.org/10.1016/j.physletb.2024.138910</ref>. The found that the black hole reduces the negative energy requirement for the warp bubble to move inward toward the black hole singularity, but increases the negative energy needed to move away from the center of the black hole. In addition, a warp drive parked at the event horizon would allow light from inside the horizon to pass through the bubble and escape back outside.

== Credit ==
Author: Luke Campbell

== References ==

[[Category:Physics]][[Category:Infrastructure]][[Category:Physics & Engineering‏‎]][[Category:Physics & Math & Engineering]][[Category:Transportation & Infrastructure‏‎]][[Category:Metric Engineering]]

Warp Drives

2026-07-30T04:15:50Z

Lwcamp: /* Conservation laws */

{{PageConstructionNotice}}

Science fiction often features spacecraft that can seemingly move across space and get between the place of departure and the destination much faster than light could have done. This appears to contradict the theory of relativity, which predicts unequivocally that nothing can move through space faster than light. Because relativity has been incredibly successful at describing nature, with its many other predictions regularly being confirmed to extraordinary accuracy and within the bounds of uncertainty of all the experiments that tested them, it gives confidence that relativity is a correct description of reality. Which seems to rather throw a wet towel on our hopes for rapid travel between stars.

However, while relativity does not allow things to move through space faster than light, it places no such restrictions on how fast space-time itself can expand, contract, or move around. This leads to the idea of a warp drive – the spacecraft remains stationary within a region of highly curved space-time, and that region moves at super-luminal speeds rather than the spacecraft.

== The Alcubierre warp drive ==

The first warp drive geometry that satisfied the Einstein field equations of relativity was proposed by Miguel Alcubierre<ref>M. Alcubierre, "The warp drive: hyper-fast travel within general relativity." Classical and Quantum Gravity. 11 (5): L73–L77 (1994). [https://arxiv.org/abs/gr-qc/0009013 arXiv:gr-qc/0009013]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1994CQGra..11L..73A 1994CQGra..11L..73A]. doi:[https://doi.org/10.1088%2F0264-9381%2F11%2F5%2F001 10.1088/0264-9381/11/5/001]. S2CID [https://api.semanticscholar.org/CorpusID:4797900 4797900].</ref>. In this geometry, a sphere of space-time moves at an arbitrary speed (potentially but not necessarily a speed much faster than light). Objects within the sphere are moved along with the sphere; an object at rest within the sphere would be moved along with the sphere indefinitely. Space is expanding at the rear boundary and contracting at the front boundary in order to keep the sphere moving. In order to satisfy the Einstein field equations, the boundary of the sphere must have a negative energy density. The challenges of space-time geometries with negative energy densities are described in our page on [[Wormholes#Exotic_energy_conditions|wormholes]], for our purposes it is enough to note that negative energy densities can pose problems if not handled carefully, there may be limits on how much negative energy you can have without nearby positive energy density, and it may not be possible to get enough negative energy to support a warp drive; although none of this is ruled out by physics – yet!

<table align=center>
<tr><td width=650>[[File:Energy_magnitude_of_Alcubierre_drive.png]]
<td width=650>[[File:Expansion_of_Alcubierre_drive.png]]
<tr><td>The magnitude of the energy of the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. (The energy density is negative everywhere in the shell, here only the magnitude is shown to aid visual comprehension.)
<td>The expansion of space in the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. Negative expansion means that space is contracting in that direction, positive expansion that it is expanding.
</table>

NOTE: What the? The energy distribution is symmetric! How does the drive know which way to go? There must be significant contributions of other components of the stress-energy tensor, and those have got to be asymmetric along the forward/backward axis. Check this when I get time.

(And what's with the ring in all the artwork, anyway?)

=== So what is going on? ===

What is it that makes a warp drive a warp drive, and how does it work?

A warp drive is a concept that comes out of the science of general relativity. General relativity collects space and time together into a single "structure" called spacetime. This structure is curved, and its curvature is determined by the material that is in it. Absent forces acting between matter, particles of matter move through spacetime on the shortest possible paths in this curved environment.
<blockquote>
"Spacetime tells matter how to move; matter tells spacetime how to curve." - John Archibald Wheeler
</blockquote>
All of this is encompassed in the Einstein field equation:
<div align=center>
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 8πG
<td rowspan="2" nowrap="nowrap" align="right"> T = R -  
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right"> R g
<tr>
<td style="border-top:solid 1px black;"> c4
<td style="border-top:solid 1px black;"> 2
</table>
</div>
Unfortunately, by convention, both the symbols G and R get used for a lot of different things. Here, symbols in bold are second rank tensors – constructs of arrays of 4×4 numbers where 4 is necessary because that is the number of dimensions of spacetime. Conventionally, these numbers are indexed by a value going from 0 to 3, with 0 being the part corresponding to the coordinate of time and 1 to 3 being the Cartesian coordinates that you might know as x, y, and z.

T is the stress-energy tensor: the (0,0) component represents the energy density at a given point; the (0,i) and (i, 0) components (with i going from 1 to 3) represent the momentum density; and the (i,j) components represent the stresses present, such as pressure or tension or shear.

The other term, R - R g/2, involves a partial description of the curvature. A full description of the curvature requires a construct of 4×4×4×4 components; too much for the stress energy tensor to fully constrain. Which is okay, because this allows curvature to exist away from where matter is, allowing gravity to have long range effects across empty space as well as for phenomena like gravitational waves. But it is enough that a given distribution of matter T gives enough constraints (along with specifying how the geometry behaves far away) that you can generally figure out what the actual gravity is.

This is the usual story that we are told. But there is an alternate way of looking at the same geometry such that the math works out the same, but it is described conceptually in a very different way. This divides spacetime up into "layers", called foliations, that represent slices of space and going from one foliation to the next is increasing in time. This is useful for computation because you can assign coordinates to your system. There are two additional constructs – the lapse function and the shift vector field. Starting at an event on one foliation, the lapse function at that position tells you that after a certain "proper time" (time perceived by an observer at that event) the time coordinate of the observer will be the proper time times the lapse function at that event. And the new spatial coordinates of the observer will be the proper time times the shift vector at that event. (Strictly speaking, this is only exactly true in the limit of the proper time approaching zero or when the lapse function and shift vectors are constant. But hopefully this gives a good enough insight to what these entities are so that we can go on to describe warp drives.)

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Lapse_shift.png]]
<tr><td>A folliation of space-time into slices of constant time coordinates (grey). At an event (black circle), the lapse function (yellow) shows how far into the future the point at that event will be projected in a unit amount of proper time experienced by an observer initially at the event. The shift vector (red) shows how that point changes coordinates after the unit amount of proper time. The cyan vector shows the new coordinates the point is projected onto. The lower black surface indicates a foliation of spacetime at the same time coordinate; after unit proper time, the events on the lower foliation get projected to the curved upper black foliation according to the shift vectors and lapse functions at each of their respective points.
</table>
</table>

Warp drives are generally restricted to spacetime geometries with a constant lapse function - all observers go forward by the same time coordinate after experiencing the same amount of proper time. Thus, all warp drives have no time dilation.
Far from the warp bubble, the shift vector vanishes – each spatial coordinate is mapped to the same spatial coordinate after any amount of proper time so long as the warp drive is not near. But the inside of the warp bubble is an extended region where the shift vector field is non-zero and constant. In this bubble, all spatial coordinates are mapped to a coordinate that shifts linearly in time with the proper time of the observer. Astute readers will see that this linear change in position with time is very similar to a velocity. This region of constant shift vector then moves with the shift vector velocity to keep stuff originally in the bubble so that it stays in the bubble. In between the inside of the bubble and the far away regions with zero shift vector, the shift vector field falls away to match one region to the other.

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Warp_shift.png]]
<tr><td>The shift vector of a one-dimensional slice through warp geometry, in red. At the edges there is no shift. At the center (inside the warp bubble) there is a region of uniform shift so that at a later time the inside of the warp bubble is rigidly translated to different spatial coordinates (in blue). The two foliations remain the same distance apart, because warp drives maintain a unit lapse function.
</table>
</table>

Traditionally, scientists would start with an interesting distribution of matter given by T and then find out what kind of curvature it created. This works well for planets and stars and black holes and cosmic strings and other things which are observed in the universe and you want to see what sort of effect they had. But starting in the 1980's, some physicists started looking at Einstein's field equations the other way around. They start with a desired geometry, find the curvature part of the field equation R - R g/2, and use that to find out what T has to be in order to get that curvature. This was originally done for [[Wormholes|wormholes]], but in 1994 Miguel Alcubierre used this same trick to figure out what could create a region of constant shift vector that moved with the shift vector - the warp drive. With this demand for a warp geometry, the required distribution of matter and energy and momentum and stress can be determined.

The original Alcubierre shift vector N→ produces a spherical warp bubble with an axi-symmetric distribution of, well, everything along the direction of motion with warp velocity vector v→.
<div align="center">
N→ = v→ f(rs)
</div>
Here, rs is the distance from the center of the warp bubble, and f(rs) can be any function that is 1 near rs = 0 and is 0 for rs well beyond the radius of the warp bubble.

=== Challenges and possible resolutions ===

Almost as soon as Alcubierre proposed his warp drive, others began picking it apart.
<ol>
<li>Pfenning and Ford<ref name="PfenningFord_2001">M. J. Pfenning and L. H. Ford, "The unphysical nature of 'Warp Drive'", Classical and Quantum Gravity. 14 (7): 1743–1751 (1997). arXiv:[https://arxiv.org/abs/gr-qc/9702026 gr-qc/9702026]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1997CQGra..14.1743P 1997CQGra..14.1743P]. doi:[https://doi.org/10.1088%2F0264-9381%2F14%2F7%2F011 10.1088/0264-9381/14/7/011]. S2CID [https://api.semanticscholar.org/CorpusID:15279207 15279207].</ref> found that if the negative energy density wall around the bubble obeys [[Wormholes#Quantum_energy_inequalities|quantum energy inequalities]], then for warp speeds of around light speed the shell thickness would be on the order of a hundred Planck lengths; a distance far smaller than any other known physical phenomenon or object.
<li>In the same work, Pfenning and Ford also found that for a bubble a hundred meters in radius, a warp speed of around the speed of light, and the required thickness of about 100 Planck lengths the energy in the bubble shell would be E ≈ -1063 kg c2. The magnitude of this latter value is ten orders of magnitude larger than the energy of the entire visible universe. They do note, however, that if quantum energy inequalities can be ignored then a warp drive with a hundred meters radius but a shell thickness of one meter would "only" have an energy magnitude of about a quarter solar mass.
<li>As with any method of faster than light travel, the warp drive could be used to make a time machine.
<li>Perhaps most seriously, if the warp drive is going faster than light speed, the negative energy regions on the outside of the shell won't be in the warp parts of the bubble. They will have to be moving through space at faster than light speed if the bubble is to maintain its integrity, which is the very problem that the warp drive was designed to avoid. The outside of the warp bubble shell that maintains the warp bubble would fall away and the warp bubble would quickly erode to sub-luminal speeds.<ref name="Van Den Broeck 2000">Chris Van Den Broeck, "On the (im)possibility of warp bubbles", arXiv:gr-qc/9906050 https://doi.org/10.48550/arXiv.gr-qc/9906050</ref>
<li>A more general problem that includes the previous one is that the geometry of the warp drive determines the distribution of energy, momentum, and stress (pressure, shear, and tension) by Einstein's field equation, which states that the stress energy tensor is directly proportional to a reduced order curvature tensor of space-time. However, the stress-energy tensor must also obey the continuity equation – basically, for the energy to change in a given volume of space that energy must come in or go out of the boundary of that volume and similarly for the momentum. So a warp drive can have the warp condition for one moment in time. But after that, the stress-energy that maintains the warp will start to move according to the forces on it and according to the natural paths that it takes through the warped space-time. There is no guarantee that the configuration of the stress-energy at later times will be sufficient to maintain the warp.
<blockquote>
From the perspective of simulating the warp drive dynamically, the key challenge is stability. One can straightforwardly generate the initial metric that describes the Alcubierre metric and infer the instantaneous matter configuration that must support it using the Einstein Equation, but a time-domain evolution of the coupled matter-gravity system necessitates that we specify the equation of state of the matter, which in turn determines how it evolves. In a perfect fluid, this involves specifying the relation between the fluid pressure and density, but in a more general case, the full relation between the
stress-energy components must be given. There is (to our knowledge) no known equation of state that would maintain the warp drive metric in a stable configuration over time - therefore, whilst one can require that initially, the warp bubble is constant, it will quickly evolve away from that state and, in most cases, the warp fluid and spacetime deformations will disperse or collapse into a central point.
 
– Clough, Dietrich, and Khan<ref>Katy Clough, Tim Dietrich, and Sebastian Khan, "What no one has seen before: gravitational waveforms from warp drive collapse", arXiv:2406.02466 [gr-qc] https://doi.org/10.48550/arXiv.2406.02466</ref>
</blockquote>
<li>In order to get the matter (or more generally, stress-energy) that forms the warp shell up to speeds that could allow the shell to exist, you need to be able to, well, accelerate that matter to warp speeds. Which gives a chicken-and-egg problem - in order to achieve superluminal travel you need to be able to get matter or energy moving superluminally in the first place<ref name="Bobrick and Martire">Alexey Bobrick and Gianni Martire, "Introducing Physical Warp Drives", arXiv:2102.06824 [gr-qc] https://doi.org/10.48550/arXiv.2102.06824</ref><ref>D. H. Coule, "No warp drive", Class. Quantum Grav. 15 (1998) 2523–2527. https://doi.org/10.1088/0264-9381/15/8/026</ref>, which rather makes warp drives unnecessary from the get-go.
<li>A warp bubble moving faster than light will not be able to get any signals in front of it as it will overtake its own signal. In fact, anyone in the interior of the bubble would not be able to send signals to the front surface of the bubble. In science-speak, a horizon forms at the front of the bubble<ref name="Natario">José Natário, "Warp drive with zero expansion", Classical and Quantum Gravity. 19 (6): 1157–1166 (2002). arXiv:[https://arxiv.org/abs/gr-qc/0110086 gr-qc/0110086]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2002CQGra..19.1157N 2002CQGra..19.1157N]. doi:[https://doi.org/10.1088%2F0264-9381%2F19%2F6%2F308 10.1088/0264-9381/19/6/308]. S2CID [https://api.semanticscholar.org/CorpusID:15859984 15859984].</ref>. As a consequence, the bubble cannot be controlled, steered, nor stopped from within the bubble itself.
<li>A Horizon will naturally emit Hawking radiation. It is estimated that the radiation will raise the temperature of the interior of the bubble to approximately 1032 K<ref name="Finazzi 2009">Stefano Finazzi, Stefano Liberati, and Carlos Barceló, "Semiclassical instability of dynamical warp drives", Phys. Rev. D 79, 124017 (12 June, 2009) https://doi.org/10.1103/PhysRevD.79.124017 , arXiv:0904.0141 [gr-qc] https://doi.org/10.48550/arXiv.0904.0141</ref> (at this level of accuracy, the result is the same if you replace temperatures in K by the same number in °C or °F). This temperature, however, may be lower if quantum inequalities do not have to be fulfilled.
<li>Semi-classical instabilities at the horizon are expected to lead to a divergence of the stress-energy tensor at the boundary in a short period of time, thus leading to the destruction of the warp bubble<ref name="Finazzi 2009"></ref>.
</ol>
Helpfully, Van Den Broeck<ref name="VanDenBroeck_2000">C. Van Den Broeck, "Alcubierre’s Warp Drive: Problems and Prospects" AIP Conference Proceedings. 504: 1105–1110 (2000). Bibcode:[https://ui.adsabs.harvard.edu/abs/2000AIPC..504.1105V 2000AIPC..504.1105V]. doi:[https://doi.org/10.1063%2F1.1290913 10.1063/1.1290913].</ref> suggested several solutions – or at least mitigations – for some of these problems
<ol>
<li>Quantum inequalities had not been shown to be true in general for highly curved space-times.
<li>A proposed warp drive geometry<ref name="VanDenBroeck_1999">C. Van Den Broeck, "A 'warp drive' with more reasonable total energy requirements". Classical and Quantum Gravity. 16 (12): 3973–3979 (1999). arXiv:[https://arxiv.org/abs/gr-qc/9905084 gr-qc/9905084]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1999CQGra..16.3973V 1999CQGra..16.3973V]. doi:[https://doi.org/10.1088%2F0264-9381%2F16%2F12%2F314 10.1088/0264-9381/16/12/314]. S2CID [https://api.semanticscholar.org/CorpusID:15466313 15466313]. </ref> (see the van Den Broeck drive, below) would be able to minimize the magnitude of the energy required to about that of the mass of our sun. While still large, it is not unphysically large.
<li>Time travel is always going to be a worry with faster than light travel. There's no neat solution to this one.
<li>It may be possible for the negative energy outside of the superluminal region to compress into the superluminal region to form a shock in space-time. The front surface of the bubble would then be a singularity. It is not clear if such a sharp jump in physical properties is possible, but neither is sure that it is impossible, either. However, while this might help with the negative energy in front of the bubble getting swept up because it cannot move faster than light, it does little to help with the material at the back of the bubble on the outside getting left behind because it cannot keep up.
</ol>
Other mitigating research has also shown
<ol start=9>
<li>Semi-classical divergence only occurs in warp drives with only one spatial dimensions. Those with two or more spatial dimensions (as in our universe) only contain points where divergences occur, and these divergences can be managed. In fact, their effects can be reduced by making the warp shape more elongated<ref>Carlos Barceló, Valentin Boyanov, Luis J. Garay, Eduardo Martín-Martínez,
and Jose M. Sánchez Velázquez, "Warp drive aerodynamics", JHEP08 (2022) 288 https://doi.org/10.1007/JHEP08(2022)288 , arXiv:2207.06458 [gr-qc] https://doi.org/10.48550/arXiv.2207.06458</ref>.
</ol>

=== Alcubierre warp interactions with light and matter ===

Space is not entirely empty. It is filled with a diffuse plasma in the form of the interstellar medium, as well as cosmic radiation, light from stars, and cosmic microwave background radiation. A warp drive propagating through space will encounter this stuff. What happens when this matter and radiation have a warp bubble pass across them?

The first analysis of matter encountering a warp bubble was performed by Pfenning and Ford<ref name="PfenningFord_2001"></ref>. They looked at the warp bubble interaction with a massive object at rest with the frame of reference of the warp drive (keep in mind that the rest frame is the same inside and outside the warp bubble; in this rest frame objects inside the bubble have no momentum even though they are, in some sense, changing location rapidly with time). When the warp bubble passes, the object experiences an acceleration in the direction of the warp bubble motion. When the warp shell passes and the object is inside the bubble, it will be moving with approximately the speed of the bubble. Pfenning and Ford analyzed this problem with a continuous distribution of shell energy that, strictly speaking, never falls to zero except at the bubble center and at spatial infinity, so unless the object passes through the center in Pfenning and Ford's description it will never quite get up to the bubble's speed. In this case, the object will move almost as fast as the object but will pass through the bubble in a finite time, after which it will again be at rest with respect to the reference frame but displaced along the direction of the warp bubble motion by some distance. With this description of the warp bubble, a spacecraft of finite size will always be moving a little slower than the warp bubble and would have to use rockets to keep up with it. in addition, the spacecraft would experience tidal forces that would cause stress on the spacecraft's structure.

Pfenning and Ford also examined cases where the shell is of a finite (possibly infinitesimal) width and falls to zero both inside and outside the bubble. In this case any matter encountering the bubble would be collected at the bow of the bubble and thereafter move along with it.

McMonigal et al. analyzed the situation for both massive particles and light moving along the axis of travel of the bubble<ref name="McMonigalEtAl_2012">B. McMonigal, G. F. Lewis, and P. O'Byrne, "Alcubierre warp drive: On the matter of matter". Physical Review D. 85 (6) 064024 (20 March 2012). arXiv:[https://arxiv.org/abs/1202.5708 1202.5708]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2012PhRvD..85f4024M 2012PhRvD..85f4024M]. doi:[https://doi.org/10.1103%2FPhysRevD.85.064024 10.1103/PhysRevD.85.064024]. S2CID [https://api.semanticscholar.org/CorpusID:3993148 3993148].</ref>. They found that light moving opposite the warp direction passed through the bubble without incident, being only somewhat delayed by passing through the bubble. A warp bubble that is warping at sub-luminal speeds can have light catch up from behind it. This light is able to pass through the bubble, and is somewhat advanced in its path by the speed of the bubble. For super-luminal warp bubbles, however, the situation is different. The bubble will catch up to light moving it its own direction that is originally in front of it. This light cannot escape forward, the bubble being too fast. Nor can it escape backward, as the light is propagating forward and the interior of the bubble is at rest. Thus, the light gets caught at the bow of the warp bubble, unable to escape for so long as the warp bubble is active. This light is strongly blue shifted to extremely energetic x-rays and gamma rays. The space behind the bubble is swept clear of forward-moving light.

The situation for matter over-run by a super-luminal warp bubble is similar. Matter moving backward passes through the bubble, being only swept a ways along the bubble's path as sign of its passing but otherwise continuing on their way. Matter at rest behaves the same way as Pfenning and Ford discovered. Meanwhile, matter moving in the same direction of the warp is overtaken and collects at the bow of the bubble, unable to leave for so long as the bubble is warping. This matter "surfing" on the bubble bow is highly accelerated to relativistic speeds, experiencing extreme time dilation.

For a sub-luminal bubble, there is a speed faster than the warp speed where particles coming up from behind overtake the bubble and pass through, out the front. Particles below this speed but still faster than the warp speed still overtake the bubble, but then bounce backwards out of the bubble from behind. They are still moving forward, but are now moving more slowly than the warp speed. Particles moving forward but slower than the bubble will be overtaken and then bounced out the front with a speed higher than the warp speed. Those particles moving backward will pass through the bubble and exit with their initial speed.

The spacecraft inside the bubble will observe that light moving backward compared to the warp direction is blue-shifted, and matter moving backward passes through with increased energy. Meanwhile, a sub-luminal bubble will see light that was moving forward as red-shifted and matter catching up to the bubble from behind will pass by with reduced energy (superluminal bubbles, of course, do not have any matter or radiation catching up to them from behind).

When a superluminal warp bubble that has been collecting matter and radiation for the duration of its journey and blue-shifting it to much higher energies is turned off, all that matter and energy is released as a blast of radiation in the direction the bubble was warping.

McMonigal et al. conclude by noting that any spacecraft in the warp bubble would need shielding to protect against the blue-shifted radiation and matter of increased energy. In addition the destination would be "blasted into oblivion" by the release of matter and radiation that had been caught in the bubble during the trip.

Clark et al.<ref>Chad Clark, William A. Hiscock, and Shane L. Larson, "Null geodesics in the Alcubierre warp drive spacetime: the view from the bridge", arXiv:gr-qc/9907019 (2018) https://doi.org/10.48550/arXiv.gr-qc/9907019</ref> examine what it would look like from within an operating warp drive. They find that light moving at 90 degrees to the direction of motion will look like it is still coming from 90 degrees. However, light from the front hemisphere is squished progressively toward the forward point the faster you go. For a warp bubble moving faster than the speed of light, there is a cone behind the warp bubble where no light can reach it. However, you don't see a black spot behind you – light from the regions in the backward hemisphere from outside that cone that can reach you is distorted toward the direction behind you to fill in that spot. Light from the forward direction is blue-shifted and light from behind is red-shifted. At speeds of around 200 times light speed, the cosmic microwave background would be blue shifted to energies similar to that of the solar photosphere, heating anything in the warp bubble as if they were close to the sun.

== Other warp geometries ==

The Alcubierre warp drive is the first and best known of the warp drives. It has been around long enough that there have been many studies done on it. The warp geometry is particularly simple, making analysis easier. But in the time that the Alcubierre warp was proposed, many other researchers have come up with other warp designs, and even generalized the idea to include a wide array of additional possibilities.

Most of the general statements about the challenges of the Alcubierre drive hold for these other drives. Many of the details about interactions with matter are also likely to hold, although these other styles of warp drive have not yet been subject to such rigorous investigation.

=== Van Den Broeck warp drive ===

If an Alcubierre warp bubble a hundred meters across requires a magnitude of energy greater than the entire energy in the observable universe, one option to reduce the magnitude of energy used is to make the warp bubble smaller. Much smaller. Pfenning and Ford<ref name="PfenningFord_2001"></ref> suggested making the warp bubble smaller than an atom. Of course, that brings up the problem of how to stuff a spacecraft in there. Van Den Broeck proposed a solution<ref name="VanDenBroeck_1999"></ref>: make the warp bubble only about the size of an atomic nucleus but expand the space inside the bubble enormously so that a spacecraft could fit in. This is somewhat like a nuclear sized wormhole that leads to a pocket universe that holds the spacecraft. Unfortunately, the metric doesn't fit neatly into any [[Wormholes#Three_dimensions|embedding diagram]] that I can figure out, but the math works out so that spatial coordinates inside the bubble are expanded by a factor of 1017 and a spacecraft would have a few hundred meters of bubble interior to putz around in. This trick manages to reduce the magnitude of energy to only about the mass energy of a few stars similar to our own. Other than that, it is otherwise a normal Alcubierre drive.

Further work<ref>S. Krasnikov, "The quantum inequalities do not forbid spacetime shortcuts", Physical Review D. 67 (10) 104013. https://doi.org/10.1103%2FPhysRevD.67.104013, arXiv:gr-qc/0207057 (2003) https://doi.org/10.48550/arXiv.gr-qc/0207057 </ref> has managed to use Van Den Broeck's trick to reduce the magnitude of the negative energy to only about a milligram.

=== Natário warp drive ===

The Alcubierre drive is not the only way to construct a warp drive. José Natário<ref name="Natario"></ref> showed that it was just one example of an entire class of warp drives. These are space-time geometries with a smooth and localized shift vector field and a unit lapse function.

Natário then went on to find in that class a set of warp drives that do not have any expansion or contraction of space at all. Instead, space encountering the front boundary of the warp bubble instead "slides" around the outside of the bubble until it gets to the corresponding place on the back and is left behind there. Another way to think of it is that space entering the bubble shell is compressed in the radial direction of the shell but is simultaneously expanded in the tangential direction so that there is no net change in volume. When it gets to the back, the opposite occurs. In the language of mathematics, the shift vector field is divergenceless.

<table align=center>
<tr><td width=360>[[File:Natario_drive_flow_of_space.png]]
<td width=445>[[File:Natario_warp_drive_horizons.png]]
<tr><td style="vertical-align: top;">A representation of the motion of space in a zero-expansion Natário drive with a vanishingly thin warp shell around the bubble.
<td>The horizons of the generalized Natário class of warp drives (including the Alcubierre warp drive and all fell-Heisenberg drives) at superluminal speed in the direction of the green arrow. No event inside the bubble can affect events outside the bubble in front of the blue line. No events outside the bubble behind the red line can affect events inside the bubble. The lines have an angle α with respect to the direction of motion with sin(α) = 1/vs for vs the speed of the bubble relative to light speed. This is analogous to the Mach cone of supersonic objects.
</table>

NOTE: There are two different things that can be meant by the term "Natário warp drive". The generalized Natário warp drive is the general theory of all warp drives that includes within it the Alcubierre, Fell-Heisenberg, and Lentz warp drives, among infinitely many others. The zero-expansion Natário warp drive is the specific implementation of the generalized Natário warp drive that has a divergenceless shift vector field.

The generalized class of warp drives developed by Natário (including both the Alcubierre and this zero-expansion warp drive) are shown to always have regions in the warp shell where the energy density is negative to at least some observers. You can't get away from it – to warp, you need negative energy density.

The Natário class of warp drives blueshift light coming in from the front and redshift light catching up from the back. For the case of a super-luminal warp bubble, there will of course be a horizon at the back of the bubble and light will not be able to get through from behind. For a bubble speed vs relative to the speed of light, light coming from straight ahead will be blueshifted up in frequency and photon energy by a factor of 1 + vs. For sub-luminal travel, light coming from behind will be redshifted by a factor 1 - vs. In general for n as the unit direction along which the light is propagating, the blueshift factor will be 1 + vs ⋅ n. If the warp shell is infinitesimally thin, light from outside that reaches the center of the bubble will not be distorted in direction although it will be frequency shifted, but observers still inside the bubble but displaced from the center will see the field of view distorted as well as frequency shifted.

Most of the difficulties that apply to the Alcubierre warp drive also apply to the generalized Natário warp drive (and thus to the zero-expansion Natário drive and Fell-Heisenberg drive as well). In particular, quantum energy inequalities, high magnitudes of the energy required, formation of time machines, instabilities related to the stress-energy making up the warp shell being unable to keep up with the bubble, the chicken-and-egg problem of needing FTL to achieve FTL, the formation of horizons making warp bubbles uncontrollable, and Hawking radiation at the horizons, are all still concerns for generic warp drives.

=== Fell-Heisenberg warp drives ===

S. D. B. Fell and L. Heisenberg investigated what sort of warp structure is necessary to produce a warp drive with positive energy density everywhere<ref name="Fell-Heisenberg">S. D. B. Fell and L> Heisenberg, "Positive energy warp drive from hidden geometric structures", Classical and Quantum Gravity 38 155020 (2021) https://doi.org/10.1088/1361-6382/ac0e47 https://arxiv.org/abs/2104.06488</ref>. They found that warp drives of the generic Natário form with irrotational shift vectors had a simple expression for the energy density and vanishing momentum (irrotational means that the shift vector can be described as the gradient of a scalar potential field or, equivalently, that the curl of the shift vector field vanishes everywhere). Fell and Heisenberg proposed several potential fields that either had positive energy density everywhere except at discontinuities, or that were continuous but had regions that violated the weak energy condition (i. e. negative energy densities in some frames of reference). One specific example they provided of a warp field had an energy that is about 10,000 times less than the mass-energy of our sun. Or only about half the mass-energy of Jupiter. This is a significant improvement over many other proposed drives.

Despite the introduction discussing warp drive configurations that satisfy the various energy conditions, the configurations described in the paper are shown to locally violate the weak and strong energy conditions. Nonetheless, the energy density is still mostly positive. Santiago, Schuster, and Visser's<ref name="Santiago Schuster Visser 2022">J. Santiago, S. Schuster, and M. Visser, "Generic warp drives violate the null energy condition", Physical Review D 105, 064038 (2022) https://doi.org/10.1103/PhysRevD.105.064038</ref> work shows that the various energy conditions must still be violated by this warp drive; to not violate these, energy density must be positive in all reference frames not just those of the co-moving observer.

Any vector field can be described by its divergence, its curl, and a constant. The constant part of a warp drive's shift vector can be discarded because then the warp drive would not be localized. It is interesting, therefore, that the Natário zero-expansion drive has zero divergence and can be described only by its curl, while the Fell-Heisenberg drive has zero curl and can be described only by its divergence. These two classes of drives thus rely on opposite choices of setting one of the two different parts describing a vector field to zero. As might be expected, the Fell-Heisenberg drive is not zero expansion – space-time will change in volume as it passes through the warp bubble (although after going all the way through it will return to its original volume). However, objects in space-time will retain their original orientation as they pass through the warp bubble.

Any vector field with zero curl can be represented as the gradient of a suitable scalar function. Thus, the Fell-Heisenberg drive can be completely described by just scalar field (one number at all points of space and time) rather than a vector field.

Previous examples of warp drives, such as the Alcubierre and zero-expansion Natário drives, tend toward a flat space-time as you go far enough away faster than would occur if there were any mass present in the warp drive. That is, the ADM mass of such drives is zero. However, one of the Fell-Heisenberg examples described a spacetime where the geometry approaches that of what you would get for a massive object at the warp location, giving one of the first examples of a warp drive with mass.

In the Fell-Heisenberg drive the momentum vanishes everywhere. Yet the energy occupies regions where the shift vector is varying rapidly. The lack of momentum means that the energy will not move to keep up with the differential expansion and movement of the spacetime. As a consequence, at later times the energy will have a different distribution than what is necessary to maintain the given warp configuration. The exact time evolution is not solved but this likely leads to collapse of warp bubble. Note that this is not a problem unique to the Fell-Heisenberg drive. Assuming a geometry that describes a warp produces a specific distribution of the stress-energy tensor that satisfies Einstein's equation. But there is no guarantee that this stress energy tensor satisfies the continuity equation, which describes how the distribution of matter and momentum change with time. The issue of stability of any warp drive is not well understood, but most warp drive geometries will be unstable in the sense that their matter that generates the warp fields will not be able to keep up with the motion of the warp. The Fell-Heisenberg drive is unique only in that, with vanishing momentum, there is no configuration that allows stable propagation.

(n. b. The Heisenberg here is Lavinia Heisenberg, not the Werner Heisenberg of quantum physics and uncertainty principle fame.)

=== Lentz warp drive ===

E. Lentz<ref name="Lentz">E. W. Lentz, "Breaking the warp barrier: hyper-fast solitons in Einstein–Maxwell-plasma theory", Classical and Quantum Gravity. 38 075015 (2021). arXiv:[https://arxiv.org/abs/2006.07125 2006.07125]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2021CQGra..38g5015L 2021CQGra..38g5015L]. doi:[https://doi.org/10.1088%2F1361-6382%2Fabe692 10.1088/1361-6382/abe692]. ISSN [https://search.worldcat.org/issn/0264-9381 0264-9381]. S2CID [https://api.semanticscholar.org/CorpusID:219635854 219635854].</ref> investigated a particular configuration of the Fell-Heisenberg class of warp drives, where the scalar field forms a selection of rhomboid regions, resulting in an energy density that is everywhere positive and a shift vector with vanishing tidal force at the center for a comfortable ride.

However, Santiago, Schuster, and Visser<ref name="Santiago Schuster Visser 2022"></ref> dispute claims that the Lentz drive satisfies the energy conditions, noting that everywhere positive energy density in one frame of reference is insufficient to establish that the energy density is positive in all reference frames; knowledge of the Cauchy stress tensor is also needed. They show that any generic warp drive will violate the strong energy condition, null energy condition, and weak energy condition.

== Conservation laws ==

All the warp drives investigated so far have geometries that quickly fall off from the highly curved region near the warp bubble to a region of spacetime that is almost flat; and becomes even more flat the farther away from the warp bubble you get. There is a word for geometries of this kind – they are called asymptotically flat. Basically, you can always go far enough away in any direction to reach a place where the geometry is sufficiently flat to meet any flatness criterion you choose.

When you have an asymptotically flat geometry, then in the far away mostly flat regions gravity is well described by a linearized theory of gravity – where the perturbations to the geometry are linear in the stress energy tensor. This means that gravity is basically Newtonian, along with a few other bells and whistles like gravitational waves and frame dragging.

When linearized gravity is a good approximation, then it is known that energy, momentum, and linear momentum are conserved. Although parts of a warp geometry are highly curved and non-linear, you can measure the conserved energy and momentum and angular momentum from far away in the linear region. This ADM energy and etcetera is the quantity that is conserved when you are worrying about conserving such things, not necessarily the energy and so forth you get in the highly curved spacetime region. In fact, it is not even simple to define where the energy and such actually is withing the highly curved spacetime region; it is non-local and different observers will disagree on how it is distributed. But everyone agrees on the ADM conserved quantities from out in nearly flat spacetime (up to the usual changes due to changing speeds, which are trivial and come up with normal objects as well).

And now we run into a bit of a conundrum: most of the warp geometries so far proposed have zero ADM mass<ref>Sebastian Schuster, Jessica Santiago,and Matt Visser, "ADM mass in warp drive spacetimes" General Relativity and Gravitation (2023) 55:14 https://doi.org/10.1007/s10714-022-03061-9</ref>, meaning that things like the Alcubierre drive technically have no energy or momentum or angular momentum at all. Despite all of the extremes of negative energy regions within the wall of the warp bubble, the actual energy that you need to spend to make one is zero. This can lead to a bit of a problem. If you have a spacecraft that has mass, that mass-energy has to be conserved. So when you turn on your warp drive and the total warp mass is zero, where does that energy go?

One resolution is just that the mass of the spacecraft is negligible and the small amount of curvature it causes can be ignored. In this case, the warp geometries discussed so far are merely an approximation that neglects this effect. In fact, some of the proposed Fell-Heisenberg drives map not onto entirely flat spacetime geometries at large distances, but onto the Schwarzschild geometry that is appropriate for systems with mass at the center. If this is the case, then the warping bubble and spacecraft system will need to worry about conservation of all their conserved properties.

Let's look at what the conservation laws mean for a spacecraft equipped with a warp drive, assuming that no matter or radiation are emitted by activating or deactivating the warp drive.

We can start with an observer floating in space. We'll call her Alice. And, approaching Alice at 100 m/s is the starship Enterprise with a mass of a cool 200,000 tons (2×108 kg). from this we know that the Enterprise has, in Alice's frame of reference, a kinetic energy of 1 TJ, a total energy (including mass and kinetic energy) of 1.8×1025 J, a momentum of 2×1010 kg m/s, and – because it is headed straight toward Alice – an angular momentum of zero. All of this is without any warp drive activated.

On the bridge of the Enterprise, Spock picks up Alice on the sensors and tells Captain Kirk. Kirk does not want to run into Alice (he'd rather date her), so he orders helm to turn on the Enterprise's warp drive and move them 1 km up so that their course takes them 1 km from Alice instead of straight into her. Sulu warps the Enterprise according to these commands, and then turns off the warp drive.

After this maneuver, keeping the Enterprise's total energy and momentum the same means that the Enterprise still has a mass of 200,000 tons and a velocity of 100 m/s in its original direction. So we can see that the warp drive must preserve the velocity vector of the thing in it (at least in flat spacetime where outside forces are not acting on it). The problem comes from angular momentum. With a known mass of 200,000 tons, a speed of 100 m/s, and a distance of closest approach of 1 km; from Alice's frame of reference the Enterprise now has an angular momentum of 2×1013 kg m2/s after the warp maneuver, while before the maneuver it has an angular momentum of zero.

From this we can see that warp drives cannot be reactionless. Arbitrary displacements using warp without emitting propellant result in violation of the law of conservation of angular momentum. As long as we restrict ourselves to accepted science based on what we know of general relativity, to move a warp drive with mass you need to accelerate it or decelerate it with rockets or other normal means of delivering momentum (such as light sails, mag sails, or pushing off of a planet). Warp drives must be inertial to comply with physics. And all of this was analyzed only from times when the warp drive was turned off; it does not depend on any assumptions about what happens when the warp is on.

(ADM mass -> 0; conservation of energy/mass issues.)

(Discuss issues of conservation of angular momentum.)

(If angular momentum conservation is ignored, discuss how momentum & energy are affected by outside forces. In a gravitational field equivalent to inertial frame ... like being in an accelerated elevator; will acquire momentum buildup while staying "at rest".)

== Warp Railroads ==

Van Den Broeck<ref name="VanDenBroeck_2000"></ref> suggested a warp railroad,

You can set up devices ahead of time along the path of the bubble that produce the negative energy regions for the warp bubble at the appropriate time without needing the negative energy to ever move with the bubble at all. At least for bubbles on a predictable schedule and route.

helps with:

-stability (elaborate)

-conservation laws (elaborate)

Mention Krasnikov<ref name="Krasnikov 1998">S. V. Krasnikov, "Hyperfast travel in general relativity" Phys. Rev. D 57 4760</ref>

Hyperwave<ref name="Pieri 2024">Lorenzo Pieri, "Hyperwave: hyper-Fast Communication within General Relativity", arXiv:2311.12069 [gr-qc] https://doi.org/10.48550/arXiv.2311.12069</ref> makes use of this warp railroad idea.

== Quantum effects ==

(quantum stuff here<ref name="Hiscock1997">W. A. Hiscock, "Quantum effects in the Alcubierre warp drive spacetime", Classical and Quantum Gravity 14 L183 https://doi.org/10.1088/0264-9381/14/11/002 https://arxiv.org/abs/gr-qc/9707024</ref><ref name="Finazzi et al 2009">S. Finazzi, S. Liberati, C. Barceló, "Semiclassical instability of dynamical warp drives", Physical Review D 79, 124017 (2009)https://doi.org/10.1103/PhysRevD.79.124017 https://arxiv.org/abs/0904.0141</ref>)

From<ref name="Pieri 2024"></ref>
<blockquote>
It should be emphasised that Quantum Inequalities have not been proven for realistic interacting quantum fields, where the they are expected to be weaker if they exist [Cadamuro, 2019], and they often require Minkowskian conditions at infinity, a condition which is not appropriate for fields inside bounded regions, as in a Casimir devices.
</blockquote>

== Warp drives and black holes ==

What happens if you drive your warp drive into a black hole?

The pop-sci reason that nothing can escape a black hole is that at the event horizon the escape velocity is faster than light and nothing can go faster than light. A warp drive can go faster than light.

The more detailed reason from general relativity is that at the event horizon space and time are rotated sufficiently that "inward" becomes "forward in time." You can no more go farther away from the singularity in a black hole once you are inside the event horizon than you can go backward in time. Any super-luminal travel can go backward in time, and warp drives can engage in super-luminal travel.

The Hawking area theorem that says that black holes can only grow, never shrink relies on the condition that the energy is everywhere positive. Warp drives have negative energy density.

So in principle, there's nothing that prevents a warp drive from taking a dip into a black hole and coming back out again.

Garattini and Zatrimaylov looked into the problem of a warp drive near (and in) a black hole<ref name="Garattini and Zatrimaylov">R. Garattini and K. Zatrimaylov, "Black holes, warp drives, and energy conditions", Physics Letters B 856 138910 (2024) https://doi.org/10.1016/j.physletb.2024.138910</ref>. The found that the black hole reduces the negative energy requirement for the warp bubble to move inward toward the black hole singularity, but increases the negative energy needed to move away from the center of the black hole. In addition, a warp drive parked at the event horizon would allow light from inside the horizon to pass through the bubble and escape back outside.

== Credit ==
Author: Luke Campbell

== References ==

[[Category:Physics]][[Category:Infrastructure]][[Category:Physics & Engineering‏‎]][[Category:Physics & Math & Engineering]][[Category:Transportation & Infrastructure‏‎]][[Category:Metric Engineering]]

Warp Drives

2026-07-30T03:50:33Z

Lwcamp: /* Conservation laws */

{{PageConstructionNotice}}

Science fiction often features spacecraft that can seemingly move across space and get between the place of departure and the destination much faster than light could have done. This appears to contradict the theory of relativity, which predicts unequivocally that nothing can move through space faster than light. Because relativity has been incredibly successful at describing nature, with its many other predictions regularly being confirmed to extraordinary accuracy and within the bounds of uncertainty of all the experiments that tested them, it gives confidence that relativity is a correct description of reality. Which seems to rather throw a wet towel on our hopes for rapid travel between stars.

However, while relativity does not allow things to move through space faster than light, it places no such restrictions on how fast space-time itself can expand, contract, or move around. This leads to the idea of a warp drive – the spacecraft remains stationary within a region of highly curved space-time, and that region moves at super-luminal speeds rather than the spacecraft.

== The Alcubierre warp drive ==

The first warp drive geometry that satisfied the Einstein field equations of relativity was proposed by Miguel Alcubierre<ref>M. Alcubierre, "The warp drive: hyper-fast travel within general relativity." Classical and Quantum Gravity. 11 (5): L73–L77 (1994). [https://arxiv.org/abs/gr-qc/0009013 arXiv:gr-qc/0009013]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1994CQGra..11L..73A 1994CQGra..11L..73A]. doi:[https://doi.org/10.1088%2F0264-9381%2F11%2F5%2F001 10.1088/0264-9381/11/5/001]. S2CID [https://api.semanticscholar.org/CorpusID:4797900 4797900].</ref>. In this geometry, a sphere of space-time moves at an arbitrary speed (potentially but not necessarily a speed much faster than light). Objects within the sphere are moved along with the sphere; an object at rest within the sphere would be moved along with the sphere indefinitely. Space is expanding at the rear boundary and contracting at the front boundary in order to keep the sphere moving. In order to satisfy the Einstein field equations, the boundary of the sphere must have a negative energy density. The challenges of space-time geometries with negative energy densities are described in our page on [[Wormholes#Exotic_energy_conditions|wormholes]], for our purposes it is enough to note that negative energy densities can pose problems if not handled carefully, there may be limits on how much negative energy you can have without nearby positive energy density, and it may not be possible to get enough negative energy to support a warp drive; although none of this is ruled out by physics – yet!

<table align=center>
<tr><td width=650>[[File:Energy_magnitude_of_Alcubierre_drive.png]]
<td width=650>[[File:Expansion_of_Alcubierre_drive.png]]
<tr><td>The magnitude of the energy of the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. (The energy density is negative everywhere in the shell, here only the magnitude is shown to aid visual comprehension.)
<td>The expansion of space in the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. Negative expansion means that space is contracting in that direction, positive expansion that it is expanding.
</table>

NOTE: What the? The energy distribution is symmetric! How does the drive know which way to go? There must be significant contributions of other components of the stress-energy tensor, and those have got to be asymmetric along the forward/backward axis. Check this when I get time.

(And what's with the ring in all the artwork, anyway?)

=== So what is going on? ===

What is it that makes a warp drive a warp drive, and how does it work?

A warp drive is a concept that comes out of the science of general relativity. General relativity collects space and time together into a single "structure" called spacetime. This structure is curved, and its curvature is determined by the material that is in it. Absent forces acting between matter, particles of matter move through spacetime on the shortest possible paths in this curved environment.
<blockquote>
"Spacetime tells matter how to move; matter tells spacetime how to curve." - John Archibald Wheeler
</blockquote>
All of this is encompassed in the Einstein field equation:
<div align=center>
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 8πG
<td rowspan="2" nowrap="nowrap" align="right"> T = R -  
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right"> R g
<tr>
<td style="border-top:solid 1px black;"> c4
<td style="border-top:solid 1px black;"> 2
</table>
</div>
Unfortunately, by convention, both the symbols G and R get used for a lot of different things. Here, symbols in bold are second rank tensors – constructs of arrays of 4×4 numbers where 4 is necessary because that is the number of dimensions of spacetime. Conventionally, these numbers are indexed by a value going from 0 to 3, with 0 being the part corresponding to the coordinate of time and 1 to 3 being the Cartesian coordinates that you might know as x, y, and z.

T is the stress-energy tensor: the (0,0) component represents the energy density at a given point; the (0,i) and (i, 0) components (with i going from 1 to 3) represent the momentum density; and the (i,j) components represent the stresses present, such as pressure or tension or shear.

The other term, R - R g/2, involves a partial description of the curvature. A full description of the curvature requires a construct of 4×4×4×4 components; too much for the stress energy tensor to fully constrain. Which is okay, because this allows curvature to exist away from where matter is, allowing gravity to have long range effects across empty space as well as for phenomena like gravitational waves. But it is enough that a given distribution of matter T gives enough constraints (along with specifying how the geometry behaves far away) that you can generally figure out what the actual gravity is.

This is the usual story that we are told. But there is an alternate way of looking at the same geometry such that the math works out the same, but it is described conceptually in a very different way. This divides spacetime up into "layers", called foliations, that represent slices of space and going from one foliation to the next is increasing in time. This is useful for computation because you can assign coordinates to your system. There are two additional constructs – the lapse function and the shift vector field. Starting at an event on one foliation, the lapse function at that position tells you that after a certain "proper time" (time perceived by an observer at that event) the time coordinate of the observer will be the proper time times the lapse function at that event. And the new spatial coordinates of the observer will be the proper time times the shift vector at that event. (Strictly speaking, this is only exactly true in the limit of the proper time approaching zero or when the lapse function and shift vectors are constant. But hopefully this gives a good enough insight to what these entities are so that we can go on to describe warp drives.)

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Lapse_shift.png]]
<tr><td>A folliation of space-time into slices of constant time coordinates (grey). At an event (black circle), the lapse function (yellow) shows how far into the future the point at that event will be projected in a unit amount of proper time experienced by an observer initially at the event. The shift vector (red) shows how that point changes coordinates after the unit amount of proper time. The cyan vector shows the new coordinates the point is projected onto. The lower black surface indicates a foliation of spacetime at the same time coordinate; after unit proper time, the events on the lower foliation get projected to the curved upper black foliation according to the shift vectors and lapse functions at each of their respective points.
</table>
</table>

Warp drives are generally restricted to spacetime geometries with a constant lapse function - all observers go forward by the same time coordinate after experiencing the same amount of proper time. Thus, all warp drives have no time dilation.
Far from the warp bubble, the shift vector vanishes – each spatial coordinate is mapped to the same spatial coordinate after any amount of proper time so long as the warp drive is not near. But the inside of the warp bubble is an extended region where the shift vector field is non-zero and constant. In this bubble, all spatial coordinates are mapped to a coordinate that shifts linearly in time with the proper time of the observer. Astute readers will see that this linear change in position with time is very similar to a velocity. This region of constant shift vector then moves with the shift vector velocity to keep stuff originally in the bubble so that it stays in the bubble. In between the inside of the bubble and the far away regions with zero shift vector, the shift vector field falls away to match one region to the other.

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Warp_shift.png]]
<tr><td>The shift vector of a one-dimensional slice through warp geometry, in red. At the edges there is no shift. At the center (inside the warp bubble) there is a region of uniform shift so that at a later time the inside of the warp bubble is rigidly translated to different spatial coordinates (in blue). The two foliations remain the same distance apart, because warp drives maintain a unit lapse function.
</table>
</table>

Traditionally, scientists would start with an interesting distribution of matter given by T and then find out what kind of curvature it created. This works well for planets and stars and black holes and cosmic strings and other things which are observed in the universe and you want to see what sort of effect they had. But starting in the 1980's, some physicists started looking at Einstein's field equations the other way around. They start with a desired geometry, find the curvature part of the field equation R - R g/2, and use that to find out what T has to be in order to get that curvature. This was originally done for [[Wormholes|wormholes]], but in 1994 Miguel Alcubierre used this same trick to figure out what could create a region of constant shift vector that moved with the shift vector - the warp drive. With this demand for a warp geometry, the required distribution of matter and energy and momentum and stress can be determined.

The original Alcubierre shift vector N→ produces a spherical warp bubble with an axi-symmetric distribution of, well, everything along the direction of motion with warp velocity vector v→.
<div align="center">
N→ = v→ f(rs)
</div>
Here, rs is the distance from the center of the warp bubble, and f(rs) can be any function that is 1 near rs = 0 and is 0 for rs well beyond the radius of the warp bubble.

=== Challenges and possible resolutions ===

Almost as soon as Alcubierre proposed his warp drive, others began picking it apart.
<ol>
<li>Pfenning and Ford<ref name="PfenningFord_2001">M. J. Pfenning and L. H. Ford, "The unphysical nature of 'Warp Drive'", Classical and Quantum Gravity. 14 (7): 1743–1751 (1997). arXiv:[https://arxiv.org/abs/gr-qc/9702026 gr-qc/9702026]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1997CQGra..14.1743P 1997CQGra..14.1743P]. doi:[https://doi.org/10.1088%2F0264-9381%2F14%2F7%2F011 10.1088/0264-9381/14/7/011]. S2CID [https://api.semanticscholar.org/CorpusID:15279207 15279207].</ref> found that if the negative energy density wall around the bubble obeys [[Wormholes#Quantum_energy_inequalities|quantum energy inequalities]], then for warp speeds of around light speed the shell thickness would be on the order of a hundred Planck lengths; a distance far smaller than any other known physical phenomenon or object.
<li>In the same work, Pfenning and Ford also found that for a bubble a hundred meters in radius, a warp speed of around the speed of light, and the required thickness of about 100 Planck lengths the energy in the bubble shell would be E ≈ -1063 kg c2. The magnitude of this latter value is ten orders of magnitude larger than the energy of the entire visible universe. They do note, however, that if quantum energy inequalities can be ignored then a warp drive with a hundred meters radius but a shell thickness of one meter would "only" have an energy magnitude of about a quarter solar mass.
<li>As with any method of faster than light travel, the warp drive could be used to make a time machine.
<li>Perhaps most seriously, if the warp drive is going faster than light speed, the negative energy regions on the outside of the shell won't be in the warp parts of the bubble. They will have to be moving through space at faster than light speed if the bubble is to maintain its integrity, which is the very problem that the warp drive was designed to avoid. The outside of the warp bubble shell that maintains the warp bubble would fall away and the warp bubble would quickly erode to sub-luminal speeds.<ref name="Van Den Broeck 2000">Chris Van Den Broeck, "On the (im)possibility of warp bubbles", arXiv:gr-qc/9906050 https://doi.org/10.48550/arXiv.gr-qc/9906050</ref>
<li>A more general problem that includes the previous one is that the geometry of the warp drive determines the distribution of energy, momentum, and stress (pressure, shear, and tension) by Einstein's field equation, which states that the stress energy tensor is directly proportional to a reduced order curvature tensor of space-time. However, the stress-energy tensor must also obey the continuity equation – basically, for the energy to change in a given volume of space that energy must come in or go out of the boundary of that volume and similarly for the momentum. So a warp drive can have the warp condition for one moment in time. But after that, the stress-energy that maintains the warp will start to move according to the forces on it and according to the natural paths that it takes through the warped space-time. There is no guarantee that the configuration of the stress-energy at later times will be sufficient to maintain the warp.
<blockquote>
From the perspective of simulating the warp drive dynamically, the key challenge is stability. One can straightforwardly generate the initial metric that describes the Alcubierre metric and infer the instantaneous matter configuration that must support it using the Einstein Equation, but a time-domain evolution of the coupled matter-gravity system necessitates that we specify the equation of state of the matter, which in turn determines how it evolves. In a perfect fluid, this involves specifying the relation between the fluid pressure and density, but in a more general case, the full relation between the
stress-energy components must be given. There is (to our knowledge) no known equation of state that would maintain the warp drive metric in a stable configuration over time - therefore, whilst one can require that initially, the warp bubble is constant, it will quickly evolve away from that state and, in most cases, the warp fluid and spacetime deformations will disperse or collapse into a central point.
 
– Clough, Dietrich, and Khan<ref>Katy Clough, Tim Dietrich, and Sebastian Khan, "What no one has seen before: gravitational waveforms from warp drive collapse", arXiv:2406.02466 [gr-qc] https://doi.org/10.48550/arXiv.2406.02466</ref>
</blockquote>
<li>In order to get the matter (or more generally, stress-energy) that forms the warp shell up to speeds that could allow the shell to exist, you need to be able to, well, accelerate that matter to warp speeds. Which gives a chicken-and-egg problem - in order to achieve superluminal travel you need to be able to get matter or energy moving superluminally in the first place<ref name="Bobrick and Martire">Alexey Bobrick and Gianni Martire, "Introducing Physical Warp Drives", arXiv:2102.06824 [gr-qc] https://doi.org/10.48550/arXiv.2102.06824</ref><ref>D. H. Coule, "No warp drive", Class. Quantum Grav. 15 (1998) 2523–2527. https://doi.org/10.1088/0264-9381/15/8/026</ref>, which rather makes warp drives unnecessary from the get-go.
<li>A warp bubble moving faster than light will not be able to get any signals in front of it as it will overtake its own signal. In fact, anyone in the interior of the bubble would not be able to send signals to the front surface of the bubble. In science-speak, a horizon forms at the front of the bubble<ref name="Natario">José Natário, "Warp drive with zero expansion", Classical and Quantum Gravity. 19 (6): 1157–1166 (2002). arXiv:[https://arxiv.org/abs/gr-qc/0110086 gr-qc/0110086]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2002CQGra..19.1157N 2002CQGra..19.1157N]. doi:[https://doi.org/10.1088%2F0264-9381%2F19%2F6%2F308 10.1088/0264-9381/19/6/308]. S2CID [https://api.semanticscholar.org/CorpusID:15859984 15859984].</ref>. As a consequence, the bubble cannot be controlled, steered, nor stopped from within the bubble itself.
<li>A Horizon will naturally emit Hawking radiation. It is estimated that the radiation will raise the temperature of the interior of the bubble to approximately 1032 K<ref name="Finazzi 2009">Stefano Finazzi, Stefano Liberati, and Carlos Barceló, "Semiclassical instability of dynamical warp drives", Phys. Rev. D 79, 124017 (12 June, 2009) https://doi.org/10.1103/PhysRevD.79.124017 , arXiv:0904.0141 [gr-qc] https://doi.org/10.48550/arXiv.0904.0141</ref> (at this level of accuracy, the result is the same if you replace temperatures in K by the same number in °C or °F). This temperature, however, may be lower if quantum inequalities do not have to be fulfilled.
<li>Semi-classical instabilities at the horizon are expected to lead to a divergence of the stress-energy tensor at the boundary in a short period of time, thus leading to the destruction of the warp bubble<ref name="Finazzi 2009"></ref>.
</ol>
Helpfully, Van Den Broeck<ref name="VanDenBroeck_2000">C. Van Den Broeck, "Alcubierre’s Warp Drive: Problems and Prospects" AIP Conference Proceedings. 504: 1105–1110 (2000). Bibcode:[https://ui.adsabs.harvard.edu/abs/2000AIPC..504.1105V 2000AIPC..504.1105V]. doi:[https://doi.org/10.1063%2F1.1290913 10.1063/1.1290913].</ref> suggested several solutions – or at least mitigations – for some of these problems
<ol>
<li>Quantum inequalities had not been shown to be true in general for highly curved space-times.
<li>A proposed warp drive geometry<ref name="VanDenBroeck_1999">C. Van Den Broeck, "A 'warp drive' with more reasonable total energy requirements". Classical and Quantum Gravity. 16 (12): 3973–3979 (1999). arXiv:[https://arxiv.org/abs/gr-qc/9905084 gr-qc/9905084]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1999CQGra..16.3973V 1999CQGra..16.3973V]. doi:[https://doi.org/10.1088%2F0264-9381%2F16%2F12%2F314 10.1088/0264-9381/16/12/314]. S2CID [https://api.semanticscholar.org/CorpusID:15466313 15466313]. </ref> (see the van Den Broeck drive, below) would be able to minimize the magnitude of the energy required to about that of the mass of our sun. While still large, it is not unphysically large.
<li>Time travel is always going to be a worry with faster than light travel. There's no neat solution to this one.
<li>It may be possible for the negative energy outside of the superluminal region to compress into the superluminal region to form a shock in space-time. The front surface of the bubble would then be a singularity. It is not clear if such a sharp jump in physical properties is possible, but neither is sure that it is impossible, either. However, while this might help with the negative energy in front of the bubble getting swept up because it cannot move faster than light, it does little to help with the material at the back of the bubble on the outside getting left behind because it cannot keep up.
</ol>
Other mitigating research has also shown
<ol start=9>
<li>Semi-classical divergence only occurs in warp drives with only one spatial dimensions. Those with two or more spatial dimensions (as in our universe) only contain points where divergences occur, and these divergences can be managed. In fact, their effects can be reduced by making the warp shape more elongated<ref>Carlos Barceló, Valentin Boyanov, Luis J. Garay, Eduardo Martín-Martínez,
and Jose M. Sánchez Velázquez, "Warp drive aerodynamics", JHEP08 (2022) 288 https://doi.org/10.1007/JHEP08(2022)288 , arXiv:2207.06458 [gr-qc] https://doi.org/10.48550/arXiv.2207.06458</ref>.
</ol>

=== Alcubierre warp interactions with light and matter ===

Space is not entirely empty. It is filled with a diffuse plasma in the form of the interstellar medium, as well as cosmic radiation, light from stars, and cosmic microwave background radiation. A warp drive propagating through space will encounter this stuff. What happens when this matter and radiation have a warp bubble pass across them?

The first analysis of matter encountering a warp bubble was performed by Pfenning and Ford<ref name="PfenningFord_2001"></ref>. They looked at the warp bubble interaction with a massive object at rest with the frame of reference of the warp drive (keep in mind that the rest frame is the same inside and outside the warp bubble; in this rest frame objects inside the bubble have no momentum even though they are, in some sense, changing location rapidly with time). When the warp bubble passes, the object experiences an acceleration in the direction of the warp bubble motion. When the warp shell passes and the object is inside the bubble, it will be moving with approximately the speed of the bubble. Pfenning and Ford analyzed this problem with a continuous distribution of shell energy that, strictly speaking, never falls to zero except at the bubble center and at spatial infinity, so unless the object passes through the center in Pfenning and Ford's description it will never quite get up to the bubble's speed. In this case, the object will move almost as fast as the object but will pass through the bubble in a finite time, after which it will again be at rest with respect to the reference frame but displaced along the direction of the warp bubble motion by some distance. With this description of the warp bubble, a spacecraft of finite size will always be moving a little slower than the warp bubble and would have to use rockets to keep up with it. in addition, the spacecraft would experience tidal forces that would cause stress on the spacecraft's structure.

Pfenning and Ford also examined cases where the shell is of a finite (possibly infinitesimal) width and falls to zero both inside and outside the bubble. In this case any matter encountering the bubble would be collected at the bow of the bubble and thereafter move along with it.

McMonigal et al. analyzed the situation for both massive particles and light moving along the axis of travel of the bubble<ref name="McMonigalEtAl_2012">B. McMonigal, G. F. Lewis, and P. O'Byrne, "Alcubierre warp drive: On the matter of matter". Physical Review D. 85 (6) 064024 (20 March 2012). arXiv:[https://arxiv.org/abs/1202.5708 1202.5708]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2012PhRvD..85f4024M 2012PhRvD..85f4024M]. doi:[https://doi.org/10.1103%2FPhysRevD.85.064024 10.1103/PhysRevD.85.064024]. S2CID [https://api.semanticscholar.org/CorpusID:3993148 3993148].</ref>. They found that light moving opposite the warp direction passed through the bubble without incident, being only somewhat delayed by passing through the bubble. A warp bubble that is warping at sub-luminal speeds can have light catch up from behind it. This light is able to pass through the bubble, and is somewhat advanced in its path by the speed of the bubble. For super-luminal warp bubbles, however, the situation is different. The bubble will catch up to light moving it its own direction that is originally in front of it. This light cannot escape forward, the bubble being too fast. Nor can it escape backward, as the light is propagating forward and the interior of the bubble is at rest. Thus, the light gets caught at the bow of the warp bubble, unable to escape for so long as the warp bubble is active. This light is strongly blue shifted to extremely energetic x-rays and gamma rays. The space behind the bubble is swept clear of forward-moving light.

The situation for matter over-run by a super-luminal warp bubble is similar. Matter moving backward passes through the bubble, being only swept a ways along the bubble's path as sign of its passing but otherwise continuing on their way. Matter at rest behaves the same way as Pfenning and Ford discovered. Meanwhile, matter moving in the same direction of the warp is overtaken and collects at the bow of the bubble, unable to leave for so long as the bubble is warping. This matter "surfing" on the bubble bow is highly accelerated to relativistic speeds, experiencing extreme time dilation.

For a sub-luminal bubble, there is a speed faster than the warp speed where particles coming up from behind overtake the bubble and pass through, out the front. Particles below this speed but still faster than the warp speed still overtake the bubble, but then bounce backwards out of the bubble from behind. They are still moving forward, but are now moving more slowly than the warp speed. Particles moving forward but slower than the bubble will be overtaken and then bounced out the front with a speed higher than the warp speed. Those particles moving backward will pass through the bubble and exit with their initial speed.

The spacecraft inside the bubble will observe that light moving backward compared to the warp direction is blue-shifted, and matter moving backward passes through with increased energy. Meanwhile, a sub-luminal bubble will see light that was moving forward as red-shifted and matter catching up to the bubble from behind will pass by with reduced energy (superluminal bubbles, of course, do not have any matter or radiation catching up to them from behind).

When a superluminal warp bubble that has been collecting matter and radiation for the duration of its journey and blue-shifting it to much higher energies is turned off, all that matter and energy is released as a blast of radiation in the direction the bubble was warping.

McMonigal et al. conclude by noting that any spacecraft in the warp bubble would need shielding to protect against the blue-shifted radiation and matter of increased energy. In addition the destination would be "blasted into oblivion" by the release of matter and radiation that had been caught in the bubble during the trip.

Clark et al.<ref>Chad Clark, William A. Hiscock, and Shane L. Larson, "Null geodesics in the Alcubierre warp drive spacetime: the view from the bridge", arXiv:gr-qc/9907019 (2018) https://doi.org/10.48550/arXiv.gr-qc/9907019</ref> examine what it would look like from within an operating warp drive. They find that light moving at 90 degrees to the direction of motion will look like it is still coming from 90 degrees. However, light from the front hemisphere is squished progressively toward the forward point the faster you go. For a warp bubble moving faster than the speed of light, there is a cone behind the warp bubble where no light can reach it. However, you don't see a black spot behind you – light from the regions in the backward hemisphere from outside that cone that can reach you is distorted toward the direction behind you to fill in that spot. Light from the forward direction is blue-shifted and light from behind is red-shifted. At speeds of around 200 times light speed, the cosmic microwave background would be blue shifted to energies similar to that of the solar photosphere, heating anything in the warp bubble as if they were close to the sun.

== Other warp geometries ==

The Alcubierre warp drive is the first and best known of the warp drives. It has been around long enough that there have been many studies done on it. The warp geometry is particularly simple, making analysis easier. But in the time that the Alcubierre warp was proposed, many other researchers have come up with other warp designs, and even generalized the idea to include a wide array of additional possibilities.

Most of the general statements about the challenges of the Alcubierre drive hold for these other drives. Many of the details about interactions with matter are also likely to hold, although these other styles of warp drive have not yet been subject to such rigorous investigation.

=== Van Den Broeck warp drive ===

If an Alcubierre warp bubble a hundred meters across requires a magnitude of energy greater than the entire energy in the observable universe, one option to reduce the magnitude of energy used is to make the warp bubble smaller. Much smaller. Pfenning and Ford<ref name="PfenningFord_2001"></ref> suggested making the warp bubble smaller than an atom. Of course, that brings up the problem of how to stuff a spacecraft in there. Van Den Broeck proposed a solution<ref name="VanDenBroeck_1999"></ref>: make the warp bubble only about the size of an atomic nucleus but expand the space inside the bubble enormously so that a spacecraft could fit in. This is somewhat like a nuclear sized wormhole that leads to a pocket universe that holds the spacecraft. Unfortunately, the metric doesn't fit neatly into any [[Wormholes#Three_dimensions|embedding diagram]] that I can figure out, but the math works out so that spatial coordinates inside the bubble are expanded by a factor of 1017 and a spacecraft would have a few hundred meters of bubble interior to putz around in. This trick manages to reduce the magnitude of energy to only about the mass energy of a few stars similar to our own. Other than that, it is otherwise a normal Alcubierre drive.

Further work<ref>S. Krasnikov, "The quantum inequalities do not forbid spacetime shortcuts", Physical Review D. 67 (10) 104013. https://doi.org/10.1103%2FPhysRevD.67.104013, arXiv:gr-qc/0207057 (2003) https://doi.org/10.48550/arXiv.gr-qc/0207057 </ref> has managed to use Van Den Broeck's trick to reduce the magnitude of the negative energy to only about a milligram.

=== Natário warp drive ===

The Alcubierre drive is not the only way to construct a warp drive. José Natário<ref name="Natario"></ref> showed that it was just one example of an entire class of warp drives. These are space-time geometries with a smooth and localized shift vector field and a unit lapse function.

Natário then went on to find in that class a set of warp drives that do not have any expansion or contraction of space at all. Instead, space encountering the front boundary of the warp bubble instead "slides" around the outside of the bubble until it gets to the corresponding place on the back and is left behind there. Another way to think of it is that space entering the bubble shell is compressed in the radial direction of the shell but is simultaneously expanded in the tangential direction so that there is no net change in volume. When it gets to the back, the opposite occurs. In the language of mathematics, the shift vector field is divergenceless.

<table align=center>
<tr><td width=360>[[File:Natario_drive_flow_of_space.png]]
<td width=445>[[File:Natario_warp_drive_horizons.png]]
<tr><td style="vertical-align: top;">A representation of the motion of space in a zero-expansion Natário drive with a vanishingly thin warp shell around the bubble.
<td>The horizons of the generalized Natário class of warp drives (including the Alcubierre warp drive and all fell-Heisenberg drives) at superluminal speed in the direction of the green arrow. No event inside the bubble can affect events outside the bubble in front of the blue line. No events outside the bubble behind the red line can affect events inside the bubble. The lines have an angle α with respect to the direction of motion with sin(α) = 1/vs for vs the speed of the bubble relative to light speed. This is analogous to the Mach cone of supersonic objects.
</table>

NOTE: There are two different things that can be meant by the term "Natário warp drive". The generalized Natário warp drive is the general theory of all warp drives that includes within it the Alcubierre, Fell-Heisenberg, and Lentz warp drives, among infinitely many others. The zero-expansion Natário warp drive is the specific implementation of the generalized Natário warp drive that has a divergenceless shift vector field.

The generalized class of warp drives developed by Natário (including both the Alcubierre and this zero-expansion warp drive) are shown to always have regions in the warp shell where the energy density is negative to at least some observers. You can't get away from it – to warp, you need negative energy density.

The Natário class of warp drives blueshift light coming in from the front and redshift light catching up from the back. For the case of a super-luminal warp bubble, there will of course be a horizon at the back of the bubble and light will not be able to get through from behind. For a bubble speed vs relative to the speed of light, light coming from straight ahead will be blueshifted up in frequency and photon energy by a factor of 1 + vs. For sub-luminal travel, light coming from behind will be redshifted by a factor 1 - vs. In general for n as the unit direction along which the light is propagating, the blueshift factor will be 1 + vs ⋅ n. If the warp shell is infinitesimally thin, light from outside that reaches the center of the bubble will not be distorted in direction although it will be frequency shifted, but observers still inside the bubble but displaced from the center will see the field of view distorted as well as frequency shifted.

Most of the difficulties that apply to the Alcubierre warp drive also apply to the generalized Natário warp drive (and thus to the zero-expansion Natário drive and Fell-Heisenberg drive as well). In particular, quantum energy inequalities, high magnitudes of the energy required, formation of time machines, instabilities related to the stress-energy making up the warp shell being unable to keep up with the bubble, the chicken-and-egg problem of needing FTL to achieve FTL, the formation of horizons making warp bubbles uncontrollable, and Hawking radiation at the horizons, are all still concerns for generic warp drives.

=== Fell-Heisenberg warp drives ===

S. D. B. Fell and L. Heisenberg investigated what sort of warp structure is necessary to produce a warp drive with positive energy density everywhere<ref name="Fell-Heisenberg">S. D. B. Fell and L> Heisenberg, "Positive energy warp drive from hidden geometric structures", Classical and Quantum Gravity 38 155020 (2021) https://doi.org/10.1088/1361-6382/ac0e47 https://arxiv.org/abs/2104.06488</ref>. They found that warp drives of the generic Natário form with irrotational shift vectors had a simple expression for the energy density and vanishing momentum (irrotational means that the shift vector can be described as the gradient of a scalar potential field or, equivalently, that the curl of the shift vector field vanishes everywhere). Fell and Heisenberg proposed several potential fields that either had positive energy density everywhere except at discontinuities, or that were continuous but had regions that violated the weak energy condition (i. e. negative energy densities in some frames of reference). One specific example they provided of a warp field had an energy that is about 10,000 times less than the mass-energy of our sun. Or only about half the mass-energy of Jupiter. This is a significant improvement over many other proposed drives.

Despite the introduction discussing warp drive configurations that satisfy the various energy conditions, the configurations described in the paper are shown to locally violate the weak and strong energy conditions. Nonetheless, the energy density is still mostly positive. Santiago, Schuster, and Visser's<ref name="Santiago Schuster Visser 2022">J. Santiago, S. Schuster, and M. Visser, "Generic warp drives violate the null energy condition", Physical Review D 105, 064038 (2022) https://doi.org/10.1103/PhysRevD.105.064038</ref> work shows that the various energy conditions must still be violated by this warp drive; to not violate these, energy density must be positive in all reference frames not just those of the co-moving observer.

Any vector field can be described by its divergence, its curl, and a constant. The constant part of a warp drive's shift vector can be discarded because then the warp drive would not be localized. It is interesting, therefore, that the Natário zero-expansion drive has zero divergence and can be described only by its curl, while the Fell-Heisenberg drive has zero curl and can be described only by its divergence. These two classes of drives thus rely on opposite choices of setting one of the two different parts describing a vector field to zero. As might be expected, the Fell-Heisenberg drive is not zero expansion – space-time will change in volume as it passes through the warp bubble (although after going all the way through it will return to its original volume). However, objects in space-time will retain their original orientation as they pass through the warp bubble.

Any vector field with zero curl can be represented as the gradient of a suitable scalar function. Thus, the Fell-Heisenberg drive can be completely described by just scalar field (one number at all points of space and time) rather than a vector field.

Previous examples of warp drives, such as the Alcubierre and zero-expansion Natário drives, tend toward a flat space-time as you go far enough away faster than would occur if there were any mass present in the warp drive. That is, the ADM mass of such drives is zero. However, one of the Fell-Heisenberg examples described a spacetime where the geometry approaches that of what you would get for a massive object at the warp location, giving one of the first examples of a warp drive with mass.

In the Fell-Heisenberg drive the momentum vanishes everywhere. Yet the energy occupies regions where the shift vector is varying rapidly. The lack of momentum means that the energy will not move to keep up with the differential expansion and movement of the spacetime. As a consequence, at later times the energy will have a different distribution than what is necessary to maintain the given warp configuration. The exact time evolution is not solved but this likely leads to collapse of warp bubble. Note that this is not a problem unique to the Fell-Heisenberg drive. Assuming a geometry that describes a warp produces a specific distribution of the stress-energy tensor that satisfies Einstein's equation. But there is no guarantee that this stress energy tensor satisfies the continuity equation, which describes how the distribution of matter and momentum change with time. The issue of stability of any warp drive is not well understood, but most warp drive geometries will be unstable in the sense that their matter that generates the warp fields will not be able to keep up with the motion of the warp. The Fell-Heisenberg drive is unique only in that, with vanishing momentum, there is no configuration that allows stable propagation.

(n. b. The Heisenberg here is Lavinia Heisenberg, not the Werner Heisenberg of quantum physics and uncertainty principle fame.)

=== Lentz warp drive ===

E. Lentz<ref name="Lentz">E. W. Lentz, "Breaking the warp barrier: hyper-fast solitons in Einstein–Maxwell-plasma theory", Classical and Quantum Gravity. 38 075015 (2021). arXiv:[https://arxiv.org/abs/2006.07125 2006.07125]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2021CQGra..38g5015L 2021CQGra..38g5015L]. doi:[https://doi.org/10.1088%2F1361-6382%2Fabe692 10.1088/1361-6382/abe692]. ISSN [https://search.worldcat.org/issn/0264-9381 0264-9381]. S2CID [https://api.semanticscholar.org/CorpusID:219635854 219635854].</ref> investigated a particular configuration of the Fell-Heisenberg class of warp drives, where the scalar field forms a selection of rhomboid regions, resulting in an energy density that is everywhere positive and a shift vector with vanishing tidal force at the center for a comfortable ride.

However, Santiago, Schuster, and Visser<ref name="Santiago Schuster Visser 2022"></ref> dispute claims that the Lentz drive satisfies the energy conditions, noting that everywhere positive energy density in one frame of reference is insufficient to establish that the energy density is positive in all reference frames; knowledge of the Cauchy stress tensor is also needed. They show that any generic warp drive will violate the strong energy condition, null energy condition, and weak energy condition.

== Conservation laws ==

All the warp drives investigated so far have geometries that quickly fall off from the highly curved region near the warp bubble to a region of spacetime that is almost flat; and becomes even more flat the farther away from the warp bubble you get. There is a word for geometries of this kind – they are called asymptotically flat. Basically, you can always go far enough away in any direction to reach a place where the geometry is sufficiently flat to meet any flatness criterion you choose.

When you have an asymptotically flat geometry, then in the far away mostly flat regions gravity is well described by a linearized theory of gravity – where the perturbations to the geometry are linear in the stress energy tensor. This means that gravity is basically Newtonian, along with a few other bells and whistles like gravitational waves and frame dragging.

When linearized gravity is a good approximation, then it is known that energy, momentum, and linear momentum are conserved. Although parts of a warp geometry are highly curved and non-linear, you can measure the conserved energy and momentum and angular momentum from far away in the linear region. This ADM energy and etcetera is the quantity that is conserved when you are worrying about conserving such things, not necessarily the energy and so forth you get in the highly curved spacetime region. In fact, it is not even simple to define where the energy and such actually is withing the highly curved spacetime region; it is non-local and different observers will disagree on how it is distributed. But everyone agrees on the ADM conserved quantities from out in nearly flat spacetime (up to the usual changes due to changing speeds, which are trivial and come up with normal objects as well).

And now we run into a bit of a conundrum: most of the warp geometries so far proposed have zero ADM mass, meaning that things like the Alcubierre drive technically have no energy or momentum or angular momentum at all. Despite all of the extremes of negative energy regions within the wall of the warp bubble, the actual energy that you need to spend to make one is zero. This can lead to a bit of a problem. If you have a spacecraft that has mass, that mass-energy has to be conserved. So when you turn on your warp drive and the total warp mass is zero, where does that energy go?

One resolution is just that the mass of the spacecraft is negligible and the small amount of curvature it causes can be ignored. In this case, the warp geometries discussed so far are merely an approximation that neglects this effect. In fact, some of the proposed Fell-Heisenberg drives map not onto entirely flat spacetime geometries at large distances, but onto the Schwarzschild geometry that is appropriate for systems with mass at the center. If this is the case, then the warping bubble and spacecraft system will need to worry about conservation of all their conserved properties.

Let's look at what the conservation laws mean for a spacecraft equipped with a warp drive, assuming that no matter or radiation are emitted by activating or deactivating the warp drive.

We can start with an observer floating in space. We'll call her Alice. And, approaching Alice at 100 m/s is the starship Enterprise with a mass of a cool 200,000 tons (2×108 kg). from this we know that the Enterprise has, in Alice's frame of reference, a kinetic energy of 1 TJ, a total energy (including mass and kinetic energy) of 1.8×1025 J, a momentum of 2×1010 kg m/s, and – because it is headed straight toward Alice – an angular momentum of zero. All of this is without any warp drive activated.

On the bridge of the Enterprise, Spock picks up Alice on the sensors and tells Captain Kirk. Kirk does not want to run into Alice (he'd rather date her), so he orders helm to turn on the Enterprise's warp drive and move them 1 km up so that their course takes them 1 km from Alice instead of straight into her. Sulu warps the Enterprise according to these commands, and then turns off the warp drive.

After this maneuver, keeping the Enterprise's total energy and momentum the same means that the Enterprise still has a mass of 200,000 tons and a velocity of 100 m/s in its original direction. So we can see that the warp drive must preserve the velocity vector of the thing in it (at least in flat spacetime where outside forces are not acting on it). The problem comes from angular momentum. With a known mass of 200,000 tons, a speed of 100 m/s, and a distance of closest approach of 1 km; from Alice's frame of reference the Enterprise now has an angular momentum of 2×1013 kg m2/s after the warp maneuver, while before the maneuver it has an angular momentum of zero.

From this we can see that warp drives cannot be reactionless. Arbitrary displacements using warp without emitting propellant result in violation of the law of conservation of angular momentum. As long as we restrict ourselves to accepted science based on what we know of general relativity, to move a warp drive with mass you need to accelerate it or decelerate it with rockets or other normal means of delivering momentum (such as light sails, mag sails, or pushing off of a planet). Warp drives must be inertial to comply with physics. And all of this was analyzed only from times when the warp drive was turned off; it does not depend on any assumptions about what happens when the warp is on.

(ADM mass -> 0; conservation of energy/mass issues.)<ref>Sebastian Schuster, Jessica Santiago,and Matt Visser, "ADM mass in warp drive spacetimes" General Relativity and Gravitation (2023) 55:14 https://doi.org/10.1007/s10714-022-03061-9</ref>

(Discuss issues of conservation of angular momentum.)

(If angular momentum conservation is ignored, discuss how momentum & energy are affected by outside forces. In a gravitational field equivalent to inertial frame ... like being in an accelerated elevator; will acquire momentum buildup while staying "at rest".)

== Warp Railroads ==

Van Den Broeck<ref name="VanDenBroeck_2000"></ref> suggested a warp railroad,

You can set up devices ahead of time along the path of the bubble that produce the negative energy regions for the warp bubble at the appropriate time without needing the negative energy to ever move with the bubble at all. At least for bubbles on a predictable schedule and route.

helps with:

-stability (elaborate)

-conservation laws (elaborate)

Mention Krasnikov<ref name="Krasnikov 1998">S. V. Krasnikov, "Hyperfast travel in general relativity" Phys. Rev. D 57 4760</ref>

Hyperwave<ref name="Pieri 2024">Lorenzo Pieri, "Hyperwave: hyper-Fast Communication within General Relativity", arXiv:2311.12069 [gr-qc] https://doi.org/10.48550/arXiv.2311.12069</ref> makes use of this warp railroad idea.

== Quantum effects ==

(quantum stuff here<ref name="Hiscock1997">W. A. Hiscock, "Quantum effects in the Alcubierre warp drive spacetime", Classical and Quantum Gravity 14 L183 https://doi.org/10.1088/0264-9381/14/11/002 https://arxiv.org/abs/gr-qc/9707024</ref><ref name="Finazzi et al 2009">S. Finazzi, S. Liberati, C. Barceló, "Semiclassical instability of dynamical warp drives", Physical Review D 79, 124017 (2009)https://doi.org/10.1103/PhysRevD.79.124017 https://arxiv.org/abs/0904.0141</ref>)

From<ref name="Pieri 2024"></ref>
<blockquote>
It should be emphasised that Quantum Inequalities have not been proven for realistic interacting quantum fields, where the they are expected to be weaker if they exist [Cadamuro, 2019], and they often require Minkowskian conditions at infinity, a condition which is not appropriate for fields inside bounded regions, as in a Casimir devices.
</blockquote>

== Warp drives and black holes ==

What happens if you drive your warp drive into a black hole?

The pop-sci reason that nothing can escape a black hole is that at the event horizon the escape velocity is faster than light and nothing can go faster than light. A warp drive can go faster than light.

The more detailed reason from general relativity is that at the event horizon space and time are rotated sufficiently that "inward" becomes "forward in time." You can no more go farther away from the singularity in a black hole once you are inside the event horizon than you can go backward in time. Any super-luminal travel can go backward in time, and warp drives can engage in super-luminal travel.

The Hawking area theorem that says that black holes can only grow, never shrink relies on the condition that the energy is everywhere positive. Warp drives have negative energy density.

So in principle, there's nothing that prevents a warp drive from taking a dip into a black hole and coming back out again.

Garattini and Zatrimaylov looked into the problem of a warp drive near (and in) a black hole<ref name="Garattini and Zatrimaylov">R. Garattini and K. Zatrimaylov, "Black holes, warp drives, and energy conditions", Physics Letters B 856 138910 (2024) https://doi.org/10.1016/j.physletb.2024.138910</ref>. The found that the black hole reduces the negative energy requirement for the warp bubble to move inward toward the black hole singularity, but increases the negative energy needed to move away from the center of the black hole. In addition, a warp drive parked at the event horizon would allow light from inside the horizon to pass through the bubble and escape back outside.

== Credit ==
Author: Luke Campbell

== References ==

[[Category:Physics]][[Category:Infrastructure]][[Category:Physics & Engineering‏‎]][[Category:Physics & Math & Engineering]][[Category:Transportation & Infrastructure‏‎]][[Category:Metric Engineering]]

Warp Drives

2026-07-30T03:49:43Z

Lwcamp: /* Conservation laws */

{{PageConstructionNotice}}

Science fiction often features spacecraft that can seemingly move across space and get between the place of departure and the destination much faster than light could have done. This appears to contradict the theory of relativity, which predicts unequivocally that nothing can move through space faster than light. Because relativity has been incredibly successful at describing nature, with its many other predictions regularly being confirmed to extraordinary accuracy and within the bounds of uncertainty of all the experiments that tested them, it gives confidence that relativity is a correct description of reality. Which seems to rather throw a wet towel on our hopes for rapid travel between stars.

However, while relativity does not allow things to move through space faster than light, it places no such restrictions on how fast space-time itself can expand, contract, or move around. This leads to the idea of a warp drive – the spacecraft remains stationary within a region of highly curved space-time, and that region moves at super-luminal speeds rather than the spacecraft.

== The Alcubierre warp drive ==

The first warp drive geometry that satisfied the Einstein field equations of relativity was proposed by Miguel Alcubierre<ref>M. Alcubierre, "The warp drive: hyper-fast travel within general relativity." Classical and Quantum Gravity. 11 (5): L73–L77 (1994). [https://arxiv.org/abs/gr-qc/0009013 arXiv:gr-qc/0009013]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1994CQGra..11L..73A 1994CQGra..11L..73A]. doi:[https://doi.org/10.1088%2F0264-9381%2F11%2F5%2F001 10.1088/0264-9381/11/5/001]. S2CID [https://api.semanticscholar.org/CorpusID:4797900 4797900].</ref>. In this geometry, a sphere of space-time moves at an arbitrary speed (potentially but not necessarily a speed much faster than light). Objects within the sphere are moved along with the sphere; an object at rest within the sphere would be moved along with the sphere indefinitely. Space is expanding at the rear boundary and contracting at the front boundary in order to keep the sphere moving. In order to satisfy the Einstein field equations, the boundary of the sphere must have a negative energy density. The challenges of space-time geometries with negative energy densities are described in our page on [[Wormholes#Exotic_energy_conditions|wormholes]], for our purposes it is enough to note that negative energy densities can pose problems if not handled carefully, there may be limits on how much negative energy you can have without nearby positive energy density, and it may not be possible to get enough negative energy to support a warp drive; although none of this is ruled out by physics – yet!

<table align=center>
<tr><td width=650>[[File:Energy_magnitude_of_Alcubierre_drive.png]]
<td width=650>[[File:Expansion_of_Alcubierre_drive.png]]
<tr><td>The magnitude of the energy of the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. (The energy density is negative everywhere in the shell, here only the magnitude is shown to aid visual comprehension.)
<td>The expansion of space in the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. Negative expansion means that space is contracting in that direction, positive expansion that it is expanding.
</table>

NOTE: What the? The energy distribution is symmetric! How does the drive know which way to go? There must be significant contributions of other components of the stress-energy tensor, and those have got to be asymmetric along the forward/backward axis. Check this when I get time.

(And what's with the ring in all the artwork, anyway?)

=== So what is going on? ===

What is it that makes a warp drive a warp drive, and how does it work?

A warp drive is a concept that comes out of the science of general relativity. General relativity collects space and time together into a single "structure" called spacetime. This structure is curved, and its curvature is determined by the material that is in it. Absent forces acting between matter, particles of matter move through spacetime on the shortest possible paths in this curved environment.
<blockquote>
"Spacetime tells matter how to move; matter tells spacetime how to curve." - John Archibald Wheeler
</blockquote>
All of this is encompassed in the Einstein field equation:
<div align=center>
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 8πG
<td rowspan="2" nowrap="nowrap" align="right"> T = R -  
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right"> R g
<tr>
<td style="border-top:solid 1px black;"> c4
<td style="border-top:solid 1px black;"> 2
</table>
</div>
Unfortunately, by convention, both the symbols G and R get used for a lot of different things. Here, symbols in bold are second rank tensors – constructs of arrays of 4×4 numbers where 4 is necessary because that is the number of dimensions of spacetime. Conventionally, these numbers are indexed by a value going from 0 to 3, with 0 being the part corresponding to the coordinate of time and 1 to 3 being the Cartesian coordinates that you might know as x, y, and z.

T is the stress-energy tensor: the (0,0) component represents the energy density at a given point; the (0,i) and (i, 0) components (with i going from 1 to 3) represent the momentum density; and the (i,j) components represent the stresses present, such as pressure or tension or shear.

The other term, R - R g/2, involves a partial description of the curvature. A full description of the curvature requires a construct of 4×4×4×4 components; too much for the stress energy tensor to fully constrain. Which is okay, because this allows curvature to exist away from where matter is, allowing gravity to have long range effects across empty space as well as for phenomena like gravitational waves. But it is enough that a given distribution of matter T gives enough constraints (along with specifying how the geometry behaves far away) that you can generally figure out what the actual gravity is.

This is the usual story that we are told. But there is an alternate way of looking at the same geometry such that the math works out the same, but it is described conceptually in a very different way. This divides spacetime up into "layers", called foliations, that represent slices of space and going from one foliation to the next is increasing in time. This is useful for computation because you can assign coordinates to your system. There are two additional constructs – the lapse function and the shift vector field. Starting at an event on one foliation, the lapse function at that position tells you that after a certain "proper time" (time perceived by an observer at that event) the time coordinate of the observer will be the proper time times the lapse function at that event. And the new spatial coordinates of the observer will be the proper time times the shift vector at that event. (Strictly speaking, this is only exactly true in the limit of the proper time approaching zero or when the lapse function and shift vectors are constant. But hopefully this gives a good enough insight to what these entities are so that we can go on to describe warp drives.)

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Lapse_shift.png]]
<tr><td>A folliation of space-time into slices of constant time coordinates (grey). At an event (black circle), the lapse function (yellow) shows how far into the future the point at that event will be projected in a unit amount of proper time experienced by an observer initially at the event. The shift vector (red) shows how that point changes coordinates after the unit amount of proper time. The cyan vector shows the new coordinates the point is projected onto. The lower black surface indicates a foliation of spacetime at the same time coordinate; after unit proper time, the events on the lower foliation get projected to the curved upper black foliation according to the shift vectors and lapse functions at each of their respective points.
</table>
</table>

Warp drives are generally restricted to spacetime geometries with a constant lapse function - all observers go forward by the same time coordinate after experiencing the same amount of proper time. Thus, all warp drives have no time dilation.
Far from the warp bubble, the shift vector vanishes – each spatial coordinate is mapped to the same spatial coordinate after any amount of proper time so long as the warp drive is not near. But the inside of the warp bubble is an extended region where the shift vector field is non-zero and constant. In this bubble, all spatial coordinates are mapped to a coordinate that shifts linearly in time with the proper time of the observer. Astute readers will see that this linear change in position with time is very similar to a velocity. This region of constant shift vector then moves with the shift vector velocity to keep stuff originally in the bubble so that it stays in the bubble. In between the inside of the bubble and the far away regions with zero shift vector, the shift vector field falls away to match one region to the other.

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Warp_shift.png]]
<tr><td>The shift vector of a one-dimensional slice through warp geometry, in red. At the edges there is no shift. At the center (inside the warp bubble) there is a region of uniform shift so that at a later time the inside of the warp bubble is rigidly translated to different spatial coordinates (in blue). The two foliations remain the same distance apart, because warp drives maintain a unit lapse function.
</table>
</table>

Traditionally, scientists would start with an interesting distribution of matter given by T and then find out what kind of curvature it created. This works well for planets and stars and black holes and cosmic strings and other things which are observed in the universe and you want to see what sort of effect they had. But starting in the 1980's, some physicists started looking at Einstein's field equations the other way around. They start with a desired geometry, find the curvature part of the field equation R - R g/2, and use that to find out what T has to be in order to get that curvature. This was originally done for [[Wormholes|wormholes]], but in 1994 Miguel Alcubierre used this same trick to figure out what could create a region of constant shift vector that moved with the shift vector - the warp drive. With this demand for a warp geometry, the required distribution of matter and energy and momentum and stress can be determined.

The original Alcubierre shift vector N→ produces a spherical warp bubble with an axi-symmetric distribution of, well, everything along the direction of motion with warp velocity vector v→.
<div align="center">
N→ = v→ f(rs)
</div>
Here, rs is the distance from the center of the warp bubble, and f(rs) can be any function that is 1 near rs = 0 and is 0 for rs well beyond the radius of the warp bubble.

=== Challenges and possible resolutions ===

Almost as soon as Alcubierre proposed his warp drive, others began picking it apart.
<ol>
<li>Pfenning and Ford<ref name="PfenningFord_2001">M. J. Pfenning and L. H. Ford, "The unphysical nature of 'Warp Drive'", Classical and Quantum Gravity. 14 (7): 1743–1751 (1997). arXiv:[https://arxiv.org/abs/gr-qc/9702026 gr-qc/9702026]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1997CQGra..14.1743P 1997CQGra..14.1743P]. doi:[https://doi.org/10.1088%2F0264-9381%2F14%2F7%2F011 10.1088/0264-9381/14/7/011]. S2CID [https://api.semanticscholar.org/CorpusID:15279207 15279207].</ref> found that if the negative energy density wall around the bubble obeys [[Wormholes#Quantum_energy_inequalities|quantum energy inequalities]], then for warp speeds of around light speed the shell thickness would be on the order of a hundred Planck lengths; a distance far smaller than any other known physical phenomenon or object.
<li>In the same work, Pfenning and Ford also found that for a bubble a hundred meters in radius, a warp speed of around the speed of light, and the required thickness of about 100 Planck lengths the energy in the bubble shell would be E ≈ -1063 kg c2. The magnitude of this latter value is ten orders of magnitude larger than the energy of the entire visible universe. They do note, however, that if quantum energy inequalities can be ignored then a warp drive with a hundred meters radius but a shell thickness of one meter would "only" have an energy magnitude of about a quarter solar mass.
<li>As with any method of faster than light travel, the warp drive could be used to make a time machine.
<li>Perhaps most seriously, if the warp drive is going faster than light speed, the negative energy regions on the outside of the shell won't be in the warp parts of the bubble. They will have to be moving through space at faster than light speed if the bubble is to maintain its integrity, which is the very problem that the warp drive was designed to avoid. The outside of the warp bubble shell that maintains the warp bubble would fall away and the warp bubble would quickly erode to sub-luminal speeds.<ref name="Van Den Broeck 2000">Chris Van Den Broeck, "On the (im)possibility of warp bubbles", arXiv:gr-qc/9906050 https://doi.org/10.48550/arXiv.gr-qc/9906050</ref>
<li>A more general problem that includes the previous one is that the geometry of the warp drive determines the distribution of energy, momentum, and stress (pressure, shear, and tension) by Einstein's field equation, which states that the stress energy tensor is directly proportional to a reduced order curvature tensor of space-time. However, the stress-energy tensor must also obey the continuity equation – basically, for the energy to change in a given volume of space that energy must come in or go out of the boundary of that volume and similarly for the momentum. So a warp drive can have the warp condition for one moment in time. But after that, the stress-energy that maintains the warp will start to move according to the forces on it and according to the natural paths that it takes through the warped space-time. There is no guarantee that the configuration of the stress-energy at later times will be sufficient to maintain the warp.
<blockquote>
From the perspective of simulating the warp drive dynamically, the key challenge is stability. One can straightforwardly generate the initial metric that describes the Alcubierre metric and infer the instantaneous matter configuration that must support it using the Einstein Equation, but a time-domain evolution of the coupled matter-gravity system necessitates that we specify the equation of state of the matter, which in turn determines how it evolves. In a perfect fluid, this involves specifying the relation between the fluid pressure and density, but in a more general case, the full relation between the
stress-energy components must be given. There is (to our knowledge) no known equation of state that would maintain the warp drive metric in a stable configuration over time - therefore, whilst one can require that initially, the warp bubble is constant, it will quickly evolve away from that state and, in most cases, the warp fluid and spacetime deformations will disperse or collapse into a central point.
 
– Clough, Dietrich, and Khan<ref>Katy Clough, Tim Dietrich, and Sebastian Khan, "What no one has seen before: gravitational waveforms from warp drive collapse", arXiv:2406.02466 [gr-qc] https://doi.org/10.48550/arXiv.2406.02466</ref>
</blockquote>
<li>In order to get the matter (or more generally, stress-energy) that forms the warp shell up to speeds that could allow the shell to exist, you need to be able to, well, accelerate that matter to warp speeds. Which gives a chicken-and-egg problem - in order to achieve superluminal travel you need to be able to get matter or energy moving superluminally in the first place<ref name="Bobrick and Martire">Alexey Bobrick and Gianni Martire, "Introducing Physical Warp Drives", arXiv:2102.06824 [gr-qc] https://doi.org/10.48550/arXiv.2102.06824</ref><ref>D. H. Coule, "No warp drive", Class. Quantum Grav. 15 (1998) 2523–2527. https://doi.org/10.1088/0264-9381/15/8/026</ref>, which rather makes warp drives unnecessary from the get-go.
<li>A warp bubble moving faster than light will not be able to get any signals in front of it as it will overtake its own signal. In fact, anyone in the interior of the bubble would not be able to send signals to the front surface of the bubble. In science-speak, a horizon forms at the front of the bubble<ref name="Natario">José Natário, "Warp drive with zero expansion", Classical and Quantum Gravity. 19 (6): 1157–1166 (2002). arXiv:[https://arxiv.org/abs/gr-qc/0110086 gr-qc/0110086]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2002CQGra..19.1157N 2002CQGra..19.1157N]. doi:[https://doi.org/10.1088%2F0264-9381%2F19%2F6%2F308 10.1088/0264-9381/19/6/308]. S2CID [https://api.semanticscholar.org/CorpusID:15859984 15859984].</ref>. As a consequence, the bubble cannot be controlled, steered, nor stopped from within the bubble itself.
<li>A Horizon will naturally emit Hawking radiation. It is estimated that the radiation will raise the temperature of the interior of the bubble to approximately 1032 K<ref name="Finazzi 2009">Stefano Finazzi, Stefano Liberati, and Carlos Barceló, "Semiclassical instability of dynamical warp drives", Phys. Rev. D 79, 124017 (12 June, 2009) https://doi.org/10.1103/PhysRevD.79.124017 , arXiv:0904.0141 [gr-qc] https://doi.org/10.48550/arXiv.0904.0141</ref> (at this level of accuracy, the result is the same if you replace temperatures in K by the same number in °C or °F). This temperature, however, may be lower if quantum inequalities do not have to be fulfilled.
<li>Semi-classical instabilities at the horizon are expected to lead to a divergence of the stress-energy tensor at the boundary in a short period of time, thus leading to the destruction of the warp bubble<ref name="Finazzi 2009"></ref>.
</ol>
Helpfully, Van Den Broeck<ref name="VanDenBroeck_2000">C. Van Den Broeck, "Alcubierre’s Warp Drive: Problems and Prospects" AIP Conference Proceedings. 504: 1105–1110 (2000). Bibcode:[https://ui.adsabs.harvard.edu/abs/2000AIPC..504.1105V 2000AIPC..504.1105V]. doi:[https://doi.org/10.1063%2F1.1290913 10.1063/1.1290913].</ref> suggested several solutions – or at least mitigations – for some of these problems
<ol>
<li>Quantum inequalities had not been shown to be true in general for highly curved space-times.
<li>A proposed warp drive geometry<ref name="VanDenBroeck_1999">C. Van Den Broeck, "A 'warp drive' with more reasonable total energy requirements". Classical and Quantum Gravity. 16 (12): 3973–3979 (1999). arXiv:[https://arxiv.org/abs/gr-qc/9905084 gr-qc/9905084]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1999CQGra..16.3973V 1999CQGra..16.3973V]. doi:[https://doi.org/10.1088%2F0264-9381%2F16%2F12%2F314 10.1088/0264-9381/16/12/314]. S2CID [https://api.semanticscholar.org/CorpusID:15466313 15466313]. </ref> (see the van Den Broeck drive, below) would be able to minimize the magnitude of the energy required to about that of the mass of our sun. While still large, it is not unphysically large.
<li>Time travel is always going to be a worry with faster than light travel. There's no neat solution to this one.
<li>It may be possible for the negative energy outside of the superluminal region to compress into the superluminal region to form a shock in space-time. The front surface of the bubble would then be a singularity. It is not clear if such a sharp jump in physical properties is possible, but neither is sure that it is impossible, either. However, while this might help with the negative energy in front of the bubble getting swept up because it cannot move faster than light, it does little to help with the material at the back of the bubble on the outside getting left behind because it cannot keep up.
</ol>
Other mitigating research has also shown
<ol start=9>
<li>Semi-classical divergence only occurs in warp drives with only one spatial dimensions. Those with two or more spatial dimensions (as in our universe) only contain points where divergences occur, and these divergences can be managed. In fact, their effects can be reduced by making the warp shape more elongated<ref>Carlos Barceló, Valentin Boyanov, Luis J. Garay, Eduardo Martín-Martínez,
and Jose M. Sánchez Velázquez, "Warp drive aerodynamics", JHEP08 (2022) 288 https://doi.org/10.1007/JHEP08(2022)288 , arXiv:2207.06458 [gr-qc] https://doi.org/10.48550/arXiv.2207.06458</ref>.
</ol>

=== Alcubierre warp interactions with light and matter ===

Space is not entirely empty. It is filled with a diffuse plasma in the form of the interstellar medium, as well as cosmic radiation, light from stars, and cosmic microwave background radiation. A warp drive propagating through space will encounter this stuff. What happens when this matter and radiation have a warp bubble pass across them?

The first analysis of matter encountering a warp bubble was performed by Pfenning and Ford<ref name="PfenningFord_2001"></ref>. They looked at the warp bubble interaction with a massive object at rest with the frame of reference of the warp drive (keep in mind that the rest frame is the same inside and outside the warp bubble; in this rest frame objects inside the bubble have no momentum even though they are, in some sense, changing location rapidly with time). When the warp bubble passes, the object experiences an acceleration in the direction of the warp bubble motion. When the warp shell passes and the object is inside the bubble, it will be moving with approximately the speed of the bubble. Pfenning and Ford analyzed this problem with a continuous distribution of shell energy that, strictly speaking, never falls to zero except at the bubble center and at spatial infinity, so unless the object passes through the center in Pfenning and Ford's description it will never quite get up to the bubble's speed. In this case, the object will move almost as fast as the object but will pass through the bubble in a finite time, after which it will again be at rest with respect to the reference frame but displaced along the direction of the warp bubble motion by some distance. With this description of the warp bubble, a spacecraft of finite size will always be moving a little slower than the warp bubble and would have to use rockets to keep up with it. in addition, the spacecraft would experience tidal forces that would cause stress on the spacecraft's structure.

Pfenning and Ford also examined cases where the shell is of a finite (possibly infinitesimal) width and falls to zero both inside and outside the bubble. In this case any matter encountering the bubble would be collected at the bow of the bubble and thereafter move along with it.

McMonigal et al. analyzed the situation for both massive particles and light moving along the axis of travel of the bubble<ref name="McMonigalEtAl_2012">B. McMonigal, G. F. Lewis, and P. O'Byrne, "Alcubierre warp drive: On the matter of matter". Physical Review D. 85 (6) 064024 (20 March 2012). arXiv:[https://arxiv.org/abs/1202.5708 1202.5708]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2012PhRvD..85f4024M 2012PhRvD..85f4024M]. doi:[https://doi.org/10.1103%2FPhysRevD.85.064024 10.1103/PhysRevD.85.064024]. S2CID [https://api.semanticscholar.org/CorpusID:3993148 3993148].</ref>. They found that light moving opposite the warp direction passed through the bubble without incident, being only somewhat delayed by passing through the bubble. A warp bubble that is warping at sub-luminal speeds can have light catch up from behind it. This light is able to pass through the bubble, and is somewhat advanced in its path by the speed of the bubble. For super-luminal warp bubbles, however, the situation is different. The bubble will catch up to light moving it its own direction that is originally in front of it. This light cannot escape forward, the bubble being too fast. Nor can it escape backward, as the light is propagating forward and the interior of the bubble is at rest. Thus, the light gets caught at the bow of the warp bubble, unable to escape for so long as the warp bubble is active. This light is strongly blue shifted to extremely energetic x-rays and gamma rays. The space behind the bubble is swept clear of forward-moving light.

The situation for matter over-run by a super-luminal warp bubble is similar. Matter moving backward passes through the bubble, being only swept a ways along the bubble's path as sign of its passing but otherwise continuing on their way. Matter at rest behaves the same way as Pfenning and Ford discovered. Meanwhile, matter moving in the same direction of the warp is overtaken and collects at the bow of the bubble, unable to leave for so long as the bubble is warping. This matter "surfing" on the bubble bow is highly accelerated to relativistic speeds, experiencing extreme time dilation.

For a sub-luminal bubble, there is a speed faster than the warp speed where particles coming up from behind overtake the bubble and pass through, out the front. Particles below this speed but still faster than the warp speed still overtake the bubble, but then bounce backwards out of the bubble from behind. They are still moving forward, but are now moving more slowly than the warp speed. Particles moving forward but slower than the bubble will be overtaken and then bounced out the front with a speed higher than the warp speed. Those particles moving backward will pass through the bubble and exit with their initial speed.

The spacecraft inside the bubble will observe that light moving backward compared to the warp direction is blue-shifted, and matter moving backward passes through with increased energy. Meanwhile, a sub-luminal bubble will see light that was moving forward as red-shifted and matter catching up to the bubble from behind will pass by with reduced energy (superluminal bubbles, of course, do not have any matter or radiation catching up to them from behind).

When a superluminal warp bubble that has been collecting matter and radiation for the duration of its journey and blue-shifting it to much higher energies is turned off, all that matter and energy is released as a blast of radiation in the direction the bubble was warping.

McMonigal et al. conclude by noting that any spacecraft in the warp bubble would need shielding to protect against the blue-shifted radiation and matter of increased energy. In addition the destination would be "blasted into oblivion" by the release of matter and radiation that had been caught in the bubble during the trip.

Clark et al.<ref>Chad Clark, William A. Hiscock, and Shane L. Larson, "Null geodesics in the Alcubierre warp drive spacetime: the view from the bridge", arXiv:gr-qc/9907019 (2018) https://doi.org/10.48550/arXiv.gr-qc/9907019</ref> examine what it would look like from within an operating warp drive. They find that light moving at 90 degrees to the direction of motion will look like it is still coming from 90 degrees. However, light from the front hemisphere is squished progressively toward the forward point the faster you go. For a warp bubble moving faster than the speed of light, there is a cone behind the warp bubble where no light can reach it. However, you don't see a black spot behind you – light from the regions in the backward hemisphere from outside that cone that can reach you is distorted toward the direction behind you to fill in that spot. Light from the forward direction is blue-shifted and light from behind is red-shifted. At speeds of around 200 times light speed, the cosmic microwave background would be blue shifted to energies similar to that of the solar photosphere, heating anything in the warp bubble as if they were close to the sun.

== Other warp geometries ==

The Alcubierre warp drive is the first and best known of the warp drives. It has been around long enough that there have been many studies done on it. The warp geometry is particularly simple, making analysis easier. But in the time that the Alcubierre warp was proposed, many other researchers have come up with other warp designs, and even generalized the idea to include a wide array of additional possibilities.

Most of the general statements about the challenges of the Alcubierre drive hold for these other drives. Many of the details about interactions with matter are also likely to hold, although these other styles of warp drive have not yet been subject to such rigorous investigation.

=== Van Den Broeck warp drive ===

If an Alcubierre warp bubble a hundred meters across requires a magnitude of energy greater than the entire energy in the observable universe, one option to reduce the magnitude of energy used is to make the warp bubble smaller. Much smaller. Pfenning and Ford<ref name="PfenningFord_2001"></ref> suggested making the warp bubble smaller than an atom. Of course, that brings up the problem of how to stuff a spacecraft in there. Van Den Broeck proposed a solution<ref name="VanDenBroeck_1999"></ref>: make the warp bubble only about the size of an atomic nucleus but expand the space inside the bubble enormously so that a spacecraft could fit in. This is somewhat like a nuclear sized wormhole that leads to a pocket universe that holds the spacecraft. Unfortunately, the metric doesn't fit neatly into any [[Wormholes#Three_dimensions|embedding diagram]] that I can figure out, but the math works out so that spatial coordinates inside the bubble are expanded by a factor of 1017 and a spacecraft would have a few hundred meters of bubble interior to putz around in. This trick manages to reduce the magnitude of energy to only about the mass energy of a few stars similar to our own. Other than that, it is otherwise a normal Alcubierre drive.

Further work<ref>S. Krasnikov, "The quantum inequalities do not forbid spacetime shortcuts", Physical Review D. 67 (10) 104013. https://doi.org/10.1103%2FPhysRevD.67.104013, arXiv:gr-qc/0207057 (2003) https://doi.org/10.48550/arXiv.gr-qc/0207057 </ref> has managed to use Van Den Broeck's trick to reduce the magnitude of the negative energy to only about a milligram.

=== Natário warp drive ===

The Alcubierre drive is not the only way to construct a warp drive. José Natário<ref name="Natario"></ref> showed that it was just one example of an entire class of warp drives. These are space-time geometries with a smooth and localized shift vector field and a unit lapse function.

Natário then went on to find in that class a set of warp drives that do not have any expansion or contraction of space at all. Instead, space encountering the front boundary of the warp bubble instead "slides" around the outside of the bubble until it gets to the corresponding place on the back and is left behind there. Another way to think of it is that space entering the bubble shell is compressed in the radial direction of the shell but is simultaneously expanded in the tangential direction so that there is no net change in volume. When it gets to the back, the opposite occurs. In the language of mathematics, the shift vector field is divergenceless.

<table align=center>
<tr><td width=360>[[File:Natario_drive_flow_of_space.png]]
<td width=445>[[File:Natario_warp_drive_horizons.png]]
<tr><td style="vertical-align: top;">A representation of the motion of space in a zero-expansion Natário drive with a vanishingly thin warp shell around the bubble.
<td>The horizons of the generalized Natário class of warp drives (including the Alcubierre warp drive and all fell-Heisenberg drives) at superluminal speed in the direction of the green arrow. No event inside the bubble can affect events outside the bubble in front of the blue line. No events outside the bubble behind the red line can affect events inside the bubble. The lines have an angle α with respect to the direction of motion with sin(α) = 1/vs for vs the speed of the bubble relative to light speed. This is analogous to the Mach cone of supersonic objects.
</table>

NOTE: There are two different things that can be meant by the term "Natário warp drive". The generalized Natário warp drive is the general theory of all warp drives that includes within it the Alcubierre, Fell-Heisenberg, and Lentz warp drives, among infinitely many others. The zero-expansion Natário warp drive is the specific implementation of the generalized Natário warp drive that has a divergenceless shift vector field.

The generalized class of warp drives developed by Natário (including both the Alcubierre and this zero-expansion warp drive) are shown to always have regions in the warp shell where the energy density is negative to at least some observers. You can't get away from it – to warp, you need negative energy density.

The Natário class of warp drives blueshift light coming in from the front and redshift light catching up from the back. For the case of a super-luminal warp bubble, there will of course be a horizon at the back of the bubble and light will not be able to get through from behind. For a bubble speed vs relative to the speed of light, light coming from straight ahead will be blueshifted up in frequency and photon energy by a factor of 1 + vs. For sub-luminal travel, light coming from behind will be redshifted by a factor 1 - vs. In general for n as the unit direction along which the light is propagating, the blueshift factor will be 1 + vs ⋅ n. If the warp shell is infinitesimally thin, light from outside that reaches the center of the bubble will not be distorted in direction although it will be frequency shifted, but observers still inside the bubble but displaced from the center will see the field of view distorted as well as frequency shifted.

Most of the difficulties that apply to the Alcubierre warp drive also apply to the generalized Natário warp drive (and thus to the zero-expansion Natário drive and Fell-Heisenberg drive as well). In particular, quantum energy inequalities, high magnitudes of the energy required, formation of time machines, instabilities related to the stress-energy making up the warp shell being unable to keep up with the bubble, the chicken-and-egg problem of needing FTL to achieve FTL, the formation of horizons making warp bubbles uncontrollable, and Hawking radiation at the horizons, are all still concerns for generic warp drives.

=== Fell-Heisenberg warp drives ===

S. D. B. Fell and L. Heisenberg investigated what sort of warp structure is necessary to produce a warp drive with positive energy density everywhere<ref name="Fell-Heisenberg">S. D. B. Fell and L> Heisenberg, "Positive energy warp drive from hidden geometric structures", Classical and Quantum Gravity 38 155020 (2021) https://doi.org/10.1088/1361-6382/ac0e47 https://arxiv.org/abs/2104.06488</ref>. They found that warp drives of the generic Natário form with irrotational shift vectors had a simple expression for the energy density and vanishing momentum (irrotational means that the shift vector can be described as the gradient of a scalar potential field or, equivalently, that the curl of the shift vector field vanishes everywhere). Fell and Heisenberg proposed several potential fields that either had positive energy density everywhere except at discontinuities, or that were continuous but had regions that violated the weak energy condition (i. e. negative energy densities in some frames of reference). One specific example they provided of a warp field had an energy that is about 10,000 times less than the mass-energy of our sun. Or only about half the mass-energy of Jupiter. This is a significant improvement over many other proposed drives.

Despite the introduction discussing warp drive configurations that satisfy the various energy conditions, the configurations described in the paper are shown to locally violate the weak and strong energy conditions. Nonetheless, the energy density is still mostly positive. Santiago, Schuster, and Visser's<ref name="Santiago Schuster Visser 2022">J. Santiago, S. Schuster, and M. Visser, "Generic warp drives violate the null energy condition", Physical Review D 105, 064038 (2022) https://doi.org/10.1103/PhysRevD.105.064038</ref> work shows that the various energy conditions must still be violated by this warp drive; to not violate these, energy density must be positive in all reference frames not just those of the co-moving observer.

Any vector field can be described by its divergence, its curl, and a constant. The constant part of a warp drive's shift vector can be discarded because then the warp drive would not be localized. It is interesting, therefore, that the Natário zero-expansion drive has zero divergence and can be described only by its curl, while the Fell-Heisenberg drive has zero curl and can be described only by its divergence. These two classes of drives thus rely on opposite choices of setting one of the two different parts describing a vector field to zero. As might be expected, the Fell-Heisenberg drive is not zero expansion – space-time will change in volume as it passes through the warp bubble (although after going all the way through it will return to its original volume). However, objects in space-time will retain their original orientation as they pass through the warp bubble.

Any vector field with zero curl can be represented as the gradient of a suitable scalar function. Thus, the Fell-Heisenberg drive can be completely described by just scalar field (one number at all points of space and time) rather than a vector field.

Previous examples of warp drives, such as the Alcubierre and zero-expansion Natário drives, tend toward a flat space-time as you go far enough away faster than would occur if there were any mass present in the warp drive. That is, the ADM mass of such drives is zero. However, one of the Fell-Heisenberg examples described a spacetime where the geometry approaches that of what you would get for a massive object at the warp location, giving one of the first examples of a warp drive with mass.

In the Fell-Heisenberg drive the momentum vanishes everywhere. Yet the energy occupies regions where the shift vector is varying rapidly. The lack of momentum means that the energy will not move to keep up with the differential expansion and movement of the spacetime. As a consequence, at later times the energy will have a different distribution than what is necessary to maintain the given warp configuration. The exact time evolution is not solved but this likely leads to collapse of warp bubble. Note that this is not a problem unique to the Fell-Heisenberg drive. Assuming a geometry that describes a warp produces a specific distribution of the stress-energy tensor that satisfies Einstein's equation. But there is no guarantee that this stress energy tensor satisfies the continuity equation, which describes how the distribution of matter and momentum change with time. The issue of stability of any warp drive is not well understood, but most warp drive geometries will be unstable in the sense that their matter that generates the warp fields will not be able to keep up with the motion of the warp. The Fell-Heisenberg drive is unique only in that, with vanishing momentum, there is no configuration that allows stable propagation.

(n. b. The Heisenberg here is Lavinia Heisenberg, not the Werner Heisenberg of quantum physics and uncertainty principle fame.)

=== Lentz warp drive ===

E. Lentz<ref name="Lentz">E. W. Lentz, "Breaking the warp barrier: hyper-fast solitons in Einstein–Maxwell-plasma theory", Classical and Quantum Gravity. 38 075015 (2021). arXiv:[https://arxiv.org/abs/2006.07125 2006.07125]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2021CQGra..38g5015L 2021CQGra..38g5015L]. doi:[https://doi.org/10.1088%2F1361-6382%2Fabe692 10.1088/1361-6382/abe692]. ISSN [https://search.worldcat.org/issn/0264-9381 0264-9381]. S2CID [https://api.semanticscholar.org/CorpusID:219635854 219635854].</ref> investigated a particular configuration of the Fell-Heisenberg class of warp drives, where the scalar field forms a selection of rhomboid regions, resulting in an energy density that is everywhere positive and a shift vector with vanishing tidal force at the center for a comfortable ride.

However, Santiago, Schuster, and Visser<ref name="Santiago Schuster Visser 2022"></ref> dispute claims that the Lentz drive satisfies the energy conditions, noting that everywhere positive energy density in one frame of reference is insufficient to establish that the energy density is positive in all reference frames; knowledge of the Cauchy stress tensor is also needed. They show that any generic warp drive will violate the strong energy condition, null energy condition, and weak energy condition.

== Conservation laws ==

All the warp drives investigated so far have geometries that quickly fall off from the highly curved region near the warp bubble to a region of spacetime that is almost flat; and becomes even more flat the farther away from the warp bubble you get. There is a word for geometries of this kind – they are called asymptotically flat. Basically, you can always go far enough away in any direction to reach a place where the geometry is sufficiently flat to meet any flatness criterion you choose.

When you have an asymptotically flat geometry, then in the far away mostly flat regions gravity is well described by a linearized theory of gravity – where the perturbations to the geometry are linear in the stress energy tensor. This means that gravity is basically Newtonian, along with a few other bells and whistles like gravitational waves and frame dragging.

When linearized gravity is a good approximation, then it is known that energy, momentum, and linear momentum are conserved. Although parts of a warp geometry are highly curved and non-linear, you can measure the conserved energy and momentum and angular momentum from far away in the linear region. This ADM energy and etcetera is the quantity that is conserved when you are worrying about conserving such things, not necessarily the energy and so forth you get in the highly curved spacetime region. In fact, it is not even simple to define where the energy and such actually is withing the highly curved spacetime region; it is non-local and different observers will disagree on how it is distributed. But everyone agrees on the ADM conserved quantities from out in nearly flat spacetime (up to the usual changes due to changing speeds, which are trivial and come up with normal objects as well).

And now we run into a bit of a conundrum: most of the warp geometries so far proposed have zero ADM mass, meaning that things like the Alcubierre drive technically have no energy or momentum or angular momentum at all. Despite all of the extremes of negative energy regions within the wall of the warp bubble, the actual energy that you need to spend to make one is zero. This can lead to a bit of a problem. If you have a spacecraft that has mass, that mass-energy has to be conserved. So when you turn on your warp drive and the total warp mass is zero, where does that energy go?

One resolution is just that the mass of the spacecraft is negligible and the small amount of curvature it causes can be ignored. In this case, the warp geometries discussed so far are merely an approximation that neglects this effect. In fact, some of the proposed Fell-Heisenberg drives map not onto entirely flat spacetime geometries at large distances, but onto the Schwarzschild geometry that is appropriate for systems with mass at the center. If this is the case, then the warping bubble and spacecraft system will need to worry about conservation of all their conserved properties.

Let's look at what the conservation laws mean for a spacecraft equipped with a warp drive, assuming that no matter or radiation are emitted by activating or deactivating the warp drive.

We can start with an observer floating in space. We'll call her Alice. And, approaching Alice at 100 m/s is the starship Enterprise with a mass of a cool 200,000 tons (2×108 kg). from this we know that the Enterprise has, in Alice's frame of reference, a kinetic energy of 1 TJ, a total energy (including mass and kinetic energy) of 1.8×1025 J, a momentum of 2×1010 kg m/s, and – because it is headed straight toward Alice – an angular momentum of zero. All of this is without any warp drive activated.

On the bridge of the Enterprise, Spock picks up Alice on the sensors and tells Captain Kirk. Kirk does not want to run into Alice (he'd rather date her), so he orders helm to turn on the Enterprise's warp drive and move them 1 km away so that their course takes them only 1 km from Alice instead of straight into her. Sulu warps the Enterprise according to these commands, and then turns off the warp drive.

After this maneuver, keeping the Enterprise's total energy and momentum the same means that the Enterprise still has a mass of 200,000 tons and a velocity of 100 m/s in its original direction. So we can see that the warp drive must preserve the velocity vector of the thing in it (at least in flat spacetime where outside forces are not acting on it). The problem comes from angular momentum. With a known mass of 200,000 tons, a speed of 100 m/s, and a distance of closest approach of 1 km; from Alice's frame of reference the Enterprise now has an angular momentum of 2×1013 kg m2/s after the warp maneuver, while before the maneuver it has an angular momentum of zero.

From this we can see that warp drives cannot be reactionless. Arbitrary displacements using warp without emitting propellant result in violation of the law of conservation of angular momentum. As long as we restrict ourselves to accepted science based on what we know of general relativity, to move a warp drive with mass you need to accelerate it or decelerate it with rockets or other normal means of delivering momentum (such as light sails, mag sails, or pushing off of a planet). Warp drives must be inertial to comply with physics. And all of this was analyzed only from times when the warp drive was turned off; it does not depend on any assumptions about what happens when the warp is on.

(ADM mass -> 0; conservation of energy/mass issues.)<ref>Sebastian Schuster, Jessica Santiago,and Matt Visser, "ADM mass in warp drive spacetimes" General Relativity and Gravitation (2023) 55:14 https://doi.org/10.1007/s10714-022-03061-9</ref>

(Discuss issues of conservation of angular momentum.)

(If angular momentum conservation is ignored, discuss how momentum & energy are affected by outside forces. In a gravitational field equivalent to inertial frame ... like being in an accelerated elevator; will acquire momentum buildup while staying "at rest".)

== Warp Railroads ==

Van Den Broeck<ref name="VanDenBroeck_2000"></ref> suggested a warp railroad,

You can set up devices ahead of time along the path of the bubble that produce the negative energy regions for the warp bubble at the appropriate time without needing the negative energy to ever move with the bubble at all. At least for bubbles on a predictable schedule and route.

helps with:

-stability (elaborate)

-conservation laws (elaborate)

Mention Krasnikov<ref name="Krasnikov 1998">S. V. Krasnikov, "Hyperfast travel in general relativity" Phys. Rev. D 57 4760</ref>

Hyperwave<ref name="Pieri 2024">Lorenzo Pieri, "Hyperwave: hyper-Fast Communication within General Relativity", arXiv:2311.12069 [gr-qc] https://doi.org/10.48550/arXiv.2311.12069</ref> makes use of this warp railroad idea.

== Quantum effects ==

(quantum stuff here<ref name="Hiscock1997">W. A. Hiscock, "Quantum effects in the Alcubierre warp drive spacetime", Classical and Quantum Gravity 14 L183 https://doi.org/10.1088/0264-9381/14/11/002 https://arxiv.org/abs/gr-qc/9707024</ref><ref name="Finazzi et al 2009">S. Finazzi, S. Liberati, C. Barceló, "Semiclassical instability of dynamical warp drives", Physical Review D 79, 124017 (2009)https://doi.org/10.1103/PhysRevD.79.124017 https://arxiv.org/abs/0904.0141</ref>)

From<ref name="Pieri 2024"></ref>
<blockquote>
It should be emphasised that Quantum Inequalities have not been proven for realistic interacting quantum fields, where the they are expected to be weaker if they exist [Cadamuro, 2019], and they often require Minkowskian conditions at infinity, a condition which is not appropriate for fields inside bounded regions, as in a Casimir devices.
</blockquote>

== Warp drives and black holes ==

What happens if you drive your warp drive into a black hole?

The pop-sci reason that nothing can escape a black hole is that at the event horizon the escape velocity is faster than light and nothing can go faster than light. A warp drive can go faster than light.

The more detailed reason from general relativity is that at the event horizon space and time are rotated sufficiently that "inward" becomes "forward in time." You can no more go farther away from the singularity in a black hole once you are inside the event horizon than you can go backward in time. Any super-luminal travel can go backward in time, and warp drives can engage in super-luminal travel.

The Hawking area theorem that says that black holes can only grow, never shrink relies on the condition that the energy is everywhere positive. Warp drives have negative energy density.

So in principle, there's nothing that prevents a warp drive from taking a dip into a black hole and coming back out again.

Garattini and Zatrimaylov looked into the problem of a warp drive near (and in) a black hole<ref name="Garattini and Zatrimaylov">R. Garattini and K. Zatrimaylov, "Black holes, warp drives, and energy conditions", Physics Letters B 856 138910 (2024) https://doi.org/10.1016/j.physletb.2024.138910</ref>. The found that the black hole reduces the negative energy requirement for the warp bubble to move inward toward the black hole singularity, but increases the negative energy needed to move away from the center of the black hole. In addition, a warp drive parked at the event horizon would allow light from inside the horizon to pass through the bubble and escape back outside.

== Credit ==
Author: Luke Campbell

== References ==

[[Category:Physics]][[Category:Infrastructure]][[Category:Physics & Engineering‏‎]][[Category:Physics & Math & Engineering]][[Category:Transportation & Infrastructure‏‎]][[Category:Metric Engineering]]

Warp Drives

2026-07-30T03:48:16Z

Lwcamp: /* Conservation laws */

{{PageConstructionNotice}}

Science fiction often features spacecraft that can seemingly move across space and get between the place of departure and the destination much faster than light could have done. This appears to contradict the theory of relativity, which predicts unequivocally that nothing can move through space faster than light. Because relativity has been incredibly successful at describing nature, with its many other predictions regularly being confirmed to extraordinary accuracy and within the bounds of uncertainty of all the experiments that tested them, it gives confidence that relativity is a correct description of reality. Which seems to rather throw a wet towel on our hopes for rapid travel between stars.

However, while relativity does not allow things to move through space faster than light, it places no such restrictions on how fast space-time itself can expand, contract, or move around. This leads to the idea of a warp drive – the spacecraft remains stationary within a region of highly curved space-time, and that region moves at super-luminal speeds rather than the spacecraft.

== The Alcubierre warp drive ==

The first warp drive geometry that satisfied the Einstein field equations of relativity was proposed by Miguel Alcubierre<ref>M. Alcubierre, "The warp drive: hyper-fast travel within general relativity." Classical and Quantum Gravity. 11 (5): L73–L77 (1994). [https://arxiv.org/abs/gr-qc/0009013 arXiv:gr-qc/0009013]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1994CQGra..11L..73A 1994CQGra..11L..73A]. doi:[https://doi.org/10.1088%2F0264-9381%2F11%2F5%2F001 10.1088/0264-9381/11/5/001]. S2CID [https://api.semanticscholar.org/CorpusID:4797900 4797900].</ref>. In this geometry, a sphere of space-time moves at an arbitrary speed (potentially but not necessarily a speed much faster than light). Objects within the sphere are moved along with the sphere; an object at rest within the sphere would be moved along with the sphere indefinitely. Space is expanding at the rear boundary and contracting at the front boundary in order to keep the sphere moving. In order to satisfy the Einstein field equations, the boundary of the sphere must have a negative energy density. The challenges of space-time geometries with negative energy densities are described in our page on [[Wormholes#Exotic_energy_conditions|wormholes]], for our purposes it is enough to note that negative energy densities can pose problems if not handled carefully, there may be limits on how much negative energy you can have without nearby positive energy density, and it may not be possible to get enough negative energy to support a warp drive; although none of this is ruled out by physics – yet!

<table align=center>
<tr><td width=650>[[File:Energy_magnitude_of_Alcubierre_drive.png]]
<td width=650>[[File:Expansion_of_Alcubierre_drive.png]]
<tr><td>The magnitude of the energy of the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. (The energy density is negative everywhere in the shell, here only the magnitude is shown to aid visual comprehension.)
<td>The expansion of space in the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. Negative expansion means that space is contracting in that direction, positive expansion that it is expanding.
</table>

NOTE: What the? The energy distribution is symmetric! How does the drive know which way to go? There must be significant contributions of other components of the stress-energy tensor, and those have got to be asymmetric along the forward/backward axis. Check this when I get time.

(And what's with the ring in all the artwork, anyway?)

=== So what is going on? ===

What is it that makes a warp drive a warp drive, and how does it work?

A warp drive is a concept that comes out of the science of general relativity. General relativity collects space and time together into a single "structure" called spacetime. This structure is curved, and its curvature is determined by the material that is in it. Absent forces acting between matter, particles of matter move through spacetime on the shortest possible paths in this curved environment.
<blockquote>
"Spacetime tells matter how to move; matter tells spacetime how to curve." - John Archibald Wheeler
</blockquote>
All of this is encompassed in the Einstein field equation:
<div align=center>
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 8πG
<td rowspan="2" nowrap="nowrap" align="right"> T = R -  
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right"> R g
<tr>
<td style="border-top:solid 1px black;"> c4
<td style="border-top:solid 1px black;"> 2
</table>
</div>
Unfortunately, by convention, both the symbols G and R get used for a lot of different things. Here, symbols in bold are second rank tensors – constructs of arrays of 4×4 numbers where 4 is necessary because that is the number of dimensions of spacetime. Conventionally, these numbers are indexed by a value going from 0 to 3, with 0 being the part corresponding to the coordinate of time and 1 to 3 being the Cartesian coordinates that you might know as x, y, and z.

T is the stress-energy tensor: the (0,0) component represents the energy density at a given point; the (0,i) and (i, 0) components (with i going from 1 to 3) represent the momentum density; and the (i,j) components represent the stresses present, such as pressure or tension or shear.

The other term, R - R g/2, involves a partial description of the curvature. A full description of the curvature requires a construct of 4×4×4×4 components; too much for the stress energy tensor to fully constrain. Which is okay, because this allows curvature to exist away from where matter is, allowing gravity to have long range effects across empty space as well as for phenomena like gravitational waves. But it is enough that a given distribution of matter T gives enough constraints (along with specifying how the geometry behaves far away) that you can generally figure out what the actual gravity is.

This is the usual story that we are told. But there is an alternate way of looking at the same geometry such that the math works out the same, but it is described conceptually in a very different way. This divides spacetime up into "layers", called foliations, that represent slices of space and going from one foliation to the next is increasing in time. This is useful for computation because you can assign coordinates to your system. There are two additional constructs – the lapse function and the shift vector field. Starting at an event on one foliation, the lapse function at that position tells you that after a certain "proper time" (time perceived by an observer at that event) the time coordinate of the observer will be the proper time times the lapse function at that event. And the new spatial coordinates of the observer will be the proper time times the shift vector at that event. (Strictly speaking, this is only exactly true in the limit of the proper time approaching zero or when the lapse function and shift vectors are constant. But hopefully this gives a good enough insight to what these entities are so that we can go on to describe warp drives.)

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Lapse_shift.png]]
<tr><td>A folliation of space-time into slices of constant time coordinates (grey). At an event (black circle), the lapse function (yellow) shows how far into the future the point at that event will be projected in a unit amount of proper time experienced by an observer initially at the event. The shift vector (red) shows how that point changes coordinates after the unit amount of proper time. The cyan vector shows the new coordinates the point is projected onto. The lower black surface indicates a foliation of spacetime at the same time coordinate; after unit proper time, the events on the lower foliation get projected to the curved upper black foliation according to the shift vectors and lapse functions at each of their respective points.
</table>
</table>

Warp drives are generally restricted to spacetime geometries with a constant lapse function - all observers go forward by the same time coordinate after experiencing the same amount of proper time. Thus, all warp drives have no time dilation.
Far from the warp bubble, the shift vector vanishes – each spatial coordinate is mapped to the same spatial coordinate after any amount of proper time so long as the warp drive is not near. But the inside of the warp bubble is an extended region where the shift vector field is non-zero and constant. In this bubble, all spatial coordinates are mapped to a coordinate that shifts linearly in time with the proper time of the observer. Astute readers will see that this linear change in position with time is very similar to a velocity. This region of constant shift vector then moves with the shift vector velocity to keep stuff originally in the bubble so that it stays in the bubble. In between the inside of the bubble and the far away regions with zero shift vector, the shift vector field falls away to match one region to the other.

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Warp_shift.png]]
<tr><td>The shift vector of a one-dimensional slice through warp geometry, in red. At the edges there is no shift. At the center (inside the warp bubble) there is a region of uniform shift so that at a later time the inside of the warp bubble is rigidly translated to different spatial coordinates (in blue). The two foliations remain the same distance apart, because warp drives maintain a unit lapse function.
</table>
</table>

Traditionally, scientists would start with an interesting distribution of matter given by T and then find out what kind of curvature it created. This works well for planets and stars and black holes and cosmic strings and other things which are observed in the universe and you want to see what sort of effect they had. But starting in the 1980's, some physicists started looking at Einstein's field equations the other way around. They start with a desired geometry, find the curvature part of the field equation R - R g/2, and use that to find out what T has to be in order to get that curvature. This was originally done for [[Wormholes|wormholes]], but in 1994 Miguel Alcubierre used this same trick to figure out what could create a region of constant shift vector that moved with the shift vector - the warp drive. With this demand for a warp geometry, the required distribution of matter and energy and momentum and stress can be determined.

The original Alcubierre shift vector N→ produces a spherical warp bubble with an axi-symmetric distribution of, well, everything along the direction of motion with warp velocity vector v→.
<div align="center">
N→ = v→ f(rs)
</div>
Here, rs is the distance from the center of the warp bubble, and f(rs) can be any function that is 1 near rs = 0 and is 0 for rs well beyond the radius of the warp bubble.

=== Challenges and possible resolutions ===

Almost as soon as Alcubierre proposed his warp drive, others began picking it apart.
<ol>
<li>Pfenning and Ford<ref name="PfenningFord_2001">M. J. Pfenning and L. H. Ford, "The unphysical nature of 'Warp Drive'", Classical and Quantum Gravity. 14 (7): 1743–1751 (1997). arXiv:[https://arxiv.org/abs/gr-qc/9702026 gr-qc/9702026]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1997CQGra..14.1743P 1997CQGra..14.1743P]. doi:[https://doi.org/10.1088%2F0264-9381%2F14%2F7%2F011 10.1088/0264-9381/14/7/011]. S2CID [https://api.semanticscholar.org/CorpusID:15279207 15279207].</ref> found that if the negative energy density wall around the bubble obeys [[Wormholes#Quantum_energy_inequalities|quantum energy inequalities]], then for warp speeds of around light speed the shell thickness would be on the order of a hundred Planck lengths; a distance far smaller than any other known physical phenomenon or object.
<li>In the same work, Pfenning and Ford also found that for a bubble a hundred meters in radius, a warp speed of around the speed of light, and the required thickness of about 100 Planck lengths the energy in the bubble shell would be E ≈ -1063 kg c2. The magnitude of this latter value is ten orders of magnitude larger than the energy of the entire visible universe. They do note, however, that if quantum energy inequalities can be ignored then a warp drive with a hundred meters radius but a shell thickness of one meter would "only" have an energy magnitude of about a quarter solar mass.
<li>As with any method of faster than light travel, the warp drive could be used to make a time machine.
<li>Perhaps most seriously, if the warp drive is going faster than light speed, the negative energy regions on the outside of the shell won't be in the warp parts of the bubble. They will have to be moving through space at faster than light speed if the bubble is to maintain its integrity, which is the very problem that the warp drive was designed to avoid. The outside of the warp bubble shell that maintains the warp bubble would fall away and the warp bubble would quickly erode to sub-luminal speeds.<ref name="Van Den Broeck 2000">Chris Van Den Broeck, "On the (im)possibility of warp bubbles", arXiv:gr-qc/9906050 https://doi.org/10.48550/arXiv.gr-qc/9906050</ref>
<li>A more general problem that includes the previous one is that the geometry of the warp drive determines the distribution of energy, momentum, and stress (pressure, shear, and tension) by Einstein's field equation, which states that the stress energy tensor is directly proportional to a reduced order curvature tensor of space-time. However, the stress-energy tensor must also obey the continuity equation – basically, for the energy to change in a given volume of space that energy must come in or go out of the boundary of that volume and similarly for the momentum. So a warp drive can have the warp condition for one moment in time. But after that, the stress-energy that maintains the warp will start to move according to the forces on it and according to the natural paths that it takes through the warped space-time. There is no guarantee that the configuration of the stress-energy at later times will be sufficient to maintain the warp.
<blockquote>
From the perspective of simulating the warp drive dynamically, the key challenge is stability. One can straightforwardly generate the initial metric that describes the Alcubierre metric and infer the instantaneous matter configuration that must support it using the Einstein Equation, but a time-domain evolution of the coupled matter-gravity system necessitates that we specify the equation of state of the matter, which in turn determines how it evolves. In a perfect fluid, this involves specifying the relation between the fluid pressure and density, but in a more general case, the full relation between the
stress-energy components must be given. There is (to our knowledge) no known equation of state that would maintain the warp drive metric in a stable configuration over time - therefore, whilst one can require that initially, the warp bubble is constant, it will quickly evolve away from that state and, in most cases, the warp fluid and spacetime deformations will disperse or collapse into a central point.
 
– Clough, Dietrich, and Khan<ref>Katy Clough, Tim Dietrich, and Sebastian Khan, "What no one has seen before: gravitational waveforms from warp drive collapse", arXiv:2406.02466 [gr-qc] https://doi.org/10.48550/arXiv.2406.02466</ref>
</blockquote>
<li>In order to get the matter (or more generally, stress-energy) that forms the warp shell up to speeds that could allow the shell to exist, you need to be able to, well, accelerate that matter to warp speeds. Which gives a chicken-and-egg problem - in order to achieve superluminal travel you need to be able to get matter or energy moving superluminally in the first place<ref name="Bobrick and Martire">Alexey Bobrick and Gianni Martire, "Introducing Physical Warp Drives", arXiv:2102.06824 [gr-qc] https://doi.org/10.48550/arXiv.2102.06824</ref><ref>D. H. Coule, "No warp drive", Class. Quantum Grav. 15 (1998) 2523–2527. https://doi.org/10.1088/0264-9381/15/8/026</ref>, which rather makes warp drives unnecessary from the get-go.
<li>A warp bubble moving faster than light will not be able to get any signals in front of it as it will overtake its own signal. In fact, anyone in the interior of the bubble would not be able to send signals to the front surface of the bubble. In science-speak, a horizon forms at the front of the bubble<ref name="Natario">José Natário, "Warp drive with zero expansion", Classical and Quantum Gravity. 19 (6): 1157–1166 (2002). arXiv:[https://arxiv.org/abs/gr-qc/0110086 gr-qc/0110086]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2002CQGra..19.1157N 2002CQGra..19.1157N]. doi:[https://doi.org/10.1088%2F0264-9381%2F19%2F6%2F308 10.1088/0264-9381/19/6/308]. S2CID [https://api.semanticscholar.org/CorpusID:15859984 15859984].</ref>. As a consequence, the bubble cannot be controlled, steered, nor stopped from within the bubble itself.
<li>A Horizon will naturally emit Hawking radiation. It is estimated that the radiation will raise the temperature of the interior of the bubble to approximately 1032 K<ref name="Finazzi 2009">Stefano Finazzi, Stefano Liberati, and Carlos Barceló, "Semiclassical instability of dynamical warp drives", Phys. Rev. D 79, 124017 (12 June, 2009) https://doi.org/10.1103/PhysRevD.79.124017 , arXiv:0904.0141 [gr-qc] https://doi.org/10.48550/arXiv.0904.0141</ref> (at this level of accuracy, the result is the same if you replace temperatures in K by the same number in °C or °F). This temperature, however, may be lower if quantum inequalities do not have to be fulfilled.
<li>Semi-classical instabilities at the horizon are expected to lead to a divergence of the stress-energy tensor at the boundary in a short period of time, thus leading to the destruction of the warp bubble<ref name="Finazzi 2009"></ref>.
</ol>
Helpfully, Van Den Broeck<ref name="VanDenBroeck_2000">C. Van Den Broeck, "Alcubierre’s Warp Drive: Problems and Prospects" AIP Conference Proceedings. 504: 1105–1110 (2000). Bibcode:[https://ui.adsabs.harvard.edu/abs/2000AIPC..504.1105V 2000AIPC..504.1105V]. doi:[https://doi.org/10.1063%2F1.1290913 10.1063/1.1290913].</ref> suggested several solutions – or at least mitigations – for some of these problems
<ol>
<li>Quantum inequalities had not been shown to be true in general for highly curved space-times.
<li>A proposed warp drive geometry<ref name="VanDenBroeck_1999">C. Van Den Broeck, "A 'warp drive' with more reasonable total energy requirements". Classical and Quantum Gravity. 16 (12): 3973–3979 (1999). arXiv:[https://arxiv.org/abs/gr-qc/9905084 gr-qc/9905084]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1999CQGra..16.3973V 1999CQGra..16.3973V]. doi:[https://doi.org/10.1088%2F0264-9381%2F16%2F12%2F314 10.1088/0264-9381/16/12/314]. S2CID [https://api.semanticscholar.org/CorpusID:15466313 15466313]. </ref> (see the van Den Broeck drive, below) would be able to minimize the magnitude of the energy required to about that of the mass of our sun. While still large, it is not unphysically large.
<li>Time travel is always going to be a worry with faster than light travel. There's no neat solution to this one.
<li>It may be possible for the negative energy outside of the superluminal region to compress into the superluminal region to form a shock in space-time. The front surface of the bubble would then be a singularity. It is not clear if such a sharp jump in physical properties is possible, but neither is sure that it is impossible, either. However, while this might help with the negative energy in front of the bubble getting swept up because it cannot move faster than light, it does little to help with the material at the back of the bubble on the outside getting left behind because it cannot keep up.
</ol>
Other mitigating research has also shown
<ol start=9>
<li>Semi-classical divergence only occurs in warp drives with only one spatial dimensions. Those with two or more spatial dimensions (as in our universe) only contain points where divergences occur, and these divergences can be managed. In fact, their effects can be reduced by making the warp shape more elongated<ref>Carlos Barceló, Valentin Boyanov, Luis J. Garay, Eduardo Martín-Martínez,
and Jose M. Sánchez Velázquez, "Warp drive aerodynamics", JHEP08 (2022) 288 https://doi.org/10.1007/JHEP08(2022)288 , arXiv:2207.06458 [gr-qc] https://doi.org/10.48550/arXiv.2207.06458</ref>.
</ol>

=== Alcubierre warp interactions with light and matter ===

Space is not entirely empty. It is filled with a diffuse plasma in the form of the interstellar medium, as well as cosmic radiation, light from stars, and cosmic microwave background radiation. A warp drive propagating through space will encounter this stuff. What happens when this matter and radiation have a warp bubble pass across them?

The first analysis of matter encountering a warp bubble was performed by Pfenning and Ford<ref name="PfenningFord_2001"></ref>. They looked at the warp bubble interaction with a massive object at rest with the frame of reference of the warp drive (keep in mind that the rest frame is the same inside and outside the warp bubble; in this rest frame objects inside the bubble have no momentum even though they are, in some sense, changing location rapidly with time). When the warp bubble passes, the object experiences an acceleration in the direction of the warp bubble motion. When the warp shell passes and the object is inside the bubble, it will be moving with approximately the speed of the bubble. Pfenning and Ford analyzed this problem with a continuous distribution of shell energy that, strictly speaking, never falls to zero except at the bubble center and at spatial infinity, so unless the object passes through the center in Pfenning and Ford's description it will never quite get up to the bubble's speed. In this case, the object will move almost as fast as the object but will pass through the bubble in a finite time, after which it will again be at rest with respect to the reference frame but displaced along the direction of the warp bubble motion by some distance. With this description of the warp bubble, a spacecraft of finite size will always be moving a little slower than the warp bubble and would have to use rockets to keep up with it. in addition, the spacecraft would experience tidal forces that would cause stress on the spacecraft's structure.

Pfenning and Ford also examined cases where the shell is of a finite (possibly infinitesimal) width and falls to zero both inside and outside the bubble. In this case any matter encountering the bubble would be collected at the bow of the bubble and thereafter move along with it.

McMonigal et al. analyzed the situation for both massive particles and light moving along the axis of travel of the bubble<ref name="McMonigalEtAl_2012">B. McMonigal, G. F. Lewis, and P. O'Byrne, "Alcubierre warp drive: On the matter of matter". Physical Review D. 85 (6) 064024 (20 March 2012). arXiv:[https://arxiv.org/abs/1202.5708 1202.5708]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2012PhRvD..85f4024M 2012PhRvD..85f4024M]. doi:[https://doi.org/10.1103%2FPhysRevD.85.064024 10.1103/PhysRevD.85.064024]. S2CID [https://api.semanticscholar.org/CorpusID:3993148 3993148].</ref>. They found that light moving opposite the warp direction passed through the bubble without incident, being only somewhat delayed by passing through the bubble. A warp bubble that is warping at sub-luminal speeds can have light catch up from behind it. This light is able to pass through the bubble, and is somewhat advanced in its path by the speed of the bubble. For super-luminal warp bubbles, however, the situation is different. The bubble will catch up to light moving it its own direction that is originally in front of it. This light cannot escape forward, the bubble being too fast. Nor can it escape backward, as the light is propagating forward and the interior of the bubble is at rest. Thus, the light gets caught at the bow of the warp bubble, unable to escape for so long as the warp bubble is active. This light is strongly blue shifted to extremely energetic x-rays and gamma rays. The space behind the bubble is swept clear of forward-moving light.

The situation for matter over-run by a super-luminal warp bubble is similar. Matter moving backward passes through the bubble, being only swept a ways along the bubble's path as sign of its passing but otherwise continuing on their way. Matter at rest behaves the same way as Pfenning and Ford discovered. Meanwhile, matter moving in the same direction of the warp is overtaken and collects at the bow of the bubble, unable to leave for so long as the bubble is warping. This matter "surfing" on the bubble bow is highly accelerated to relativistic speeds, experiencing extreme time dilation.

For a sub-luminal bubble, there is a speed faster than the warp speed where particles coming up from behind overtake the bubble and pass through, out the front. Particles below this speed but still faster than the warp speed still overtake the bubble, but then bounce backwards out of the bubble from behind. They are still moving forward, but are now moving more slowly than the warp speed. Particles moving forward but slower than the bubble will be overtaken and then bounced out the front with a speed higher than the warp speed. Those particles moving backward will pass through the bubble and exit with their initial speed.

The spacecraft inside the bubble will observe that light moving backward compared to the warp direction is blue-shifted, and matter moving backward passes through with increased energy. Meanwhile, a sub-luminal bubble will see light that was moving forward as red-shifted and matter catching up to the bubble from behind will pass by with reduced energy (superluminal bubbles, of course, do not have any matter or radiation catching up to them from behind).

When a superluminal warp bubble that has been collecting matter and radiation for the duration of its journey and blue-shifting it to much higher energies is turned off, all that matter and energy is released as a blast of radiation in the direction the bubble was warping.

McMonigal et al. conclude by noting that any spacecraft in the warp bubble would need shielding to protect against the blue-shifted radiation and matter of increased energy. In addition the destination would be "blasted into oblivion" by the release of matter and radiation that had been caught in the bubble during the trip.

Clark et al.<ref>Chad Clark, William A. Hiscock, and Shane L. Larson, "Null geodesics in the Alcubierre warp drive spacetime: the view from the bridge", arXiv:gr-qc/9907019 (2018) https://doi.org/10.48550/arXiv.gr-qc/9907019</ref> examine what it would look like from within an operating warp drive. They find that light moving at 90 degrees to the direction of motion will look like it is still coming from 90 degrees. However, light from the front hemisphere is squished progressively toward the forward point the faster you go. For a warp bubble moving faster than the speed of light, there is a cone behind the warp bubble where no light can reach it. However, you don't see a black spot behind you – light from the regions in the backward hemisphere from outside that cone that can reach you is distorted toward the direction behind you to fill in that spot. Light from the forward direction is blue-shifted and light from behind is red-shifted. At speeds of around 200 times light speed, the cosmic microwave background would be blue shifted to energies similar to that of the solar photosphere, heating anything in the warp bubble as if they were close to the sun.

== Other warp geometries ==

The Alcubierre warp drive is the first and best known of the warp drives. It has been around long enough that there have been many studies done on it. The warp geometry is particularly simple, making analysis easier. But in the time that the Alcubierre warp was proposed, many other researchers have come up with other warp designs, and even generalized the idea to include a wide array of additional possibilities.

Most of the general statements about the challenges of the Alcubierre drive hold for these other drives. Many of the details about interactions with matter are also likely to hold, although these other styles of warp drive have not yet been subject to such rigorous investigation.

=== Van Den Broeck warp drive ===

If an Alcubierre warp bubble a hundred meters across requires a magnitude of energy greater than the entire energy in the observable universe, one option to reduce the magnitude of energy used is to make the warp bubble smaller. Much smaller. Pfenning and Ford<ref name="PfenningFord_2001"></ref> suggested making the warp bubble smaller than an atom. Of course, that brings up the problem of how to stuff a spacecraft in there. Van Den Broeck proposed a solution<ref name="VanDenBroeck_1999"></ref>: make the warp bubble only about the size of an atomic nucleus but expand the space inside the bubble enormously so that a spacecraft could fit in. This is somewhat like a nuclear sized wormhole that leads to a pocket universe that holds the spacecraft. Unfortunately, the metric doesn't fit neatly into any [[Wormholes#Three_dimensions|embedding diagram]] that I can figure out, but the math works out so that spatial coordinates inside the bubble are expanded by a factor of 1017 and a spacecraft would have a few hundred meters of bubble interior to putz around in. This trick manages to reduce the magnitude of energy to only about the mass energy of a few stars similar to our own. Other than that, it is otherwise a normal Alcubierre drive.

Further work<ref>S. Krasnikov, "The quantum inequalities do not forbid spacetime shortcuts", Physical Review D. 67 (10) 104013. https://doi.org/10.1103%2FPhysRevD.67.104013, arXiv:gr-qc/0207057 (2003) https://doi.org/10.48550/arXiv.gr-qc/0207057 </ref> has managed to use Van Den Broeck's trick to reduce the magnitude of the negative energy to only about a milligram.

=== Natário warp drive ===

The Alcubierre drive is not the only way to construct a warp drive. José Natário<ref name="Natario"></ref> showed that it was just one example of an entire class of warp drives. These are space-time geometries with a smooth and localized shift vector field and a unit lapse function.

Natário then went on to find in that class a set of warp drives that do not have any expansion or contraction of space at all. Instead, space encountering the front boundary of the warp bubble instead "slides" around the outside of the bubble until it gets to the corresponding place on the back and is left behind there. Another way to think of it is that space entering the bubble shell is compressed in the radial direction of the shell but is simultaneously expanded in the tangential direction so that there is no net change in volume. When it gets to the back, the opposite occurs. In the language of mathematics, the shift vector field is divergenceless.

<table align=center>
<tr><td width=360>[[File:Natario_drive_flow_of_space.png]]
<td width=445>[[File:Natario_warp_drive_horizons.png]]
<tr><td style="vertical-align: top;">A representation of the motion of space in a zero-expansion Natário drive with a vanishingly thin warp shell around the bubble.
<td>The horizons of the generalized Natário class of warp drives (including the Alcubierre warp drive and all fell-Heisenberg drives) at superluminal speed in the direction of the green arrow. No event inside the bubble can affect events outside the bubble in front of the blue line. No events outside the bubble behind the red line can affect events inside the bubble. The lines have an angle α with respect to the direction of motion with sin(α) = 1/vs for vs the speed of the bubble relative to light speed. This is analogous to the Mach cone of supersonic objects.
</table>

NOTE: There are two different things that can be meant by the term "Natário warp drive". The generalized Natário warp drive is the general theory of all warp drives that includes within it the Alcubierre, Fell-Heisenberg, and Lentz warp drives, among infinitely many others. The zero-expansion Natário warp drive is the specific implementation of the generalized Natário warp drive that has a divergenceless shift vector field.

The generalized class of warp drives developed by Natário (including both the Alcubierre and this zero-expansion warp drive) are shown to always have regions in the warp shell where the energy density is negative to at least some observers. You can't get away from it – to warp, you need negative energy density.

The Natário class of warp drives blueshift light coming in from the front and redshift light catching up from the back. For the case of a super-luminal warp bubble, there will of course be a horizon at the back of the bubble and light will not be able to get through from behind. For a bubble speed vs relative to the speed of light, light coming from straight ahead will be blueshifted up in frequency and photon energy by a factor of 1 + vs. For sub-luminal travel, light coming from behind will be redshifted by a factor 1 - vs. In general for n as the unit direction along which the light is propagating, the blueshift factor will be 1 + vs ⋅ n. If the warp shell is infinitesimally thin, light from outside that reaches the center of the bubble will not be distorted in direction although it will be frequency shifted, but observers still inside the bubble but displaced from the center will see the field of view distorted as well as frequency shifted.

Most of the difficulties that apply to the Alcubierre warp drive also apply to the generalized Natário warp drive (and thus to the zero-expansion Natário drive and Fell-Heisenberg drive as well). In particular, quantum energy inequalities, high magnitudes of the energy required, formation of time machines, instabilities related to the stress-energy making up the warp shell being unable to keep up with the bubble, the chicken-and-egg problem of needing FTL to achieve FTL, the formation of horizons making warp bubbles uncontrollable, and Hawking radiation at the horizons, are all still concerns for generic warp drives.

=== Fell-Heisenberg warp drives ===

S. D. B. Fell and L. Heisenberg investigated what sort of warp structure is necessary to produce a warp drive with positive energy density everywhere<ref name="Fell-Heisenberg">S. D. B. Fell and L> Heisenberg, "Positive energy warp drive from hidden geometric structures", Classical and Quantum Gravity 38 155020 (2021) https://doi.org/10.1088/1361-6382/ac0e47 https://arxiv.org/abs/2104.06488</ref>. They found that warp drives of the generic Natário form with irrotational shift vectors had a simple expression for the energy density and vanishing momentum (irrotational means that the shift vector can be described as the gradient of a scalar potential field or, equivalently, that the curl of the shift vector field vanishes everywhere). Fell and Heisenberg proposed several potential fields that either had positive energy density everywhere except at discontinuities, or that were continuous but had regions that violated the weak energy condition (i. e. negative energy densities in some frames of reference). One specific example they provided of a warp field had an energy that is about 10,000 times less than the mass-energy of our sun. Or only about half the mass-energy of Jupiter. This is a significant improvement over many other proposed drives.

Despite the introduction discussing warp drive configurations that satisfy the various energy conditions, the configurations described in the paper are shown to locally violate the weak and strong energy conditions. Nonetheless, the energy density is still mostly positive. Santiago, Schuster, and Visser's<ref name="Santiago Schuster Visser 2022">J. Santiago, S. Schuster, and M. Visser, "Generic warp drives violate the null energy condition", Physical Review D 105, 064038 (2022) https://doi.org/10.1103/PhysRevD.105.064038</ref> work shows that the various energy conditions must still be violated by this warp drive; to not violate these, energy density must be positive in all reference frames not just those of the co-moving observer.

Any vector field can be described by its divergence, its curl, and a constant. The constant part of a warp drive's shift vector can be discarded because then the warp drive would not be localized. It is interesting, therefore, that the Natário zero-expansion drive has zero divergence and can be described only by its curl, while the Fell-Heisenberg drive has zero curl and can be described only by its divergence. These two classes of drives thus rely on opposite choices of setting one of the two different parts describing a vector field to zero. As might be expected, the Fell-Heisenberg drive is not zero expansion – space-time will change in volume as it passes through the warp bubble (although after going all the way through it will return to its original volume). However, objects in space-time will retain their original orientation as they pass through the warp bubble.

Any vector field with zero curl can be represented as the gradient of a suitable scalar function. Thus, the Fell-Heisenberg drive can be completely described by just scalar field (one number at all points of space and time) rather than a vector field.

Previous examples of warp drives, such as the Alcubierre and zero-expansion Natário drives, tend toward a flat space-time as you go far enough away faster than would occur if there were any mass present in the warp drive. That is, the ADM mass of such drives is zero. However, one of the Fell-Heisenberg examples described a spacetime where the geometry approaches that of what you would get for a massive object at the warp location, giving one of the first examples of a warp drive with mass.

In the Fell-Heisenberg drive the momentum vanishes everywhere. Yet the energy occupies regions where the shift vector is varying rapidly. The lack of momentum means that the energy will not move to keep up with the differential expansion and movement of the spacetime. As a consequence, at later times the energy will have a different distribution than what is necessary to maintain the given warp configuration. The exact time evolution is not solved but this likely leads to collapse of warp bubble. Note that this is not a problem unique to the Fell-Heisenberg drive. Assuming a geometry that describes a warp produces a specific distribution of the stress-energy tensor that satisfies Einstein's equation. But there is no guarantee that this stress energy tensor satisfies the continuity equation, which describes how the distribution of matter and momentum change with time. The issue of stability of any warp drive is not well understood, but most warp drive geometries will be unstable in the sense that their matter that generates the warp fields will not be able to keep up with the motion of the warp. The Fell-Heisenberg drive is unique only in that, with vanishing momentum, there is no configuration that allows stable propagation.

(n. b. The Heisenberg here is Lavinia Heisenberg, not the Werner Heisenberg of quantum physics and uncertainty principle fame.)

=== Lentz warp drive ===

E. Lentz<ref name="Lentz">E. W. Lentz, "Breaking the warp barrier: hyper-fast solitons in Einstein–Maxwell-plasma theory", Classical and Quantum Gravity. 38 075015 (2021). arXiv:[https://arxiv.org/abs/2006.07125 2006.07125]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2021CQGra..38g5015L 2021CQGra..38g5015L]. doi:[https://doi.org/10.1088%2F1361-6382%2Fabe692 10.1088/1361-6382/abe692]. ISSN [https://search.worldcat.org/issn/0264-9381 0264-9381]. S2CID [https://api.semanticscholar.org/CorpusID:219635854 219635854].</ref> investigated a particular configuration of the Fell-Heisenberg class of warp drives, where the scalar field forms a selection of rhomboid regions, resulting in an energy density that is everywhere positive and a shift vector with vanishing tidal force at the center for a comfortable ride.

However, Santiago, Schuster, and Visser<ref name="Santiago Schuster Visser 2022"></ref> dispute claims that the Lentz drive satisfies the energy conditions, noting that everywhere positive energy density in one frame of reference is insufficient to establish that the energy density is positive in all reference frames; knowledge of the Cauchy stress tensor is also needed. They show that any generic warp drive will violate the strong energy condition, null energy condition, and weak energy condition.

== Conservation laws ==

All the warp drives investigated so far have geometries that quickly fall off from the highly curved region near the warp bubble to a region of spacetime that is almost flat; and becomes even more flat the farther away from the warp bubble you get. There is a word for geometries of this kind – they are called asymptotically flat. Basically, you can always go far enough away in any direction to reach a place where the geometry is sufficiently flat to meet any flatness criterion you choose.

When you have an asymptotically flat geometry, then in the far away mostly flat regions gravity is well described by a linearized theory of gravity – where the perturbations to the geometry are linear in the stress energy tensor. This means that gravity is basically Newtonian, along with a few other bells and whistles like gravitational waves and frame dragging.

When linearized gravity is a good approximation, then it is known that energy, momentum, and linear momentum are conserved. Although parts of a warp geometry are highly curved and non-linear, you can measure the conserved energy and momentum and angular momentum from far away in the linear region. This ADM energy and etcetera is the quantity that is conserved when you are worrying about conserving such things, not necessarily the energy and so forth you get in the highly curved spacetime region. In fact, it is not even simple to define where the energy and such actually is withing the highly curved spacetime region; it is non-local and different observers will disagree on how it is distributed. But everyone agrees on the ADM conserved quantities from out in nearly flat spacetime (up to the usual changes due to changing speeds, which are trivial and come up with normal objects as well).

And now we run into a bit of a conundrum: most of the warp geometries so far proposed have zero ADM mass, meaning that things like the Alcubierre drive technically have no energy or momentum or angular momentum at all. Despite all of the extremes of negative energy regions within the wall of the warp bubble, the actual energy that you need to spend to make one is zero. This can lead to a bit of a problem. If you have a spacecraft that has mass, that mass-energy has to be conserved. So when you turn on your warp drive and the total warp mass is zero, where does that energy go?

One resolution is just that the mass of the spacecraft is negligible and the small amount of curvature it causes can be ignored. In this case, the warp geometries discussed so far are merely an approximation that neglects this effect. In fact, some of the proposed Fell-Heisenberg drives map not onto entirely flat spacetime geometries at large distances, but onto the Schwarzschild geometry that is appropriate for systems with mass at the center. If this is the case, then the warping bubble and spacecraft system will need to worry about conservation of all their conserved properties.

Let's look at what the conservation laws mean for a spacecraft equipped with a warp drive, assuming that no matter or radiation are emitted by activating or deactivating the warp drive.

We can start with an observer floating in space. We'll call her Alice. And, approaching Alice at 100 m/s is the starship Enterprise with a mass of a cool 200,000 tons (2×108 kg). from this we know that the Enterprise has, in Alice's frame of reference, a kinetic energy of 1 TJ, a total energy (including mass and kinetic energy) of 1.8×1025 J, a momentum of 2×1010 kg m/s, and – because it is headed straight toward Alice – an angular momentum of zero. All of this is without any warp drive activated.

On the bridge of the Enterprise, Spock picks up Alice on the sensors and tells Captain Kirk. Kirk does not want to run into Alice (he'd rather date her), so he orders helm to turn on the Enterprise's warp drive and move them 1 km away so that their course takes them only 1 km from Alice instead of straight into her. Sulu warps the Enterprise according to these commands, and then turns off the warp drive.

After this maneuver, keeping the Enterprise's total energy and momentum the same means that the Enterprise still has a mass of 200,000 tons and a velocity of 100 m/s in its original direction. So we can see that the warp drive must preserve the velocity vector of the thing in it 9at least in flat spacetime where outside forces are not acting on it). The problem comes from angular momentum. With a known mass of 200,000 tons, a speed of 100 m/s, and a distance of closest approach of 1 km; from Alice's frame of reference the Enterprise now has an angular momentum of 2×1013 kg m2/s after the warp maneuver, while before the maneuver it has an angular momentum of zero.

From this we can see that warp drives cannot be reactionless. Arbitrary displacements using warp without emitting propellant result in violation of the law of conservation of angular momentum. As long as we restrict ourselves to accepted science based on what we know of general relativity, to move a warp drive with mass you need to accelerate it or decelerate it with rockets or other normal means of delivering momentum (such as light sails, mag sails, or pushing off of a planet). Warp drives must be inertial to comply with physics.

(ADM mass -> 0; conservation of energy/mass issues.)<ref>Sebastian Schuster, Jessica Santiago,and Matt Visser, "ADM mass in warp drive spacetimes" General Relativity and Gravitation (2023) 55:14 https://doi.org/10.1007/s10714-022-03061-9</ref>

(Discuss issues of conservation of angular momentum.)

(If angular momentum conservation is ignored, discuss how momentum & energy are affected by outside forces. In a gravitational field equivalent to inertial frame ... like being in an accelerated elevator; will acquire momentum buildup while staying "at rest".)

== Warp Railroads ==

Van Den Broeck<ref name="VanDenBroeck_2000"></ref> suggested a warp railroad,

You can set up devices ahead of time along the path of the bubble that produce the negative energy regions for the warp bubble at the appropriate time without needing the negative energy to ever move with the bubble at all. At least for bubbles on a predictable schedule and route.

helps with:

-stability (elaborate)

-conservation laws (elaborate)

Mention Krasnikov<ref name="Krasnikov 1998">S. V. Krasnikov, "Hyperfast travel in general relativity" Phys. Rev. D 57 4760</ref>

Hyperwave<ref name="Pieri 2024">Lorenzo Pieri, "Hyperwave: hyper-Fast Communication within General Relativity", arXiv:2311.12069 [gr-qc] https://doi.org/10.48550/arXiv.2311.12069</ref> makes use of this warp railroad idea.

== Quantum effects ==

(quantum stuff here<ref name="Hiscock1997">W. A. Hiscock, "Quantum effects in the Alcubierre warp drive spacetime", Classical and Quantum Gravity 14 L183 https://doi.org/10.1088/0264-9381/14/11/002 https://arxiv.org/abs/gr-qc/9707024</ref><ref name="Finazzi et al 2009">S. Finazzi, S. Liberati, C. Barceló, "Semiclassical instability of dynamical warp drives", Physical Review D 79, 124017 (2009)https://doi.org/10.1103/PhysRevD.79.124017 https://arxiv.org/abs/0904.0141</ref>)

From<ref name="Pieri 2024"></ref>
<blockquote>
It should be emphasised that Quantum Inequalities have not been proven for realistic interacting quantum fields, where the they are expected to be weaker if they exist [Cadamuro, 2019], and they often require Minkowskian conditions at infinity, a condition which is not appropriate for fields inside bounded regions, as in a Casimir devices.
</blockquote>

== Warp drives and black holes ==

What happens if you drive your warp drive into a black hole?

The pop-sci reason that nothing can escape a black hole is that at the event horizon the escape velocity is faster than light and nothing can go faster than light. A warp drive can go faster than light.

The more detailed reason from general relativity is that at the event horizon space and time are rotated sufficiently that "inward" becomes "forward in time." You can no more go farther away from the singularity in a black hole once you are inside the event horizon than you can go backward in time. Any super-luminal travel can go backward in time, and warp drives can engage in super-luminal travel.

The Hawking area theorem that says that black holes can only grow, never shrink relies on the condition that the energy is everywhere positive. Warp drives have negative energy density.

So in principle, there's nothing that prevents a warp drive from taking a dip into a black hole and coming back out again.

Garattini and Zatrimaylov looked into the problem of a warp drive near (and in) a black hole<ref name="Garattini and Zatrimaylov">R. Garattini and K. Zatrimaylov, "Black holes, warp drives, and energy conditions", Physics Letters B 856 138910 (2024) https://doi.org/10.1016/j.physletb.2024.138910</ref>. The found that the black hole reduces the negative energy requirement for the warp bubble to move inward toward the black hole singularity, but increases the negative energy needed to move away from the center of the black hole. In addition, a warp drive parked at the event horizon would allow light from inside the horizon to pass through the bubble and escape back outside.

== Credit ==
Author: Luke Campbell

== References ==

[[Category:Physics]][[Category:Infrastructure]][[Category:Physics & Engineering‏‎]][[Category:Physics & Math & Engineering]][[Category:Transportation & Infrastructure‏‎]][[Category:Metric Engineering]]

Warp Drives

2026-07-30T02:44:14Z

Lwcamp: /* Lentz warp drive */

{{PageConstructionNotice}}

Science fiction often features spacecraft that can seemingly move across space and get between the place of departure and the destination much faster than light could have done. This appears to contradict the theory of relativity, which predicts unequivocally that nothing can move through space faster than light. Because relativity has been incredibly successful at describing nature, with its many other predictions regularly being confirmed to extraordinary accuracy and within the bounds of uncertainty of all the experiments that tested them, it gives confidence that relativity is a correct description of reality. Which seems to rather throw a wet towel on our hopes for rapid travel between stars.

However, while relativity does not allow things to move through space faster than light, it places no such restrictions on how fast space-time itself can expand, contract, or move around. This leads to the idea of a warp drive – the spacecraft remains stationary within a region of highly curved space-time, and that region moves at super-luminal speeds rather than the spacecraft.

== The Alcubierre warp drive ==

The first warp drive geometry that satisfied the Einstein field equations of relativity was proposed by Miguel Alcubierre<ref>M. Alcubierre, "The warp drive: hyper-fast travel within general relativity." Classical and Quantum Gravity. 11 (5): L73–L77 (1994). [https://arxiv.org/abs/gr-qc/0009013 arXiv:gr-qc/0009013]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1994CQGra..11L..73A 1994CQGra..11L..73A]. doi:[https://doi.org/10.1088%2F0264-9381%2F11%2F5%2F001 10.1088/0264-9381/11/5/001]. S2CID [https://api.semanticscholar.org/CorpusID:4797900 4797900].</ref>. In this geometry, a sphere of space-time moves at an arbitrary speed (potentially but not necessarily a speed much faster than light). Objects within the sphere are moved along with the sphere; an object at rest within the sphere would be moved along with the sphere indefinitely. Space is expanding at the rear boundary and contracting at the front boundary in order to keep the sphere moving. In order to satisfy the Einstein field equations, the boundary of the sphere must have a negative energy density. The challenges of space-time geometries with negative energy densities are described in our page on [[Wormholes#Exotic_energy_conditions|wormholes]], for our purposes it is enough to note that negative energy densities can pose problems if not handled carefully, there may be limits on how much negative energy you can have without nearby positive energy density, and it may not be possible to get enough negative energy to support a warp drive; although none of this is ruled out by physics – yet!

<table align=center>
<tr><td width=650>[[File:Energy_magnitude_of_Alcubierre_drive.png]]
<td width=650>[[File:Expansion_of_Alcubierre_drive.png]]
<tr><td>The magnitude of the energy of the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. (The energy density is negative everywhere in the shell, here only the magnitude is shown to aid visual comprehension.)
<td>The expansion of space in the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. Negative expansion means that space is contracting in that direction, positive expansion that it is expanding.
</table>

NOTE: What the? The energy distribution is symmetric! How does the drive know which way to go? There must be significant contributions of other components of the stress-energy tensor, and those have got to be asymmetric along the forward/backward axis. Check this when I get time.

(And what's with the ring in all the artwork, anyway?)

=== So what is going on? ===

What is it that makes a warp drive a warp drive, and how does it work?

A warp drive is a concept that comes out of the science of general relativity. General relativity collects space and time together into a single "structure" called spacetime. This structure is curved, and its curvature is determined by the material that is in it. Absent forces acting between matter, particles of matter move through spacetime on the shortest possible paths in this curved environment.
<blockquote>
"Spacetime tells matter how to move; matter tells spacetime how to curve." - John Archibald Wheeler
</blockquote>
All of this is encompassed in the Einstein field equation:
<div align=center>
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 8πG
<td rowspan="2" nowrap="nowrap" align="right"> T = R -  
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right"> R g
<tr>
<td style="border-top:solid 1px black;"> c4
<td style="border-top:solid 1px black;"> 2
</table>
</div>
Unfortunately, by convention, both the symbols G and R get used for a lot of different things. Here, symbols in bold are second rank tensors – constructs of arrays of 4×4 numbers where 4 is necessary because that is the number of dimensions of spacetime. Conventionally, these numbers are indexed by a value going from 0 to 3, with 0 being the part corresponding to the coordinate of time and 1 to 3 being the Cartesian coordinates that you might know as x, y, and z.

T is the stress-energy tensor: the (0,0) component represents the energy density at a given point; the (0,i) and (i, 0) components (with i going from 1 to 3) represent the momentum density; and the (i,j) components represent the stresses present, such as pressure or tension or shear.

The other term, R - R g/2, involves a partial description of the curvature. A full description of the curvature requires a construct of 4×4×4×4 components; too much for the stress energy tensor to fully constrain. Which is okay, because this allows curvature to exist away from where matter is, allowing gravity to have long range effects across empty space as well as for phenomena like gravitational waves. But it is enough that a given distribution of matter T gives enough constraints (along with specifying how the geometry behaves far away) that you can generally figure out what the actual gravity is.

This is the usual story that we are told. But there is an alternate way of looking at the same geometry such that the math works out the same, but it is described conceptually in a very different way. This divides spacetime up into "layers", called foliations, that represent slices of space and going from one foliation to the next is increasing in time. This is useful for computation because you can assign coordinates to your system. There are two additional constructs – the lapse function and the shift vector field. Starting at an event on one foliation, the lapse function at that position tells you that after a certain "proper time" (time perceived by an observer at that event) the time coordinate of the observer will be the proper time times the lapse function at that event. And the new spatial coordinates of the observer will be the proper time times the shift vector at that event. (Strictly speaking, this is only exactly true in the limit of the proper time approaching zero or when the lapse function and shift vectors are constant. But hopefully this gives a good enough insight to what these entities are so that we can go on to describe warp drives.)

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Lapse_shift.png]]
<tr><td>A folliation of space-time into slices of constant time coordinates (grey). At an event (black circle), the lapse function (yellow) shows how far into the future the point at that event will be projected in a unit amount of proper time experienced by an observer initially at the event. The shift vector (red) shows how that point changes coordinates after the unit amount of proper time. The cyan vector shows the new coordinates the point is projected onto. The lower black surface indicates a foliation of spacetime at the same time coordinate; after unit proper time, the events on the lower foliation get projected to the curved upper black foliation according to the shift vectors and lapse functions at each of their respective points.
</table>
</table>

Warp drives are generally restricted to spacetime geometries with a constant lapse function - all observers go forward by the same time coordinate after experiencing the same amount of proper time. Thus, all warp drives have no time dilation.
Far from the warp bubble, the shift vector vanishes – each spatial coordinate is mapped to the same spatial coordinate after any amount of proper time so long as the warp drive is not near. But the inside of the warp bubble is an extended region where the shift vector field is non-zero and constant. In this bubble, all spatial coordinates are mapped to a coordinate that shifts linearly in time with the proper time of the observer. Astute readers will see that this linear change in position with time is very similar to a velocity. This region of constant shift vector then moves with the shift vector velocity to keep stuff originally in the bubble so that it stays in the bubble. In between the inside of the bubble and the far away regions with zero shift vector, the shift vector field falls away to match one region to the other.

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Warp_shift.png]]
<tr><td>The shift vector of a one-dimensional slice through warp geometry, in red. At the edges there is no shift. At the center (inside the warp bubble) there is a region of uniform shift so that at a later time the inside of the warp bubble is rigidly translated to different spatial coordinates (in blue). The two foliations remain the same distance apart, because warp drives maintain a unit lapse function.
</table>
</table>

Traditionally, scientists would start with an interesting distribution of matter given by T and then find out what kind of curvature it created. This works well for planets and stars and black holes and cosmic strings and other things which are observed in the universe and you want to see what sort of effect they had. But starting in the 1980's, some physicists started looking at Einstein's field equations the other way around. They start with a desired geometry, find the curvature part of the field equation R - R g/2, and use that to find out what T has to be in order to get that curvature. This was originally done for [[Wormholes|wormholes]], but in 1994 Miguel Alcubierre used this same trick to figure out what could create a region of constant shift vector that moved with the shift vector - the warp drive. With this demand for a warp geometry, the required distribution of matter and energy and momentum and stress can be determined.

The original Alcubierre shift vector N→ produces a spherical warp bubble with an axi-symmetric distribution of, well, everything along the direction of motion with warp velocity vector v→.
<div align="center">
N→ = v→ f(rs)
</div>
Here, rs is the distance from the center of the warp bubble, and f(rs) can be any function that is 1 near rs = 0 and is 0 for rs well beyond the radius of the warp bubble.

=== Challenges and possible resolutions ===

Almost as soon as Alcubierre proposed his warp drive, others began picking it apart.
<ol>
<li>Pfenning and Ford<ref name="PfenningFord_2001">M. J. Pfenning and L. H. Ford, "The unphysical nature of 'Warp Drive'", Classical and Quantum Gravity. 14 (7): 1743–1751 (1997). arXiv:[https://arxiv.org/abs/gr-qc/9702026 gr-qc/9702026]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1997CQGra..14.1743P 1997CQGra..14.1743P]. doi:[https://doi.org/10.1088%2F0264-9381%2F14%2F7%2F011 10.1088/0264-9381/14/7/011]. S2CID [https://api.semanticscholar.org/CorpusID:15279207 15279207].</ref> found that if the negative energy density wall around the bubble obeys [[Wormholes#Quantum_energy_inequalities|quantum energy inequalities]], then for warp speeds of around light speed the shell thickness would be on the order of a hundred Planck lengths; a distance far smaller than any other known physical phenomenon or object.
<li>In the same work, Pfenning and Ford also found that for a bubble a hundred meters in radius, a warp speed of around the speed of light, and the required thickness of about 100 Planck lengths the energy in the bubble shell would be E ≈ -1063 kg c2. The magnitude of this latter value is ten orders of magnitude larger than the energy of the entire visible universe. They do note, however, that if quantum energy inequalities can be ignored then a warp drive with a hundred meters radius but a shell thickness of one meter would "only" have an energy magnitude of about a quarter solar mass.
<li>As with any method of faster than light travel, the warp drive could be used to make a time machine.
<li>Perhaps most seriously, if the warp drive is going faster than light speed, the negative energy regions on the outside of the shell won't be in the warp parts of the bubble. They will have to be moving through space at faster than light speed if the bubble is to maintain its integrity, which is the very problem that the warp drive was designed to avoid. The outside of the warp bubble shell that maintains the warp bubble would fall away and the warp bubble would quickly erode to sub-luminal speeds.<ref name="Van Den Broeck 2000">Chris Van Den Broeck, "On the (im)possibility of warp bubbles", arXiv:gr-qc/9906050 https://doi.org/10.48550/arXiv.gr-qc/9906050</ref>
<li>A more general problem that includes the previous one is that the geometry of the warp drive determines the distribution of energy, momentum, and stress (pressure, shear, and tension) by Einstein's field equation, which states that the stress energy tensor is directly proportional to a reduced order curvature tensor of space-time. However, the stress-energy tensor must also obey the continuity equation – basically, for the energy to change in a given volume of space that energy must come in or go out of the boundary of that volume and similarly for the momentum. So a warp drive can have the warp condition for one moment in time. But after that, the stress-energy that maintains the warp will start to move according to the forces on it and according to the natural paths that it takes through the warped space-time. There is no guarantee that the configuration of the stress-energy at later times will be sufficient to maintain the warp.
<blockquote>
From the perspective of simulating the warp drive dynamically, the key challenge is stability. One can straightforwardly generate the initial metric that describes the Alcubierre metric and infer the instantaneous matter configuration that must support it using the Einstein Equation, but a time-domain evolution of the coupled matter-gravity system necessitates that we specify the equation of state of the matter, which in turn determines how it evolves. In a perfect fluid, this involves specifying the relation between the fluid pressure and density, but in a more general case, the full relation between the
stress-energy components must be given. There is (to our knowledge) no known equation of state that would maintain the warp drive metric in a stable configuration over time - therefore, whilst one can require that initially, the warp bubble is constant, it will quickly evolve away from that state and, in most cases, the warp fluid and spacetime deformations will disperse or collapse into a central point.
 
– Clough, Dietrich, and Khan<ref>Katy Clough, Tim Dietrich, and Sebastian Khan, "What no one has seen before: gravitational waveforms from warp drive collapse", arXiv:2406.02466 [gr-qc] https://doi.org/10.48550/arXiv.2406.02466</ref>
</blockquote>
<li>In order to get the matter (or more generally, stress-energy) that forms the warp shell up to speeds that could allow the shell to exist, you need to be able to, well, accelerate that matter to warp speeds. Which gives a chicken-and-egg problem - in order to achieve superluminal travel you need to be able to get matter or energy moving superluminally in the first place<ref name="Bobrick and Martire">Alexey Bobrick and Gianni Martire, "Introducing Physical Warp Drives", arXiv:2102.06824 [gr-qc] https://doi.org/10.48550/arXiv.2102.06824</ref><ref>D. H. Coule, "No warp drive", Class. Quantum Grav. 15 (1998) 2523–2527. https://doi.org/10.1088/0264-9381/15/8/026</ref>, which rather makes warp drives unnecessary from the get-go.
<li>A warp bubble moving faster than light will not be able to get any signals in front of it as it will overtake its own signal. In fact, anyone in the interior of the bubble would not be able to send signals to the front surface of the bubble. In science-speak, a horizon forms at the front of the bubble<ref name="Natario">José Natário, "Warp drive with zero expansion", Classical and Quantum Gravity. 19 (6): 1157–1166 (2002). arXiv:[https://arxiv.org/abs/gr-qc/0110086 gr-qc/0110086]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2002CQGra..19.1157N 2002CQGra..19.1157N]. doi:[https://doi.org/10.1088%2F0264-9381%2F19%2F6%2F308 10.1088/0264-9381/19/6/308]. S2CID [https://api.semanticscholar.org/CorpusID:15859984 15859984].</ref>. As a consequence, the bubble cannot be controlled, steered, nor stopped from within the bubble itself.
<li>A Horizon will naturally emit Hawking radiation. It is estimated that the radiation will raise the temperature of the interior of the bubble to approximately 1032 K<ref name="Finazzi 2009">Stefano Finazzi, Stefano Liberati, and Carlos Barceló, "Semiclassical instability of dynamical warp drives", Phys. Rev. D 79, 124017 (12 June, 2009) https://doi.org/10.1103/PhysRevD.79.124017 , arXiv:0904.0141 [gr-qc] https://doi.org/10.48550/arXiv.0904.0141</ref> (at this level of accuracy, the result is the same if you replace temperatures in K by the same number in °C or °F). This temperature, however, may be lower if quantum inequalities do not have to be fulfilled.
<li>Semi-classical instabilities at the horizon are expected to lead to a divergence of the stress-energy tensor at the boundary in a short period of time, thus leading to the destruction of the warp bubble<ref name="Finazzi 2009"></ref>.
</ol>
Helpfully, Van Den Broeck<ref name="VanDenBroeck_2000">C. Van Den Broeck, "Alcubierre’s Warp Drive: Problems and Prospects" AIP Conference Proceedings. 504: 1105–1110 (2000). Bibcode:[https://ui.adsabs.harvard.edu/abs/2000AIPC..504.1105V 2000AIPC..504.1105V]. doi:[https://doi.org/10.1063%2F1.1290913 10.1063/1.1290913].</ref> suggested several solutions – or at least mitigations – for some of these problems
<ol>
<li>Quantum inequalities had not been shown to be true in general for highly curved space-times.
<li>A proposed warp drive geometry<ref name="VanDenBroeck_1999">C. Van Den Broeck, "A 'warp drive' with more reasonable total energy requirements". Classical and Quantum Gravity. 16 (12): 3973–3979 (1999). arXiv:[https://arxiv.org/abs/gr-qc/9905084 gr-qc/9905084]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1999CQGra..16.3973V 1999CQGra..16.3973V]. doi:[https://doi.org/10.1088%2F0264-9381%2F16%2F12%2F314 10.1088/0264-9381/16/12/314]. S2CID [https://api.semanticscholar.org/CorpusID:15466313 15466313]. </ref> (see the van Den Broeck drive, below) would be able to minimize the magnitude of the energy required to about that of the mass of our sun. While still large, it is not unphysically large.
<li>Time travel is always going to be a worry with faster than light travel. There's no neat solution to this one.
<li>It may be possible for the negative energy outside of the superluminal region to compress into the superluminal region to form a shock in space-time. The front surface of the bubble would then be a singularity. It is not clear if such a sharp jump in physical properties is possible, but neither is sure that it is impossible, either. However, while this might help with the negative energy in front of the bubble getting swept up because it cannot move faster than light, it does little to help with the material at the back of the bubble on the outside getting left behind because it cannot keep up.
</ol>
Other mitigating research has also shown
<ol start=9>
<li>Semi-classical divergence only occurs in warp drives with only one spatial dimensions. Those with two or more spatial dimensions (as in our universe) only contain points where divergences occur, and these divergences can be managed. In fact, their effects can be reduced by making the warp shape more elongated<ref>Carlos Barceló, Valentin Boyanov, Luis J. Garay, Eduardo Martín-Martínez,
and Jose M. Sánchez Velázquez, "Warp drive aerodynamics", JHEP08 (2022) 288 https://doi.org/10.1007/JHEP08(2022)288 , arXiv:2207.06458 [gr-qc] https://doi.org/10.48550/arXiv.2207.06458</ref>.
</ol>

=== Alcubierre warp interactions with light and matter ===

Space is not entirely empty. It is filled with a diffuse plasma in the form of the interstellar medium, as well as cosmic radiation, light from stars, and cosmic microwave background radiation. A warp drive propagating through space will encounter this stuff. What happens when this matter and radiation have a warp bubble pass across them?

The first analysis of matter encountering a warp bubble was performed by Pfenning and Ford<ref name="PfenningFord_2001"></ref>. They looked at the warp bubble interaction with a massive object at rest with the frame of reference of the warp drive (keep in mind that the rest frame is the same inside and outside the warp bubble; in this rest frame objects inside the bubble have no momentum even though they are, in some sense, changing location rapidly with time). When the warp bubble passes, the object experiences an acceleration in the direction of the warp bubble motion. When the warp shell passes and the object is inside the bubble, it will be moving with approximately the speed of the bubble. Pfenning and Ford analyzed this problem with a continuous distribution of shell energy that, strictly speaking, never falls to zero except at the bubble center and at spatial infinity, so unless the object passes through the center in Pfenning and Ford's description it will never quite get up to the bubble's speed. In this case, the object will move almost as fast as the object but will pass through the bubble in a finite time, after which it will again be at rest with respect to the reference frame but displaced along the direction of the warp bubble motion by some distance. With this description of the warp bubble, a spacecraft of finite size will always be moving a little slower than the warp bubble and would have to use rockets to keep up with it. in addition, the spacecraft would experience tidal forces that would cause stress on the spacecraft's structure.

Pfenning and Ford also examined cases where the shell is of a finite (possibly infinitesimal) width and falls to zero both inside and outside the bubble. In this case any matter encountering the bubble would be collected at the bow of the bubble and thereafter move along with it.

McMonigal et al. analyzed the situation for both massive particles and light moving along the axis of travel of the bubble<ref name="McMonigalEtAl_2012">B. McMonigal, G. F. Lewis, and P. O'Byrne, "Alcubierre warp drive: On the matter of matter". Physical Review D. 85 (6) 064024 (20 March 2012). arXiv:[https://arxiv.org/abs/1202.5708 1202.5708]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2012PhRvD..85f4024M 2012PhRvD..85f4024M]. doi:[https://doi.org/10.1103%2FPhysRevD.85.064024 10.1103/PhysRevD.85.064024]. S2CID [https://api.semanticscholar.org/CorpusID:3993148 3993148].</ref>. They found that light moving opposite the warp direction passed through the bubble without incident, being only somewhat delayed by passing through the bubble. A warp bubble that is warping at sub-luminal speeds can have light catch up from behind it. This light is able to pass through the bubble, and is somewhat advanced in its path by the speed of the bubble. For super-luminal warp bubbles, however, the situation is different. The bubble will catch up to light moving it its own direction that is originally in front of it. This light cannot escape forward, the bubble being too fast. Nor can it escape backward, as the light is propagating forward and the interior of the bubble is at rest. Thus, the light gets caught at the bow of the warp bubble, unable to escape for so long as the warp bubble is active. This light is strongly blue shifted to extremely energetic x-rays and gamma rays. The space behind the bubble is swept clear of forward-moving light.

The situation for matter over-run by a super-luminal warp bubble is similar. Matter moving backward passes through the bubble, being only swept a ways along the bubble's path as sign of its passing but otherwise continuing on their way. Matter at rest behaves the same way as Pfenning and Ford discovered. Meanwhile, matter moving in the same direction of the warp is overtaken and collects at the bow of the bubble, unable to leave for so long as the bubble is warping. This matter "surfing" on the bubble bow is highly accelerated to relativistic speeds, experiencing extreme time dilation.

For a sub-luminal bubble, there is a speed faster than the warp speed where particles coming up from behind overtake the bubble and pass through, out the front. Particles below this speed but still faster than the warp speed still overtake the bubble, but then bounce backwards out of the bubble from behind. They are still moving forward, but are now moving more slowly than the warp speed. Particles moving forward but slower than the bubble will be overtaken and then bounced out the front with a speed higher than the warp speed. Those particles moving backward will pass through the bubble and exit with their initial speed.

The spacecraft inside the bubble will observe that light moving backward compared to the warp direction is blue-shifted, and matter moving backward passes through with increased energy. Meanwhile, a sub-luminal bubble will see light that was moving forward as red-shifted and matter catching up to the bubble from behind will pass by with reduced energy (superluminal bubbles, of course, do not have any matter or radiation catching up to them from behind).

When a superluminal warp bubble that has been collecting matter and radiation for the duration of its journey and blue-shifting it to much higher energies is turned off, all that matter and energy is released as a blast of radiation in the direction the bubble was warping.

McMonigal et al. conclude by noting that any spacecraft in the warp bubble would need shielding to protect against the blue-shifted radiation and matter of increased energy. In addition the destination would be "blasted into oblivion" by the release of matter and radiation that had been caught in the bubble during the trip.

Clark et al.<ref>Chad Clark, William A. Hiscock, and Shane L. Larson, "Null geodesics in the Alcubierre warp drive spacetime: the view from the bridge", arXiv:gr-qc/9907019 (2018) https://doi.org/10.48550/arXiv.gr-qc/9907019</ref> examine what it would look like from within an operating warp drive. They find that light moving at 90 degrees to the direction of motion will look like it is still coming from 90 degrees. However, light from the front hemisphere is squished progressively toward the forward point the faster you go. For a warp bubble moving faster than the speed of light, there is a cone behind the warp bubble where no light can reach it. However, you don't see a black spot behind you – light from the regions in the backward hemisphere from outside that cone that can reach you is distorted toward the direction behind you to fill in that spot. Light from the forward direction is blue-shifted and light from behind is red-shifted. At speeds of around 200 times light speed, the cosmic microwave background would be blue shifted to energies similar to that of the solar photosphere, heating anything in the warp bubble as if they were close to the sun.

== Other warp geometries ==

The Alcubierre warp drive is the first and best known of the warp drives. It has been around long enough that there have been many studies done on it. The warp geometry is particularly simple, making analysis easier. But in the time that the Alcubierre warp was proposed, many other researchers have come up with other warp designs, and even generalized the idea to include a wide array of additional possibilities.

Most of the general statements about the challenges of the Alcubierre drive hold for these other drives. Many of the details about interactions with matter are also likely to hold, although these other styles of warp drive have not yet been subject to such rigorous investigation.

=== Van Den Broeck warp drive ===

If an Alcubierre warp bubble a hundred meters across requires a magnitude of energy greater than the entire energy in the observable universe, one option to reduce the magnitude of energy used is to make the warp bubble smaller. Much smaller. Pfenning and Ford<ref name="PfenningFord_2001"></ref> suggested making the warp bubble smaller than an atom. Of course, that brings up the problem of how to stuff a spacecraft in there. Van Den Broeck proposed a solution<ref name="VanDenBroeck_1999"></ref>: make the warp bubble only about the size of an atomic nucleus but expand the space inside the bubble enormously so that a spacecraft could fit in. This is somewhat like a nuclear sized wormhole that leads to a pocket universe that holds the spacecraft. Unfortunately, the metric doesn't fit neatly into any [[Wormholes#Three_dimensions|embedding diagram]] that I can figure out, but the math works out so that spatial coordinates inside the bubble are expanded by a factor of 1017 and a spacecraft would have a few hundred meters of bubble interior to putz around in. This trick manages to reduce the magnitude of energy to only about the mass energy of a few stars similar to our own. Other than that, it is otherwise a normal Alcubierre drive.

Further work<ref>S. Krasnikov, "The quantum inequalities do not forbid spacetime shortcuts", Physical Review D. 67 (10) 104013. https://doi.org/10.1103%2FPhysRevD.67.104013, arXiv:gr-qc/0207057 (2003) https://doi.org/10.48550/arXiv.gr-qc/0207057 </ref> has managed to use Van Den Broeck's trick to reduce the magnitude of the negative energy to only about a milligram.

=== Natário warp drive ===

The Alcubierre drive is not the only way to construct a warp drive. José Natário<ref name="Natario"></ref> showed that it was just one example of an entire class of warp drives. These are space-time geometries with a smooth and localized shift vector field and a unit lapse function.

Natário then went on to find in that class a set of warp drives that do not have any expansion or contraction of space at all. Instead, space encountering the front boundary of the warp bubble instead "slides" around the outside of the bubble until it gets to the corresponding place on the back and is left behind there. Another way to think of it is that space entering the bubble shell is compressed in the radial direction of the shell but is simultaneously expanded in the tangential direction so that there is no net change in volume. When it gets to the back, the opposite occurs. In the language of mathematics, the shift vector field is divergenceless.

<table align=center>
<tr><td width=360>[[File:Natario_drive_flow_of_space.png]]
<td width=445>[[File:Natario_warp_drive_horizons.png]]
<tr><td style="vertical-align: top;">A representation of the motion of space in a zero-expansion Natário drive with a vanishingly thin warp shell around the bubble.
<td>The horizons of the generalized Natário class of warp drives (including the Alcubierre warp drive and all fell-Heisenberg drives) at superluminal speed in the direction of the green arrow. No event inside the bubble can affect events outside the bubble in front of the blue line. No events outside the bubble behind the red line can affect events inside the bubble. The lines have an angle α with respect to the direction of motion with sin(α) = 1/vs for vs the speed of the bubble relative to light speed. This is analogous to the Mach cone of supersonic objects.
</table>

NOTE: There are two different things that can be meant by the term "Natário warp drive". The generalized Natário warp drive is the general theory of all warp drives that includes within it the Alcubierre, Fell-Heisenberg, and Lentz warp drives, among infinitely many others. The zero-expansion Natário warp drive is the specific implementation of the generalized Natário warp drive that has a divergenceless shift vector field.

The generalized class of warp drives developed by Natário (including both the Alcubierre and this zero-expansion warp drive) are shown to always have regions in the warp shell where the energy density is negative to at least some observers. You can't get away from it – to warp, you need negative energy density.

The Natário class of warp drives blueshift light coming in from the front and redshift light catching up from the back. For the case of a super-luminal warp bubble, there will of course be a horizon at the back of the bubble and light will not be able to get through from behind. For a bubble speed vs relative to the speed of light, light coming from straight ahead will be blueshifted up in frequency and photon energy by a factor of 1 + vs. For sub-luminal travel, light coming from behind will be redshifted by a factor 1 - vs. In general for n as the unit direction along which the light is propagating, the blueshift factor will be 1 + vs ⋅ n. If the warp shell is infinitesimally thin, light from outside that reaches the center of the bubble will not be distorted in direction although it will be frequency shifted, but observers still inside the bubble but displaced from the center will see the field of view distorted as well as frequency shifted.

Most of the difficulties that apply to the Alcubierre warp drive also apply to the generalized Natário warp drive (and thus to the zero-expansion Natário drive and Fell-Heisenberg drive as well). In particular, quantum energy inequalities, high magnitudes of the energy required, formation of time machines, instabilities related to the stress-energy making up the warp shell being unable to keep up with the bubble, the chicken-and-egg problem of needing FTL to achieve FTL, the formation of horizons making warp bubbles uncontrollable, and Hawking radiation at the horizons, are all still concerns for generic warp drives.

=== Fell-Heisenberg warp drives ===

S. D. B. Fell and L. Heisenberg investigated what sort of warp structure is necessary to produce a warp drive with positive energy density everywhere<ref name="Fell-Heisenberg">S. D. B. Fell and L> Heisenberg, "Positive energy warp drive from hidden geometric structures", Classical and Quantum Gravity 38 155020 (2021) https://doi.org/10.1088/1361-6382/ac0e47 https://arxiv.org/abs/2104.06488</ref>. They found that warp drives of the generic Natário form with irrotational shift vectors had a simple expression for the energy density and vanishing momentum (irrotational means that the shift vector can be described as the gradient of a scalar potential field or, equivalently, that the curl of the shift vector field vanishes everywhere). Fell and Heisenberg proposed several potential fields that either had positive energy density everywhere except at discontinuities, or that were continuous but had regions that violated the weak energy condition (i. e. negative energy densities in some frames of reference). One specific example they provided of a warp field had an energy that is about 10,000 times less than the mass-energy of our sun. Or only about half the mass-energy of Jupiter. This is a significant improvement over many other proposed drives.

Despite the introduction discussing warp drive configurations that satisfy the various energy conditions, the configurations described in the paper are shown to locally violate the weak and strong energy conditions. Nonetheless, the energy density is still mostly positive. Santiago, Schuster, and Visser's<ref name="Santiago Schuster Visser 2022">J. Santiago, S. Schuster, and M. Visser, "Generic warp drives violate the null energy condition", Physical Review D 105, 064038 (2022) https://doi.org/10.1103/PhysRevD.105.064038</ref> work shows that the various energy conditions must still be violated by this warp drive; to not violate these, energy density must be positive in all reference frames not just those of the co-moving observer.

Any vector field can be described by its divergence, its curl, and a constant. The constant part of a warp drive's shift vector can be discarded because then the warp drive would not be localized. It is interesting, therefore, that the Natário zero-expansion drive has zero divergence and can be described only by its curl, while the Fell-Heisenberg drive has zero curl and can be described only by its divergence. These two classes of drives thus rely on opposite choices of setting one of the two different parts describing a vector field to zero. As might be expected, the Fell-Heisenberg drive is not zero expansion – space-time will change in volume as it passes through the warp bubble (although after going all the way through it will return to its original volume). However, objects in space-time will retain their original orientation as they pass through the warp bubble.

Any vector field with zero curl can be represented as the gradient of a suitable scalar function. Thus, the Fell-Heisenberg drive can be completely described by just scalar field (one number at all points of space and time) rather than a vector field.

Previous examples of warp drives, such as the Alcubierre and zero-expansion Natário drives, tend toward a flat space-time as you go far enough away faster than would occur if there were any mass present in the warp drive. That is, the ADM mass of such drives is zero. However, one of the Fell-Heisenberg examples described a spacetime where the geometry approaches that of what you would get for a massive object at the warp location, giving one of the first examples of a warp drive with mass.

In the Fell-Heisenberg drive the momentum vanishes everywhere. Yet the energy occupies regions where the shift vector is varying rapidly. The lack of momentum means that the energy will not move to keep up with the differential expansion and movement of the spacetime. As a consequence, at later times the energy will have a different distribution than what is necessary to maintain the given warp configuration. The exact time evolution is not solved but this likely leads to collapse of warp bubble. Note that this is not a problem unique to the Fell-Heisenberg drive. Assuming a geometry that describes a warp produces a specific distribution of the stress-energy tensor that satisfies Einstein's equation. But there is no guarantee that this stress energy tensor satisfies the continuity equation, which describes how the distribution of matter and momentum change with time. The issue of stability of any warp drive is not well understood, but most warp drive geometries will be unstable in the sense that their matter that generates the warp fields will not be able to keep up with the motion of the warp. The Fell-Heisenberg drive is unique only in that, with vanishing momentum, there is no configuration that allows stable propagation.

(n. b. The Heisenberg here is Lavinia Heisenberg, not the Werner Heisenberg of quantum physics and uncertainty principle fame.)

=== Lentz warp drive ===

E. Lentz<ref name="Lentz">E. W. Lentz, "Breaking the warp barrier: hyper-fast solitons in Einstein–Maxwell-plasma theory", Classical and Quantum Gravity. 38 075015 (2021). arXiv:[https://arxiv.org/abs/2006.07125 2006.07125]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2021CQGra..38g5015L 2021CQGra..38g5015L]. doi:[https://doi.org/10.1088%2F1361-6382%2Fabe692 10.1088/1361-6382/abe692]. ISSN [https://search.worldcat.org/issn/0264-9381 0264-9381]. S2CID [https://api.semanticscholar.org/CorpusID:219635854 219635854].</ref> investigated a particular configuration of the Fell-Heisenberg class of warp drives, where the scalar field forms a selection of rhomboid regions, resulting in an energy density that is everywhere positive and a shift vector with vanishing tidal force at the center for a comfortable ride.

However, Santiago, Schuster, and Visser<ref name="Santiago Schuster Visser 2022"></ref> dispute claims that the Lentz drive satisfies the energy conditions, noting that everywhere positive energy density in one frame of reference is insufficient to establish that the energy density is positive in all reference frames; knowledge of the Cauchy stress tensor is also needed. They show that any generic warp drive will violate the strong energy condition, null energy condition, and weak energy condition.

== Conservation laws ==

(Discussion of asymptotic flatness: what is it? Where does it apply?)

(The "big four" conservation laws and how they relate to asymptotic flatness.)

(ADM mass -> 0; conservation of energy/mass issues.)<ref>Sebastian Schuster, Jessica Santiago,and Matt Visser, "ADM mass in warp drive spacetimes" General Relativity and Gravitation (2023) 55:14 https://doi.org/10.1007/s10714-022-03061-9</ref>

(Discuss issues of conservation of angular momentum.)

(If angular momentum conservation is ignored, discuss how momentum & energy are affected by outside forces. In a gravitational field equivalent to inertial frame ... like being in an accelerated elevator; will acquire momentum buildup while staying "at rest".)

== Warp Railroads ==

Van Den Broeck<ref name="VanDenBroeck_2000"></ref> suggested a warp railroad,

You can set up devices ahead of time along the path of the bubble that produce the negative energy regions for the warp bubble at the appropriate time without needing the negative energy to ever move with the bubble at all. At least for bubbles on a predictable schedule and route.

helps with:

-stability (elaborate)

-conservation laws (elaborate)

Mention Krasnikov<ref name="Krasnikov 1998">S. V. Krasnikov, "Hyperfast travel in general relativity" Phys. Rev. D 57 4760</ref>

Hyperwave<ref name="Pieri 2024">Lorenzo Pieri, "Hyperwave: hyper-Fast Communication within General Relativity", arXiv:2311.12069 [gr-qc] https://doi.org/10.48550/arXiv.2311.12069</ref> makes use of this warp railroad idea.

== Quantum effects ==

(quantum stuff here<ref name="Hiscock1997">W. A. Hiscock, "Quantum effects in the Alcubierre warp drive spacetime", Classical and Quantum Gravity 14 L183 https://doi.org/10.1088/0264-9381/14/11/002 https://arxiv.org/abs/gr-qc/9707024</ref><ref name="Finazzi et al 2009">S. Finazzi, S. Liberati, C. Barceló, "Semiclassical instability of dynamical warp drives", Physical Review D 79, 124017 (2009)https://doi.org/10.1103/PhysRevD.79.124017 https://arxiv.org/abs/0904.0141</ref>)

From<ref name="Pieri 2024"></ref>
<blockquote>
It should be emphasised that Quantum Inequalities have not been proven for realistic interacting quantum fields, where the they are expected to be weaker if they exist [Cadamuro, 2019], and they often require Minkowskian conditions at infinity, a condition which is not appropriate for fields inside bounded regions, as in a Casimir devices.
</blockquote>

== Warp drives and black holes ==

What happens if you drive your warp drive into a black hole?

The pop-sci reason that nothing can escape a black hole is that at the event horizon the escape velocity is faster than light and nothing can go faster than light. A warp drive can go faster than light.

The more detailed reason from general relativity is that at the event horizon space and time are rotated sufficiently that "inward" becomes "forward in time." You can no more go farther away from the singularity in a black hole once you are inside the event horizon than you can go backward in time. Any super-luminal travel can go backward in time, and warp drives can engage in super-luminal travel.

The Hawking area theorem that says that black holes can only grow, never shrink relies on the condition that the energy is everywhere positive. Warp drives have negative energy density.

So in principle, there's nothing that prevents a warp drive from taking a dip into a black hole and coming back out again.

Garattini and Zatrimaylov looked into the problem of a warp drive near (and in) a black hole<ref name="Garattini and Zatrimaylov">R. Garattini and K. Zatrimaylov, "Black holes, warp drives, and energy conditions", Physics Letters B 856 138910 (2024) https://doi.org/10.1016/j.physletb.2024.138910</ref>. The found that the black hole reduces the negative energy requirement for the warp bubble to move inward toward the black hole singularity, but increases the negative energy needed to move away from the center of the black hole. In addition, a warp drive parked at the event horizon would allow light from inside the horizon to pass through the bubble and escape back outside.

== Credit ==
Author: Luke Campbell

== References ==

[[Category:Physics]][[Category:Infrastructure]][[Category:Physics & Engineering‏‎]][[Category:Physics & Math & Engineering]][[Category:Transportation & Infrastructure‏‎]][[Category:Metric Engineering]]

Warp Drives

2026-07-30T02:44:01Z

Lwcamp: /* Fell-Heisenberg warp drives */

{{PageConstructionNotice}}

Science fiction often features spacecraft that can seemingly move across space and get between the place of departure and the destination much faster than light could have done. This appears to contradict the theory of relativity, which predicts unequivocally that nothing can move through space faster than light. Because relativity has been incredibly successful at describing nature, with its many other predictions regularly being confirmed to extraordinary accuracy and within the bounds of uncertainty of all the experiments that tested them, it gives confidence that relativity is a correct description of reality. Which seems to rather throw a wet towel on our hopes for rapid travel between stars.

However, while relativity does not allow things to move through space faster than light, it places no such restrictions on how fast space-time itself can expand, contract, or move around. This leads to the idea of a warp drive – the spacecraft remains stationary within a region of highly curved space-time, and that region moves at super-luminal speeds rather than the spacecraft.

== The Alcubierre warp drive ==

The first warp drive geometry that satisfied the Einstein field equations of relativity was proposed by Miguel Alcubierre<ref>M. Alcubierre, "The warp drive: hyper-fast travel within general relativity." Classical and Quantum Gravity. 11 (5): L73–L77 (1994). [https://arxiv.org/abs/gr-qc/0009013 arXiv:gr-qc/0009013]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1994CQGra..11L..73A 1994CQGra..11L..73A]. doi:[https://doi.org/10.1088%2F0264-9381%2F11%2F5%2F001 10.1088/0264-9381/11/5/001]. S2CID [https://api.semanticscholar.org/CorpusID:4797900 4797900].</ref>. In this geometry, a sphere of space-time moves at an arbitrary speed (potentially but not necessarily a speed much faster than light). Objects within the sphere are moved along with the sphere; an object at rest within the sphere would be moved along with the sphere indefinitely. Space is expanding at the rear boundary and contracting at the front boundary in order to keep the sphere moving. In order to satisfy the Einstein field equations, the boundary of the sphere must have a negative energy density. The challenges of space-time geometries with negative energy densities are described in our page on [[Wormholes#Exotic_energy_conditions|wormholes]], for our purposes it is enough to note that negative energy densities can pose problems if not handled carefully, there may be limits on how much negative energy you can have without nearby positive energy density, and it may not be possible to get enough negative energy to support a warp drive; although none of this is ruled out by physics – yet!

<table align=center>
<tr><td width=650>[[File:Energy_magnitude_of_Alcubierre_drive.png]]
<td width=650>[[File:Expansion_of_Alcubierre_drive.png]]
<tr><td>The magnitude of the energy of the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. (The energy density is negative everywhere in the shell, here only the magnitude is shown to aid visual comprehension.)
<td>The expansion of space in the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. Negative expansion means that space is contracting in that direction, positive expansion that it is expanding.
</table>

NOTE: What the? The energy distribution is symmetric! How does the drive know which way to go? There must be significant contributions of other components of the stress-energy tensor, and those have got to be asymmetric along the forward/backward axis. Check this when I get time.

(And what's with the ring in all the artwork, anyway?)

=== So what is going on? ===

What is it that makes a warp drive a warp drive, and how does it work?

A warp drive is a concept that comes out of the science of general relativity. General relativity collects space and time together into a single "structure" called spacetime. This structure is curved, and its curvature is determined by the material that is in it. Absent forces acting between matter, particles of matter move through spacetime on the shortest possible paths in this curved environment.
<blockquote>
"Spacetime tells matter how to move; matter tells spacetime how to curve." - John Archibald Wheeler
</blockquote>
All of this is encompassed in the Einstein field equation:
<div align=center>
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 8πG
<td rowspan="2" nowrap="nowrap" align="right"> T = R -  
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right"> R g
<tr>
<td style="border-top:solid 1px black;"> c4
<td style="border-top:solid 1px black;"> 2
</table>
</div>
Unfortunately, by convention, both the symbols G and R get used for a lot of different things. Here, symbols in bold are second rank tensors – constructs of arrays of 4×4 numbers where 4 is necessary because that is the number of dimensions of spacetime. Conventionally, these numbers are indexed by a value going from 0 to 3, with 0 being the part corresponding to the coordinate of time and 1 to 3 being the Cartesian coordinates that you might know as x, y, and z.

T is the stress-energy tensor: the (0,0) component represents the energy density at a given point; the (0,i) and (i, 0) components (with i going from 1 to 3) represent the momentum density; and the (i,j) components represent the stresses present, such as pressure or tension or shear.

The other term, R - R g/2, involves a partial description of the curvature. A full description of the curvature requires a construct of 4×4×4×4 components; too much for the stress energy tensor to fully constrain. Which is okay, because this allows curvature to exist away from where matter is, allowing gravity to have long range effects across empty space as well as for phenomena like gravitational waves. But it is enough that a given distribution of matter T gives enough constraints (along with specifying how the geometry behaves far away) that you can generally figure out what the actual gravity is.

This is the usual story that we are told. But there is an alternate way of looking at the same geometry such that the math works out the same, but it is described conceptually in a very different way. This divides spacetime up into "layers", called foliations, that represent slices of space and going from one foliation to the next is increasing in time. This is useful for computation because you can assign coordinates to your system. There are two additional constructs – the lapse function and the shift vector field. Starting at an event on one foliation, the lapse function at that position tells you that after a certain "proper time" (time perceived by an observer at that event) the time coordinate of the observer will be the proper time times the lapse function at that event. And the new spatial coordinates of the observer will be the proper time times the shift vector at that event. (Strictly speaking, this is only exactly true in the limit of the proper time approaching zero or when the lapse function and shift vectors are constant. But hopefully this gives a good enough insight to what these entities are so that we can go on to describe warp drives.)

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Lapse_shift.png]]
<tr><td>A folliation of space-time into slices of constant time coordinates (grey). At an event (black circle), the lapse function (yellow) shows how far into the future the point at that event will be projected in a unit amount of proper time experienced by an observer initially at the event. The shift vector (red) shows how that point changes coordinates after the unit amount of proper time. The cyan vector shows the new coordinates the point is projected onto. The lower black surface indicates a foliation of spacetime at the same time coordinate; after unit proper time, the events on the lower foliation get projected to the curved upper black foliation according to the shift vectors and lapse functions at each of their respective points.
</table>
</table>

Warp drives are generally restricted to spacetime geometries with a constant lapse function - all observers go forward by the same time coordinate after experiencing the same amount of proper time. Thus, all warp drives have no time dilation.
Far from the warp bubble, the shift vector vanishes – each spatial coordinate is mapped to the same spatial coordinate after any amount of proper time so long as the warp drive is not near. But the inside of the warp bubble is an extended region where the shift vector field is non-zero and constant. In this bubble, all spatial coordinates are mapped to a coordinate that shifts linearly in time with the proper time of the observer. Astute readers will see that this linear change in position with time is very similar to a velocity. This region of constant shift vector then moves with the shift vector velocity to keep stuff originally in the bubble so that it stays in the bubble. In between the inside of the bubble and the far away regions with zero shift vector, the shift vector field falls away to match one region to the other.

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Warp_shift.png]]
<tr><td>The shift vector of a one-dimensional slice through warp geometry, in red. At the edges there is no shift. At the center (inside the warp bubble) there is a region of uniform shift so that at a later time the inside of the warp bubble is rigidly translated to different spatial coordinates (in blue). The two foliations remain the same distance apart, because warp drives maintain a unit lapse function.
</table>
</table>

Traditionally, scientists would start with an interesting distribution of matter given by T and then find out what kind of curvature it created. This works well for planets and stars and black holes and cosmic strings and other things which are observed in the universe and you want to see what sort of effect they had. But starting in the 1980's, some physicists started looking at Einstein's field equations the other way around. They start with a desired geometry, find the curvature part of the field equation R - R g/2, and use that to find out what T has to be in order to get that curvature. This was originally done for [[Wormholes|wormholes]], but in 1994 Miguel Alcubierre used this same trick to figure out what could create a region of constant shift vector that moved with the shift vector - the warp drive. With this demand for a warp geometry, the required distribution of matter and energy and momentum and stress can be determined.

The original Alcubierre shift vector N→ produces a spherical warp bubble with an axi-symmetric distribution of, well, everything along the direction of motion with warp velocity vector v→.
<div align="center">
N→ = v→ f(rs)
</div>
Here, rs is the distance from the center of the warp bubble, and f(rs) can be any function that is 1 near rs = 0 and is 0 for rs well beyond the radius of the warp bubble.

=== Challenges and possible resolutions ===

Almost as soon as Alcubierre proposed his warp drive, others began picking it apart.
<ol>
<li>Pfenning and Ford<ref name="PfenningFord_2001">M. J. Pfenning and L. H. Ford, "The unphysical nature of 'Warp Drive'", Classical and Quantum Gravity. 14 (7): 1743–1751 (1997). arXiv:[https://arxiv.org/abs/gr-qc/9702026 gr-qc/9702026]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1997CQGra..14.1743P 1997CQGra..14.1743P]. doi:[https://doi.org/10.1088%2F0264-9381%2F14%2F7%2F011 10.1088/0264-9381/14/7/011]. S2CID [https://api.semanticscholar.org/CorpusID:15279207 15279207].</ref> found that if the negative energy density wall around the bubble obeys [[Wormholes#Quantum_energy_inequalities|quantum energy inequalities]], then for warp speeds of around light speed the shell thickness would be on the order of a hundred Planck lengths; a distance far smaller than any other known physical phenomenon or object.
<li>In the same work, Pfenning and Ford also found that for a bubble a hundred meters in radius, a warp speed of around the speed of light, and the required thickness of about 100 Planck lengths the energy in the bubble shell would be E ≈ -1063 kg c2. The magnitude of this latter value is ten orders of magnitude larger than the energy of the entire visible universe. They do note, however, that if quantum energy inequalities can be ignored then a warp drive with a hundred meters radius but a shell thickness of one meter would "only" have an energy magnitude of about a quarter solar mass.
<li>As with any method of faster than light travel, the warp drive could be used to make a time machine.
<li>Perhaps most seriously, if the warp drive is going faster than light speed, the negative energy regions on the outside of the shell won't be in the warp parts of the bubble. They will have to be moving through space at faster than light speed if the bubble is to maintain its integrity, which is the very problem that the warp drive was designed to avoid. The outside of the warp bubble shell that maintains the warp bubble would fall away and the warp bubble would quickly erode to sub-luminal speeds.<ref name="Van Den Broeck 2000">Chris Van Den Broeck, "On the (im)possibility of warp bubbles", arXiv:gr-qc/9906050 https://doi.org/10.48550/arXiv.gr-qc/9906050</ref>
<li>A more general problem that includes the previous one is that the geometry of the warp drive determines the distribution of energy, momentum, and stress (pressure, shear, and tension) by Einstein's field equation, which states that the stress energy tensor is directly proportional to a reduced order curvature tensor of space-time. However, the stress-energy tensor must also obey the continuity equation – basically, for the energy to change in a given volume of space that energy must come in or go out of the boundary of that volume and similarly for the momentum. So a warp drive can have the warp condition for one moment in time. But after that, the stress-energy that maintains the warp will start to move according to the forces on it and according to the natural paths that it takes through the warped space-time. There is no guarantee that the configuration of the stress-energy at later times will be sufficient to maintain the warp.
<blockquote>
From the perspective of simulating the warp drive dynamically, the key challenge is stability. One can straightforwardly generate the initial metric that describes the Alcubierre metric and infer the instantaneous matter configuration that must support it using the Einstein Equation, but a time-domain evolution of the coupled matter-gravity system necessitates that we specify the equation of state of the matter, which in turn determines how it evolves. In a perfect fluid, this involves specifying the relation between the fluid pressure and density, but in a more general case, the full relation between the
stress-energy components must be given. There is (to our knowledge) no known equation of state that would maintain the warp drive metric in a stable configuration over time - therefore, whilst one can require that initially, the warp bubble is constant, it will quickly evolve away from that state and, in most cases, the warp fluid and spacetime deformations will disperse or collapse into a central point.
 
– Clough, Dietrich, and Khan<ref>Katy Clough, Tim Dietrich, and Sebastian Khan, "What no one has seen before: gravitational waveforms from warp drive collapse", arXiv:2406.02466 [gr-qc] https://doi.org/10.48550/arXiv.2406.02466</ref>
</blockquote>
<li>In order to get the matter (or more generally, stress-energy) that forms the warp shell up to speeds that could allow the shell to exist, you need to be able to, well, accelerate that matter to warp speeds. Which gives a chicken-and-egg problem - in order to achieve superluminal travel you need to be able to get matter or energy moving superluminally in the first place<ref name="Bobrick and Martire">Alexey Bobrick and Gianni Martire, "Introducing Physical Warp Drives", arXiv:2102.06824 [gr-qc] https://doi.org/10.48550/arXiv.2102.06824</ref><ref>D. H. Coule, "No warp drive", Class. Quantum Grav. 15 (1998) 2523–2527. https://doi.org/10.1088/0264-9381/15/8/026</ref>, which rather makes warp drives unnecessary from the get-go.
<li>A warp bubble moving faster than light will not be able to get any signals in front of it as it will overtake its own signal. In fact, anyone in the interior of the bubble would not be able to send signals to the front surface of the bubble. In science-speak, a horizon forms at the front of the bubble<ref name="Natario">José Natário, "Warp drive with zero expansion", Classical and Quantum Gravity. 19 (6): 1157–1166 (2002). arXiv:[https://arxiv.org/abs/gr-qc/0110086 gr-qc/0110086]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2002CQGra..19.1157N 2002CQGra..19.1157N]. doi:[https://doi.org/10.1088%2F0264-9381%2F19%2F6%2F308 10.1088/0264-9381/19/6/308]. S2CID [https://api.semanticscholar.org/CorpusID:15859984 15859984].</ref>. As a consequence, the bubble cannot be controlled, steered, nor stopped from within the bubble itself.
<li>A Horizon will naturally emit Hawking radiation. It is estimated that the radiation will raise the temperature of the interior of the bubble to approximately 1032 K<ref name="Finazzi 2009">Stefano Finazzi, Stefano Liberati, and Carlos Barceló, "Semiclassical instability of dynamical warp drives", Phys. Rev. D 79, 124017 (12 June, 2009) https://doi.org/10.1103/PhysRevD.79.124017 , arXiv:0904.0141 [gr-qc] https://doi.org/10.48550/arXiv.0904.0141</ref> (at this level of accuracy, the result is the same if you replace temperatures in K by the same number in °C or °F). This temperature, however, may be lower if quantum inequalities do not have to be fulfilled.
<li>Semi-classical instabilities at the horizon are expected to lead to a divergence of the stress-energy tensor at the boundary in a short period of time, thus leading to the destruction of the warp bubble<ref name="Finazzi 2009"></ref>.
</ol>
Helpfully, Van Den Broeck<ref name="VanDenBroeck_2000">C. Van Den Broeck, "Alcubierre’s Warp Drive: Problems and Prospects" AIP Conference Proceedings. 504: 1105–1110 (2000). Bibcode:[https://ui.adsabs.harvard.edu/abs/2000AIPC..504.1105V 2000AIPC..504.1105V]. doi:[https://doi.org/10.1063%2F1.1290913 10.1063/1.1290913].</ref> suggested several solutions – or at least mitigations – for some of these problems
<ol>
<li>Quantum inequalities had not been shown to be true in general for highly curved space-times.
<li>A proposed warp drive geometry<ref name="VanDenBroeck_1999">C. Van Den Broeck, "A 'warp drive' with more reasonable total energy requirements". Classical and Quantum Gravity. 16 (12): 3973–3979 (1999). arXiv:[https://arxiv.org/abs/gr-qc/9905084 gr-qc/9905084]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1999CQGra..16.3973V 1999CQGra..16.3973V]. doi:[https://doi.org/10.1088%2F0264-9381%2F16%2F12%2F314 10.1088/0264-9381/16/12/314]. S2CID [https://api.semanticscholar.org/CorpusID:15466313 15466313]. </ref> (see the van Den Broeck drive, below) would be able to minimize the magnitude of the energy required to about that of the mass of our sun. While still large, it is not unphysically large.
<li>Time travel is always going to be a worry with faster than light travel. There's no neat solution to this one.
<li>It may be possible for the negative energy outside of the superluminal region to compress into the superluminal region to form a shock in space-time. The front surface of the bubble would then be a singularity. It is not clear if such a sharp jump in physical properties is possible, but neither is sure that it is impossible, either. However, while this might help with the negative energy in front of the bubble getting swept up because it cannot move faster than light, it does little to help with the material at the back of the bubble on the outside getting left behind because it cannot keep up.
</ol>
Other mitigating research has also shown
<ol start=9>
<li>Semi-classical divergence only occurs in warp drives with only one spatial dimensions. Those with two or more spatial dimensions (as in our universe) only contain points where divergences occur, and these divergences can be managed. In fact, their effects can be reduced by making the warp shape more elongated<ref>Carlos Barceló, Valentin Boyanov, Luis J. Garay, Eduardo Martín-Martínez,
and Jose M. Sánchez Velázquez, "Warp drive aerodynamics", JHEP08 (2022) 288 https://doi.org/10.1007/JHEP08(2022)288 , arXiv:2207.06458 [gr-qc] https://doi.org/10.48550/arXiv.2207.06458</ref>.
</ol>

=== Alcubierre warp interactions with light and matter ===

Space is not entirely empty. It is filled with a diffuse plasma in the form of the interstellar medium, as well as cosmic radiation, light from stars, and cosmic microwave background radiation. A warp drive propagating through space will encounter this stuff. What happens when this matter and radiation have a warp bubble pass across them?

The first analysis of matter encountering a warp bubble was performed by Pfenning and Ford<ref name="PfenningFord_2001"></ref>. They looked at the warp bubble interaction with a massive object at rest with the frame of reference of the warp drive (keep in mind that the rest frame is the same inside and outside the warp bubble; in this rest frame objects inside the bubble have no momentum even though they are, in some sense, changing location rapidly with time). When the warp bubble passes, the object experiences an acceleration in the direction of the warp bubble motion. When the warp shell passes and the object is inside the bubble, it will be moving with approximately the speed of the bubble. Pfenning and Ford analyzed this problem with a continuous distribution of shell energy that, strictly speaking, never falls to zero except at the bubble center and at spatial infinity, so unless the object passes through the center in Pfenning and Ford's description it will never quite get up to the bubble's speed. In this case, the object will move almost as fast as the object but will pass through the bubble in a finite time, after which it will again be at rest with respect to the reference frame but displaced along the direction of the warp bubble motion by some distance. With this description of the warp bubble, a spacecraft of finite size will always be moving a little slower than the warp bubble and would have to use rockets to keep up with it. in addition, the spacecraft would experience tidal forces that would cause stress on the spacecraft's structure.

Pfenning and Ford also examined cases where the shell is of a finite (possibly infinitesimal) width and falls to zero both inside and outside the bubble. In this case any matter encountering the bubble would be collected at the bow of the bubble and thereafter move along with it.

McMonigal et al. analyzed the situation for both massive particles and light moving along the axis of travel of the bubble<ref name="McMonigalEtAl_2012">B. McMonigal, G. F. Lewis, and P. O'Byrne, "Alcubierre warp drive: On the matter of matter". Physical Review D. 85 (6) 064024 (20 March 2012). arXiv:[https://arxiv.org/abs/1202.5708 1202.5708]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2012PhRvD..85f4024M 2012PhRvD..85f4024M]. doi:[https://doi.org/10.1103%2FPhysRevD.85.064024 10.1103/PhysRevD.85.064024]. S2CID [https://api.semanticscholar.org/CorpusID:3993148 3993148].</ref>. They found that light moving opposite the warp direction passed through the bubble without incident, being only somewhat delayed by passing through the bubble. A warp bubble that is warping at sub-luminal speeds can have light catch up from behind it. This light is able to pass through the bubble, and is somewhat advanced in its path by the speed of the bubble. For super-luminal warp bubbles, however, the situation is different. The bubble will catch up to light moving it its own direction that is originally in front of it. This light cannot escape forward, the bubble being too fast. Nor can it escape backward, as the light is propagating forward and the interior of the bubble is at rest. Thus, the light gets caught at the bow of the warp bubble, unable to escape for so long as the warp bubble is active. This light is strongly blue shifted to extremely energetic x-rays and gamma rays. The space behind the bubble is swept clear of forward-moving light.

The situation for matter over-run by a super-luminal warp bubble is similar. Matter moving backward passes through the bubble, being only swept a ways along the bubble's path as sign of its passing but otherwise continuing on their way. Matter at rest behaves the same way as Pfenning and Ford discovered. Meanwhile, matter moving in the same direction of the warp is overtaken and collects at the bow of the bubble, unable to leave for so long as the bubble is warping. This matter "surfing" on the bubble bow is highly accelerated to relativistic speeds, experiencing extreme time dilation.

For a sub-luminal bubble, there is a speed faster than the warp speed where particles coming up from behind overtake the bubble and pass through, out the front. Particles below this speed but still faster than the warp speed still overtake the bubble, but then bounce backwards out of the bubble from behind. They are still moving forward, but are now moving more slowly than the warp speed. Particles moving forward but slower than the bubble will be overtaken and then bounced out the front with a speed higher than the warp speed. Those particles moving backward will pass through the bubble and exit with their initial speed.

The spacecraft inside the bubble will observe that light moving backward compared to the warp direction is blue-shifted, and matter moving backward passes through with increased energy. Meanwhile, a sub-luminal bubble will see light that was moving forward as red-shifted and matter catching up to the bubble from behind will pass by with reduced energy (superluminal bubbles, of course, do not have any matter or radiation catching up to them from behind).

When a superluminal warp bubble that has been collecting matter and radiation for the duration of its journey and blue-shifting it to much higher energies is turned off, all that matter and energy is released as a blast of radiation in the direction the bubble was warping.

McMonigal et al. conclude by noting that any spacecraft in the warp bubble would need shielding to protect against the blue-shifted radiation and matter of increased energy. In addition the destination would be "blasted into oblivion" by the release of matter and radiation that had been caught in the bubble during the trip.

Clark et al.<ref>Chad Clark, William A. Hiscock, and Shane L. Larson, "Null geodesics in the Alcubierre warp drive spacetime: the view from the bridge", arXiv:gr-qc/9907019 (2018) https://doi.org/10.48550/arXiv.gr-qc/9907019</ref> examine what it would look like from within an operating warp drive. They find that light moving at 90 degrees to the direction of motion will look like it is still coming from 90 degrees. However, light from the front hemisphere is squished progressively toward the forward point the faster you go. For a warp bubble moving faster than the speed of light, there is a cone behind the warp bubble where no light can reach it. However, you don't see a black spot behind you – light from the regions in the backward hemisphere from outside that cone that can reach you is distorted toward the direction behind you to fill in that spot. Light from the forward direction is blue-shifted and light from behind is red-shifted. At speeds of around 200 times light speed, the cosmic microwave background would be blue shifted to energies similar to that of the solar photosphere, heating anything in the warp bubble as if they were close to the sun.

== Other warp geometries ==

The Alcubierre warp drive is the first and best known of the warp drives. It has been around long enough that there have been many studies done on it. The warp geometry is particularly simple, making analysis easier. But in the time that the Alcubierre warp was proposed, many other researchers have come up with other warp designs, and even generalized the idea to include a wide array of additional possibilities.

Most of the general statements about the challenges of the Alcubierre drive hold for these other drives. Many of the details about interactions with matter are also likely to hold, although these other styles of warp drive have not yet been subject to such rigorous investigation.

=== Van Den Broeck warp drive ===

If an Alcubierre warp bubble a hundred meters across requires a magnitude of energy greater than the entire energy in the observable universe, one option to reduce the magnitude of energy used is to make the warp bubble smaller. Much smaller. Pfenning and Ford<ref name="PfenningFord_2001"></ref> suggested making the warp bubble smaller than an atom. Of course, that brings up the problem of how to stuff a spacecraft in there. Van Den Broeck proposed a solution<ref name="VanDenBroeck_1999"></ref>: make the warp bubble only about the size of an atomic nucleus but expand the space inside the bubble enormously so that a spacecraft could fit in. This is somewhat like a nuclear sized wormhole that leads to a pocket universe that holds the spacecraft. Unfortunately, the metric doesn't fit neatly into any [[Wormholes#Three_dimensions|embedding diagram]] that I can figure out, but the math works out so that spatial coordinates inside the bubble are expanded by a factor of 1017 and a spacecraft would have a few hundred meters of bubble interior to putz around in. This trick manages to reduce the magnitude of energy to only about the mass energy of a few stars similar to our own. Other than that, it is otherwise a normal Alcubierre drive.

Further work<ref>S. Krasnikov, "The quantum inequalities do not forbid spacetime shortcuts", Physical Review D. 67 (10) 104013. https://doi.org/10.1103%2FPhysRevD.67.104013, arXiv:gr-qc/0207057 (2003) https://doi.org/10.48550/arXiv.gr-qc/0207057 </ref> has managed to use Van Den Broeck's trick to reduce the magnitude of the negative energy to only about a milligram.

=== Natário warp drive ===

The Alcubierre drive is not the only way to construct a warp drive. José Natário<ref name="Natario"></ref> showed that it was just one example of an entire class of warp drives. These are space-time geometries with a smooth and localized shift vector field and a unit lapse function.

Natário then went on to find in that class a set of warp drives that do not have any expansion or contraction of space at all. Instead, space encountering the front boundary of the warp bubble instead "slides" around the outside of the bubble until it gets to the corresponding place on the back and is left behind there. Another way to think of it is that space entering the bubble shell is compressed in the radial direction of the shell but is simultaneously expanded in the tangential direction so that there is no net change in volume. When it gets to the back, the opposite occurs. In the language of mathematics, the shift vector field is divergenceless.

<table align=center>
<tr><td width=360>[[File:Natario_drive_flow_of_space.png]]
<td width=445>[[File:Natario_warp_drive_horizons.png]]
<tr><td style="vertical-align: top;">A representation of the motion of space in a zero-expansion Natário drive with a vanishingly thin warp shell around the bubble.
<td>The horizons of the generalized Natário class of warp drives (including the Alcubierre warp drive and all fell-Heisenberg drives) at superluminal speed in the direction of the green arrow. No event inside the bubble can affect events outside the bubble in front of the blue line. No events outside the bubble behind the red line can affect events inside the bubble. The lines have an angle α with respect to the direction of motion with sin(α) = 1/vs for vs the speed of the bubble relative to light speed. This is analogous to the Mach cone of supersonic objects.
</table>

NOTE: There are two different things that can be meant by the term "Natário warp drive". The generalized Natário warp drive is the general theory of all warp drives that includes within it the Alcubierre, Fell-Heisenberg, and Lentz warp drives, among infinitely many others. The zero-expansion Natário warp drive is the specific implementation of the generalized Natário warp drive that has a divergenceless shift vector field.

The generalized class of warp drives developed by Natário (including both the Alcubierre and this zero-expansion warp drive) are shown to always have regions in the warp shell where the energy density is negative to at least some observers. You can't get away from it – to warp, you need negative energy density.

The Natário class of warp drives blueshift light coming in from the front and redshift light catching up from the back. For the case of a super-luminal warp bubble, there will of course be a horizon at the back of the bubble and light will not be able to get through from behind. For a bubble speed vs relative to the speed of light, light coming from straight ahead will be blueshifted up in frequency and photon energy by a factor of 1 + vs. For sub-luminal travel, light coming from behind will be redshifted by a factor 1 - vs. In general for n as the unit direction along which the light is propagating, the blueshift factor will be 1 + vs ⋅ n. If the warp shell is infinitesimally thin, light from outside that reaches the center of the bubble will not be distorted in direction although it will be frequency shifted, but observers still inside the bubble but displaced from the center will see the field of view distorted as well as frequency shifted.

Most of the difficulties that apply to the Alcubierre warp drive also apply to the generalized Natário warp drive (and thus to the zero-expansion Natário drive and Fell-Heisenberg drive as well). In particular, quantum energy inequalities, high magnitudes of the energy required, formation of time machines, instabilities related to the stress-energy making up the warp shell being unable to keep up with the bubble, the chicken-and-egg problem of needing FTL to achieve FTL, the formation of horizons making warp bubbles uncontrollable, and Hawking radiation at the horizons, are all still concerns for generic warp drives.

=== Fell-Heisenberg warp drives ===

S. D. B. Fell and L. Heisenberg investigated what sort of warp structure is necessary to produce a warp drive with positive energy density everywhere<ref name="Fell-Heisenberg">S. D. B. Fell and L> Heisenberg, "Positive energy warp drive from hidden geometric structures", Classical and Quantum Gravity 38 155020 (2021) https://doi.org/10.1088/1361-6382/ac0e47 https://arxiv.org/abs/2104.06488</ref>. They found that warp drives of the generic Natário form with irrotational shift vectors had a simple expression for the energy density and vanishing momentum (irrotational means that the shift vector can be described as the gradient of a scalar potential field or, equivalently, that the curl of the shift vector field vanishes everywhere). Fell and Heisenberg proposed several potential fields that either had positive energy density everywhere except at discontinuities, or that were continuous but had regions that violated the weak energy condition (i. e. negative energy densities in some frames of reference). One specific example they provided of a warp field had an energy that is about 10,000 times less than the mass-energy of our sun. Or only about half the mass-energy of Jupiter. This is a significant improvement over many other proposed drives.

Despite the introduction discussing warp drive configurations that satisfy the various energy conditions, the configurations described in the paper are shown to locally violate the weak and strong energy conditions. Nonetheless, the energy density is still mostly positive. Santiago, Schuster, and Visser's<ref name="Santiago Schuster Visser 2022">J. Santiago, S. Schuster, and M. Visser, "Generic warp drives violate the null energy condition", Physical Review D 105, 064038 (2022) https://doi.org/10.1103/PhysRevD.105.064038</ref> work shows that the various energy conditions must still be violated by this warp drive; to not violate these, energy density must be positive in all reference frames not just those of the co-moving observer.

Any vector field can be described by its divergence, its curl, and a constant. The constant part of a warp drive's shift vector can be discarded because then the warp drive would not be localized. It is interesting, therefore, that the Natário zero-expansion drive has zero divergence and can be described only by its curl, while the Fell-Heisenberg drive has zero curl and can be described only by its divergence. These two classes of drives thus rely on opposite choices of setting one of the two different parts describing a vector field to zero. As might be expected, the Fell-Heisenberg drive is not zero expansion – space-time will change in volume as it passes through the warp bubble (although after going all the way through it will return to its original volume). However, objects in space-time will retain their original orientation as they pass through the warp bubble.

Any vector field with zero curl can be represented as the gradient of a suitable scalar function. Thus, the Fell-Heisenberg drive can be completely described by just scalar field (one number at all points of space and time) rather than a vector field.

Previous examples of warp drives, such as the Alcubierre and zero-expansion Natário drives, tend toward a flat space-time as you go far enough away faster than would occur if there were any mass present in the warp drive. That is, the ADM mass of such drives is zero. However, one of the Fell-Heisenberg examples described a spacetime where the geometry approaches that of what you would get for a massive object at the warp location, giving one of the first examples of a warp drive with mass.

In the Fell-Heisenberg drive the momentum vanishes everywhere. Yet the energy occupies regions where the shift vector is varying rapidly. The lack of momentum means that the energy will not move to keep up with the differential expansion and movement of the spacetime. As a consequence, at later times the energy will have a different distribution than what is necessary to maintain the given warp configuration. The exact time evolution is not solved but this likely leads to collapse of warp bubble. Note that this is not a problem unique to the Fell-Heisenberg drive. Assuming a geometry that describes a warp produces a specific distribution of the stress-energy tensor that satisfies Einstein's equation. But there is no guarantee that this stress energy tensor satisfies the continuity equation, which describes how the distribution of matter and momentum change with time. The issue of stability of any warp drive is not well understood, but most warp drive geometries will be unstable in the sense that their matter that generates the warp fields will not be able to keep up with the motion of the warp. The Fell-Heisenberg drive is unique only in that, with vanishing momentum, there is no configuration that allows stable propagation.

(n. b. The Heisenberg here is Lavinia Heisenberg, not the Werner Heisenberg of quantum physics and uncertainty principle fame.)

== Lentz warp drive ==

E. Lentz<ref name="Lentz">E. W. Lentz, "Breaking the warp barrier: hyper-fast solitons in Einstein–Maxwell-plasma theory", Classical and Quantum Gravity. 38 075015 (2021). arXiv:[https://arxiv.org/abs/2006.07125 2006.07125]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2021CQGra..38g5015L 2021CQGra..38g5015L]. doi:[https://doi.org/10.1088%2F1361-6382%2Fabe692 10.1088/1361-6382/abe692]. ISSN [https://search.worldcat.org/issn/0264-9381 0264-9381]. S2CID [https://api.semanticscholar.org/CorpusID:219635854 219635854].</ref> investigated a particular configuration of the Fell-Heisenberg class of warp drives, where the scalar field forms a selection of rhomboid regions, resulting in an energy density that is everywhere positive and a shift vector with vanishing tidal force at the center for a comfortable ride.

However, Santiago, Schuster, and Visser<ref name="Santiago Schuster Visser 2022"></ref> dispute claims that the Lentz drive satisfies the energy conditions, noting that everywhere positive energy density in one frame of reference is insufficient to establish that the energy density is positive in all reference frames; knowledge of the Cauchy stress tensor is also needed. They show that any generic warp drive will violate the strong energy condition, null energy condition, and weak energy condition.

== Conservation laws ==

(Discussion of asymptotic flatness: what is it? Where does it apply?)

(The "big four" conservation laws and how they relate to asymptotic flatness.)

(ADM mass -> 0; conservation of energy/mass issues.)<ref>Sebastian Schuster, Jessica Santiago,and Matt Visser, "ADM mass in warp drive spacetimes" General Relativity and Gravitation (2023) 55:14 https://doi.org/10.1007/s10714-022-03061-9</ref>

(Discuss issues of conservation of angular momentum.)

(If angular momentum conservation is ignored, discuss how momentum & energy are affected by outside forces. In a gravitational field equivalent to inertial frame ... like being in an accelerated elevator; will acquire momentum buildup while staying "at rest".)

== Warp Railroads ==

Van Den Broeck<ref name="VanDenBroeck_2000"></ref> suggested a warp railroad,

You can set up devices ahead of time along the path of the bubble that produce the negative energy regions for the warp bubble at the appropriate time without needing the negative energy to ever move with the bubble at all. At least for bubbles on a predictable schedule and route.

helps with:

-stability (elaborate)

-conservation laws (elaborate)

Mention Krasnikov<ref name="Krasnikov 1998">S. V. Krasnikov, "Hyperfast travel in general relativity" Phys. Rev. D 57 4760</ref>

Hyperwave<ref name="Pieri 2024">Lorenzo Pieri, "Hyperwave: hyper-Fast Communication within General Relativity", arXiv:2311.12069 [gr-qc] https://doi.org/10.48550/arXiv.2311.12069</ref> makes use of this warp railroad idea.

== Quantum effects ==

(quantum stuff here<ref name="Hiscock1997">W. A. Hiscock, "Quantum effects in the Alcubierre warp drive spacetime", Classical and Quantum Gravity 14 L183 https://doi.org/10.1088/0264-9381/14/11/002 https://arxiv.org/abs/gr-qc/9707024</ref><ref name="Finazzi et al 2009">S. Finazzi, S. Liberati, C. Barceló, "Semiclassical instability of dynamical warp drives", Physical Review D 79, 124017 (2009)https://doi.org/10.1103/PhysRevD.79.124017 https://arxiv.org/abs/0904.0141</ref>)

From<ref name="Pieri 2024"></ref>
<blockquote>
It should be emphasised that Quantum Inequalities have not been proven for realistic interacting quantum fields, where the they are expected to be weaker if they exist [Cadamuro, 2019], and they often require Minkowskian conditions at infinity, a condition which is not appropriate for fields inside bounded regions, as in a Casimir devices.
</blockquote>

== Warp drives and black holes ==

What happens if you drive your warp drive into a black hole?

The pop-sci reason that nothing can escape a black hole is that at the event horizon the escape velocity is faster than light and nothing can go faster than light. A warp drive can go faster than light.

The more detailed reason from general relativity is that at the event horizon space and time are rotated sufficiently that "inward" becomes "forward in time." You can no more go farther away from the singularity in a black hole once you are inside the event horizon than you can go backward in time. Any super-luminal travel can go backward in time, and warp drives can engage in super-luminal travel.

The Hawking area theorem that says that black holes can only grow, never shrink relies on the condition that the energy is everywhere positive. Warp drives have negative energy density.

So in principle, there's nothing that prevents a warp drive from taking a dip into a black hole and coming back out again.

Garattini and Zatrimaylov looked into the problem of a warp drive near (and in) a black hole<ref name="Garattini and Zatrimaylov">R. Garattini and K. Zatrimaylov, "Black holes, warp drives, and energy conditions", Physics Letters B 856 138910 (2024) https://doi.org/10.1016/j.physletb.2024.138910</ref>. The found that the black hole reduces the negative energy requirement for the warp bubble to move inward toward the black hole singularity, but increases the negative energy needed to move away from the center of the black hole. In addition, a warp drive parked at the event horizon would allow light from inside the horizon to pass through the bubble and escape back outside.

== Credit ==
Author: Luke Campbell

== References ==

[[Category:Physics]][[Category:Infrastructure]][[Category:Physics & Engineering‏‎]][[Category:Physics & Math & Engineering]][[Category:Transportation & Infrastructure‏‎]][[Category:Metric Engineering]]

Warp Drives

2026-07-30T02:43:41Z

Lwcamp: /* Natário warp drive */

{{PageConstructionNotice}}

Science fiction often features spacecraft that can seemingly move across space and get between the place of departure and the destination much faster than light could have done. This appears to contradict the theory of relativity, which predicts unequivocally that nothing can move through space faster than light. Because relativity has been incredibly successful at describing nature, with its many other predictions regularly being confirmed to extraordinary accuracy and within the bounds of uncertainty of all the experiments that tested them, it gives confidence that relativity is a correct description of reality. Which seems to rather throw a wet towel on our hopes for rapid travel between stars.

However, while relativity does not allow things to move through space faster than light, it places no such restrictions on how fast space-time itself can expand, contract, or move around. This leads to the idea of a warp drive – the spacecraft remains stationary within a region of highly curved space-time, and that region moves at super-luminal speeds rather than the spacecraft.

== The Alcubierre warp drive ==

The first warp drive geometry that satisfied the Einstein field equations of relativity was proposed by Miguel Alcubierre<ref>M. Alcubierre, "The warp drive: hyper-fast travel within general relativity." Classical and Quantum Gravity. 11 (5): L73–L77 (1994). [https://arxiv.org/abs/gr-qc/0009013 arXiv:gr-qc/0009013]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1994CQGra..11L..73A 1994CQGra..11L..73A]. doi:[https://doi.org/10.1088%2F0264-9381%2F11%2F5%2F001 10.1088/0264-9381/11/5/001]. S2CID [https://api.semanticscholar.org/CorpusID:4797900 4797900].</ref>. In this geometry, a sphere of space-time moves at an arbitrary speed (potentially but not necessarily a speed much faster than light). Objects within the sphere are moved along with the sphere; an object at rest within the sphere would be moved along with the sphere indefinitely. Space is expanding at the rear boundary and contracting at the front boundary in order to keep the sphere moving. In order to satisfy the Einstein field equations, the boundary of the sphere must have a negative energy density. The challenges of space-time geometries with negative energy densities are described in our page on [[Wormholes#Exotic_energy_conditions|wormholes]], for our purposes it is enough to note that negative energy densities can pose problems if not handled carefully, there may be limits on how much negative energy you can have without nearby positive energy density, and it may not be possible to get enough negative energy to support a warp drive; although none of this is ruled out by physics – yet!

<table align=center>
<tr><td width=650>[[File:Energy_magnitude_of_Alcubierre_drive.png]]
<td width=650>[[File:Expansion_of_Alcubierre_drive.png]]
<tr><td>The magnitude of the energy of the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. (The energy density is negative everywhere in the shell, here only the magnitude is shown to aid visual comprehension.)
<td>The expansion of space in the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. Negative expansion means that space is contracting in that direction, positive expansion that it is expanding.
</table>

NOTE: What the? The energy distribution is symmetric! How does the drive know which way to go? There must be significant contributions of other components of the stress-energy tensor, and those have got to be asymmetric along the forward/backward axis. Check this when I get time.

(And what's with the ring in all the artwork, anyway?)

=== So what is going on? ===

What is it that makes a warp drive a warp drive, and how does it work?

A warp drive is a concept that comes out of the science of general relativity. General relativity collects space and time together into a single "structure" called spacetime. This structure is curved, and its curvature is determined by the material that is in it. Absent forces acting between matter, particles of matter move through spacetime on the shortest possible paths in this curved environment.
<blockquote>
"Spacetime tells matter how to move; matter tells spacetime how to curve." - John Archibald Wheeler
</blockquote>
All of this is encompassed in the Einstein field equation:
<div align=center>
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 8πG
<td rowspan="2" nowrap="nowrap" align="right"> T = R -  
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right"> R g
<tr>
<td style="border-top:solid 1px black;"> c4
<td style="border-top:solid 1px black;"> 2
</table>
</div>
Unfortunately, by convention, both the symbols G and R get used for a lot of different things. Here, symbols in bold are second rank tensors – constructs of arrays of 4×4 numbers where 4 is necessary because that is the number of dimensions of spacetime. Conventionally, these numbers are indexed by a value going from 0 to 3, with 0 being the part corresponding to the coordinate of time and 1 to 3 being the Cartesian coordinates that you might know as x, y, and z.

T is the stress-energy tensor: the (0,0) component represents the energy density at a given point; the (0,i) and (i, 0) components (with i going from 1 to 3) represent the momentum density; and the (i,j) components represent the stresses present, such as pressure or tension or shear.

The other term, R - R g/2, involves a partial description of the curvature. A full description of the curvature requires a construct of 4×4×4×4 components; too much for the stress energy tensor to fully constrain. Which is okay, because this allows curvature to exist away from where matter is, allowing gravity to have long range effects across empty space as well as for phenomena like gravitational waves. But it is enough that a given distribution of matter T gives enough constraints (along with specifying how the geometry behaves far away) that you can generally figure out what the actual gravity is.

This is the usual story that we are told. But there is an alternate way of looking at the same geometry such that the math works out the same, but it is described conceptually in a very different way. This divides spacetime up into "layers", called foliations, that represent slices of space and going from one foliation to the next is increasing in time. This is useful for computation because you can assign coordinates to your system. There are two additional constructs – the lapse function and the shift vector field. Starting at an event on one foliation, the lapse function at that position tells you that after a certain "proper time" (time perceived by an observer at that event) the time coordinate of the observer will be the proper time times the lapse function at that event. And the new spatial coordinates of the observer will be the proper time times the shift vector at that event. (Strictly speaking, this is only exactly true in the limit of the proper time approaching zero or when the lapse function and shift vectors are constant. But hopefully this gives a good enough insight to what these entities are so that we can go on to describe warp drives.)

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Lapse_shift.png]]
<tr><td>A folliation of space-time into slices of constant time coordinates (grey). At an event (black circle), the lapse function (yellow) shows how far into the future the point at that event will be projected in a unit amount of proper time experienced by an observer initially at the event. The shift vector (red) shows how that point changes coordinates after the unit amount of proper time. The cyan vector shows the new coordinates the point is projected onto. The lower black surface indicates a foliation of spacetime at the same time coordinate; after unit proper time, the events on the lower foliation get projected to the curved upper black foliation according to the shift vectors and lapse functions at each of their respective points.
</table>
</table>

Warp drives are generally restricted to spacetime geometries with a constant lapse function - all observers go forward by the same time coordinate after experiencing the same amount of proper time. Thus, all warp drives have no time dilation.
Far from the warp bubble, the shift vector vanishes – each spatial coordinate is mapped to the same spatial coordinate after any amount of proper time so long as the warp drive is not near. But the inside of the warp bubble is an extended region where the shift vector field is non-zero and constant. In this bubble, all spatial coordinates are mapped to a coordinate that shifts linearly in time with the proper time of the observer. Astute readers will see that this linear change in position with time is very similar to a velocity. This region of constant shift vector then moves with the shift vector velocity to keep stuff originally in the bubble so that it stays in the bubble. In between the inside of the bubble and the far away regions with zero shift vector, the shift vector field falls away to match one region to the other.

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Warp_shift.png]]
<tr><td>The shift vector of a one-dimensional slice through warp geometry, in red. At the edges there is no shift. At the center (inside the warp bubble) there is a region of uniform shift so that at a later time the inside of the warp bubble is rigidly translated to different spatial coordinates (in blue). The two foliations remain the same distance apart, because warp drives maintain a unit lapse function.
</table>
</table>

Traditionally, scientists would start with an interesting distribution of matter given by T and then find out what kind of curvature it created. This works well for planets and stars and black holes and cosmic strings and other things which are observed in the universe and you want to see what sort of effect they had. But starting in the 1980's, some physicists started looking at Einstein's field equations the other way around. They start with a desired geometry, find the curvature part of the field equation R - R g/2, and use that to find out what T has to be in order to get that curvature. This was originally done for [[Wormholes|wormholes]], but in 1994 Miguel Alcubierre used this same trick to figure out what could create a region of constant shift vector that moved with the shift vector - the warp drive. With this demand for a warp geometry, the required distribution of matter and energy and momentum and stress can be determined.

The original Alcubierre shift vector N→ produces a spherical warp bubble with an axi-symmetric distribution of, well, everything along the direction of motion with warp velocity vector v→.
<div align="center">
N→ = v→ f(rs)
</div>
Here, rs is the distance from the center of the warp bubble, and f(rs) can be any function that is 1 near rs = 0 and is 0 for rs well beyond the radius of the warp bubble.

=== Challenges and possible resolutions ===

Almost as soon as Alcubierre proposed his warp drive, others began picking it apart.
<ol>
<li>Pfenning and Ford<ref name="PfenningFord_2001">M. J. Pfenning and L. H. Ford, "The unphysical nature of 'Warp Drive'", Classical and Quantum Gravity. 14 (7): 1743–1751 (1997). arXiv:[https://arxiv.org/abs/gr-qc/9702026 gr-qc/9702026]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1997CQGra..14.1743P 1997CQGra..14.1743P]. doi:[https://doi.org/10.1088%2F0264-9381%2F14%2F7%2F011 10.1088/0264-9381/14/7/011]. S2CID [https://api.semanticscholar.org/CorpusID:15279207 15279207].</ref> found that if the negative energy density wall around the bubble obeys [[Wormholes#Quantum_energy_inequalities|quantum energy inequalities]], then for warp speeds of around light speed the shell thickness would be on the order of a hundred Planck lengths; a distance far smaller than any other known physical phenomenon or object.
<li>In the same work, Pfenning and Ford also found that for a bubble a hundred meters in radius, a warp speed of around the speed of light, and the required thickness of about 100 Planck lengths the energy in the bubble shell would be E ≈ -1063 kg c2. The magnitude of this latter value is ten orders of magnitude larger than the energy of the entire visible universe. They do note, however, that if quantum energy inequalities can be ignored then a warp drive with a hundred meters radius but a shell thickness of one meter would "only" have an energy magnitude of about a quarter solar mass.
<li>As with any method of faster than light travel, the warp drive could be used to make a time machine.
<li>Perhaps most seriously, if the warp drive is going faster than light speed, the negative energy regions on the outside of the shell won't be in the warp parts of the bubble. They will have to be moving through space at faster than light speed if the bubble is to maintain its integrity, which is the very problem that the warp drive was designed to avoid. The outside of the warp bubble shell that maintains the warp bubble would fall away and the warp bubble would quickly erode to sub-luminal speeds.<ref name="Van Den Broeck 2000">Chris Van Den Broeck, "On the (im)possibility of warp bubbles", arXiv:gr-qc/9906050 https://doi.org/10.48550/arXiv.gr-qc/9906050</ref>
<li>A more general problem that includes the previous one is that the geometry of the warp drive determines the distribution of energy, momentum, and stress (pressure, shear, and tension) by Einstein's field equation, which states that the stress energy tensor is directly proportional to a reduced order curvature tensor of space-time. However, the stress-energy tensor must also obey the continuity equation – basically, for the energy to change in a given volume of space that energy must come in or go out of the boundary of that volume and similarly for the momentum. So a warp drive can have the warp condition for one moment in time. But after that, the stress-energy that maintains the warp will start to move according to the forces on it and according to the natural paths that it takes through the warped space-time. There is no guarantee that the configuration of the stress-energy at later times will be sufficient to maintain the warp.
<blockquote>
From the perspective of simulating the warp drive dynamically, the key challenge is stability. One can straightforwardly generate the initial metric that describes the Alcubierre metric and infer the instantaneous matter configuration that must support it using the Einstein Equation, but a time-domain evolution of the coupled matter-gravity system necessitates that we specify the equation of state of the matter, which in turn determines how it evolves. In a perfect fluid, this involves specifying the relation between the fluid pressure and density, but in a more general case, the full relation between the
stress-energy components must be given. There is (to our knowledge) no known equation of state that would maintain the warp drive metric in a stable configuration over time - therefore, whilst one can require that initially, the warp bubble is constant, it will quickly evolve away from that state and, in most cases, the warp fluid and spacetime deformations will disperse or collapse into a central point.
 
– Clough, Dietrich, and Khan<ref>Katy Clough, Tim Dietrich, and Sebastian Khan, "What no one has seen before: gravitational waveforms from warp drive collapse", arXiv:2406.02466 [gr-qc] https://doi.org/10.48550/arXiv.2406.02466</ref>
</blockquote>
<li>In order to get the matter (or more generally, stress-energy) that forms the warp shell up to speeds that could allow the shell to exist, you need to be able to, well, accelerate that matter to warp speeds. Which gives a chicken-and-egg problem - in order to achieve superluminal travel you need to be able to get matter or energy moving superluminally in the first place<ref name="Bobrick and Martire">Alexey Bobrick and Gianni Martire, "Introducing Physical Warp Drives", arXiv:2102.06824 [gr-qc] https://doi.org/10.48550/arXiv.2102.06824</ref><ref>D. H. Coule, "No warp drive", Class. Quantum Grav. 15 (1998) 2523–2527. https://doi.org/10.1088/0264-9381/15/8/026</ref>, which rather makes warp drives unnecessary from the get-go.
<li>A warp bubble moving faster than light will not be able to get any signals in front of it as it will overtake its own signal. In fact, anyone in the interior of the bubble would not be able to send signals to the front surface of the bubble. In science-speak, a horizon forms at the front of the bubble<ref name="Natario">José Natário, "Warp drive with zero expansion", Classical and Quantum Gravity. 19 (6): 1157–1166 (2002). arXiv:[https://arxiv.org/abs/gr-qc/0110086 gr-qc/0110086]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2002CQGra..19.1157N 2002CQGra..19.1157N]. doi:[https://doi.org/10.1088%2F0264-9381%2F19%2F6%2F308 10.1088/0264-9381/19/6/308]. S2CID [https://api.semanticscholar.org/CorpusID:15859984 15859984].</ref>. As a consequence, the bubble cannot be controlled, steered, nor stopped from within the bubble itself.
<li>A Horizon will naturally emit Hawking radiation. It is estimated that the radiation will raise the temperature of the interior of the bubble to approximately 1032 K<ref name="Finazzi 2009">Stefano Finazzi, Stefano Liberati, and Carlos Barceló, "Semiclassical instability of dynamical warp drives", Phys. Rev. D 79, 124017 (12 June, 2009) https://doi.org/10.1103/PhysRevD.79.124017 , arXiv:0904.0141 [gr-qc] https://doi.org/10.48550/arXiv.0904.0141</ref> (at this level of accuracy, the result is the same if you replace temperatures in K by the same number in °C or °F). This temperature, however, may be lower if quantum inequalities do not have to be fulfilled.
<li>Semi-classical instabilities at the horizon are expected to lead to a divergence of the stress-energy tensor at the boundary in a short period of time, thus leading to the destruction of the warp bubble<ref name="Finazzi 2009"></ref>.
</ol>
Helpfully, Van Den Broeck<ref name="VanDenBroeck_2000">C. Van Den Broeck, "Alcubierre’s Warp Drive: Problems and Prospects" AIP Conference Proceedings. 504: 1105–1110 (2000). Bibcode:[https://ui.adsabs.harvard.edu/abs/2000AIPC..504.1105V 2000AIPC..504.1105V]. doi:[https://doi.org/10.1063%2F1.1290913 10.1063/1.1290913].</ref> suggested several solutions – or at least mitigations – for some of these problems
<ol>
<li>Quantum inequalities had not been shown to be true in general for highly curved space-times.
<li>A proposed warp drive geometry<ref name="VanDenBroeck_1999">C. Van Den Broeck, "A 'warp drive' with more reasonable total energy requirements". Classical and Quantum Gravity. 16 (12): 3973–3979 (1999). arXiv:[https://arxiv.org/abs/gr-qc/9905084 gr-qc/9905084]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1999CQGra..16.3973V 1999CQGra..16.3973V]. doi:[https://doi.org/10.1088%2F0264-9381%2F16%2F12%2F314 10.1088/0264-9381/16/12/314]. S2CID [https://api.semanticscholar.org/CorpusID:15466313 15466313]. </ref> (see the van Den Broeck drive, below) would be able to minimize the magnitude of the energy required to about that of the mass of our sun. While still large, it is not unphysically large.
<li>Time travel is always going to be a worry with faster than light travel. There's no neat solution to this one.
<li>It may be possible for the negative energy outside of the superluminal region to compress into the superluminal region to form a shock in space-time. The front surface of the bubble would then be a singularity. It is not clear if such a sharp jump in physical properties is possible, but neither is sure that it is impossible, either. However, while this might help with the negative energy in front of the bubble getting swept up because it cannot move faster than light, it does little to help with the material at the back of the bubble on the outside getting left behind because it cannot keep up.
</ol>
Other mitigating research has also shown
<ol start=9>
<li>Semi-classical divergence only occurs in warp drives with only one spatial dimensions. Those with two or more spatial dimensions (as in our universe) only contain points where divergences occur, and these divergences can be managed. In fact, their effects can be reduced by making the warp shape more elongated<ref>Carlos Barceló, Valentin Boyanov, Luis J. Garay, Eduardo Martín-Martínez,
and Jose M. Sánchez Velázquez, "Warp drive aerodynamics", JHEP08 (2022) 288 https://doi.org/10.1007/JHEP08(2022)288 , arXiv:2207.06458 [gr-qc] https://doi.org/10.48550/arXiv.2207.06458</ref>.
</ol>

=== Alcubierre warp interactions with light and matter ===

Space is not entirely empty. It is filled with a diffuse plasma in the form of the interstellar medium, as well as cosmic radiation, light from stars, and cosmic microwave background radiation. A warp drive propagating through space will encounter this stuff. What happens when this matter and radiation have a warp bubble pass across them?

The first analysis of matter encountering a warp bubble was performed by Pfenning and Ford<ref name="PfenningFord_2001"></ref>. They looked at the warp bubble interaction with a massive object at rest with the frame of reference of the warp drive (keep in mind that the rest frame is the same inside and outside the warp bubble; in this rest frame objects inside the bubble have no momentum even though they are, in some sense, changing location rapidly with time). When the warp bubble passes, the object experiences an acceleration in the direction of the warp bubble motion. When the warp shell passes and the object is inside the bubble, it will be moving with approximately the speed of the bubble. Pfenning and Ford analyzed this problem with a continuous distribution of shell energy that, strictly speaking, never falls to zero except at the bubble center and at spatial infinity, so unless the object passes through the center in Pfenning and Ford's description it will never quite get up to the bubble's speed. In this case, the object will move almost as fast as the object but will pass through the bubble in a finite time, after which it will again be at rest with respect to the reference frame but displaced along the direction of the warp bubble motion by some distance. With this description of the warp bubble, a spacecraft of finite size will always be moving a little slower than the warp bubble and would have to use rockets to keep up with it. in addition, the spacecraft would experience tidal forces that would cause stress on the spacecraft's structure.

Pfenning and Ford also examined cases where the shell is of a finite (possibly infinitesimal) width and falls to zero both inside and outside the bubble. In this case any matter encountering the bubble would be collected at the bow of the bubble and thereafter move along with it.

McMonigal et al. analyzed the situation for both massive particles and light moving along the axis of travel of the bubble<ref name="McMonigalEtAl_2012">B. McMonigal, G. F. Lewis, and P. O'Byrne, "Alcubierre warp drive: On the matter of matter". Physical Review D. 85 (6) 064024 (20 March 2012). arXiv:[https://arxiv.org/abs/1202.5708 1202.5708]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2012PhRvD..85f4024M 2012PhRvD..85f4024M]. doi:[https://doi.org/10.1103%2FPhysRevD.85.064024 10.1103/PhysRevD.85.064024]. S2CID [https://api.semanticscholar.org/CorpusID:3993148 3993148].</ref>. They found that light moving opposite the warp direction passed through the bubble without incident, being only somewhat delayed by passing through the bubble. A warp bubble that is warping at sub-luminal speeds can have light catch up from behind it. This light is able to pass through the bubble, and is somewhat advanced in its path by the speed of the bubble. For super-luminal warp bubbles, however, the situation is different. The bubble will catch up to light moving it its own direction that is originally in front of it. This light cannot escape forward, the bubble being too fast. Nor can it escape backward, as the light is propagating forward and the interior of the bubble is at rest. Thus, the light gets caught at the bow of the warp bubble, unable to escape for so long as the warp bubble is active. This light is strongly blue shifted to extremely energetic x-rays and gamma rays. The space behind the bubble is swept clear of forward-moving light.

The situation for matter over-run by a super-luminal warp bubble is similar. Matter moving backward passes through the bubble, being only swept a ways along the bubble's path as sign of its passing but otherwise continuing on their way. Matter at rest behaves the same way as Pfenning and Ford discovered. Meanwhile, matter moving in the same direction of the warp is overtaken and collects at the bow of the bubble, unable to leave for so long as the bubble is warping. This matter "surfing" on the bubble bow is highly accelerated to relativistic speeds, experiencing extreme time dilation.

For a sub-luminal bubble, there is a speed faster than the warp speed where particles coming up from behind overtake the bubble and pass through, out the front. Particles below this speed but still faster than the warp speed still overtake the bubble, but then bounce backwards out of the bubble from behind. They are still moving forward, but are now moving more slowly than the warp speed. Particles moving forward but slower than the bubble will be overtaken and then bounced out the front with a speed higher than the warp speed. Those particles moving backward will pass through the bubble and exit with their initial speed.

The spacecraft inside the bubble will observe that light moving backward compared to the warp direction is blue-shifted, and matter moving backward passes through with increased energy. Meanwhile, a sub-luminal bubble will see light that was moving forward as red-shifted and matter catching up to the bubble from behind will pass by with reduced energy (superluminal bubbles, of course, do not have any matter or radiation catching up to them from behind).

When a superluminal warp bubble that has been collecting matter and radiation for the duration of its journey and blue-shifting it to much higher energies is turned off, all that matter and energy is released as a blast of radiation in the direction the bubble was warping.

McMonigal et al. conclude by noting that any spacecraft in the warp bubble would need shielding to protect against the blue-shifted radiation and matter of increased energy. In addition the destination would be "blasted into oblivion" by the release of matter and radiation that had been caught in the bubble during the trip.

Clark et al.<ref>Chad Clark, William A. Hiscock, and Shane L. Larson, "Null geodesics in the Alcubierre warp drive spacetime: the view from the bridge", arXiv:gr-qc/9907019 (2018) https://doi.org/10.48550/arXiv.gr-qc/9907019</ref> examine what it would look like from within an operating warp drive. They find that light moving at 90 degrees to the direction of motion will look like it is still coming from 90 degrees. However, light from the front hemisphere is squished progressively toward the forward point the faster you go. For a warp bubble moving faster than the speed of light, there is a cone behind the warp bubble where no light can reach it. However, you don't see a black spot behind you – light from the regions in the backward hemisphere from outside that cone that can reach you is distorted toward the direction behind you to fill in that spot. Light from the forward direction is blue-shifted and light from behind is red-shifted. At speeds of around 200 times light speed, the cosmic microwave background would be blue shifted to energies similar to that of the solar photosphere, heating anything in the warp bubble as if they were close to the sun.

== Other warp geometries ==

The Alcubierre warp drive is the first and best known of the warp drives. It has been around long enough that there have been many studies done on it. The warp geometry is particularly simple, making analysis easier. But in the time that the Alcubierre warp was proposed, many other researchers have come up with other warp designs, and even generalized the idea to include a wide array of additional possibilities.

Most of the general statements about the challenges of the Alcubierre drive hold for these other drives. Many of the details about interactions with matter are also likely to hold, although these other styles of warp drive have not yet been subject to such rigorous investigation.

=== Van Den Broeck warp drive ===

If an Alcubierre warp bubble a hundred meters across requires a magnitude of energy greater than the entire energy in the observable universe, one option to reduce the magnitude of energy used is to make the warp bubble smaller. Much smaller. Pfenning and Ford<ref name="PfenningFord_2001"></ref> suggested making the warp bubble smaller than an atom. Of course, that brings up the problem of how to stuff a spacecraft in there. Van Den Broeck proposed a solution<ref name="VanDenBroeck_1999"></ref>: make the warp bubble only about the size of an atomic nucleus but expand the space inside the bubble enormously so that a spacecraft could fit in. This is somewhat like a nuclear sized wormhole that leads to a pocket universe that holds the spacecraft. Unfortunately, the metric doesn't fit neatly into any [[Wormholes#Three_dimensions|embedding diagram]] that I can figure out, but the math works out so that spatial coordinates inside the bubble are expanded by a factor of 1017 and a spacecraft would have a few hundred meters of bubble interior to putz around in. This trick manages to reduce the magnitude of energy to only about the mass energy of a few stars similar to our own. Other than that, it is otherwise a normal Alcubierre drive.

Further work<ref>S. Krasnikov, "The quantum inequalities do not forbid spacetime shortcuts", Physical Review D. 67 (10) 104013. https://doi.org/10.1103%2FPhysRevD.67.104013, arXiv:gr-qc/0207057 (2003) https://doi.org/10.48550/arXiv.gr-qc/0207057 </ref> has managed to use Van Den Broeck's trick to reduce the magnitude of the negative energy to only about a milligram.

=== Natário warp drive ===

The Alcubierre drive is not the only way to construct a warp drive. José Natário<ref name="Natario"></ref> showed that it was just one example of an entire class of warp drives. These are space-time geometries with a smooth and localized shift vector field and a unit lapse function.

Natário then went on to find in that class a set of warp drives that do not have any expansion or contraction of space at all. Instead, space encountering the front boundary of the warp bubble instead "slides" around the outside of the bubble until it gets to the corresponding place on the back and is left behind there. Another way to think of it is that space entering the bubble shell is compressed in the radial direction of the shell but is simultaneously expanded in the tangential direction so that there is no net change in volume. When it gets to the back, the opposite occurs. In the language of mathematics, the shift vector field is divergenceless.

<table align=center>
<tr><td width=360>[[File:Natario_drive_flow_of_space.png]]
<td width=445>[[File:Natario_warp_drive_horizons.png]]
<tr><td style="vertical-align: top;">A representation of the motion of space in a zero-expansion Natário drive with a vanishingly thin warp shell around the bubble.
<td>The horizons of the generalized Natário class of warp drives (including the Alcubierre warp drive and all fell-Heisenberg drives) at superluminal speed in the direction of the green arrow. No event inside the bubble can affect events outside the bubble in front of the blue line. No events outside the bubble behind the red line can affect events inside the bubble. The lines have an angle α with respect to the direction of motion with sin(α) = 1/vs for vs the speed of the bubble relative to light speed. This is analogous to the Mach cone of supersonic objects.
</table>

NOTE: There are two different things that can be meant by the term "Natário warp drive". The generalized Natário warp drive is the general theory of all warp drives that includes within it the Alcubierre, Fell-Heisenberg, and Lentz warp drives, among infinitely many others. The zero-expansion Natário warp drive is the specific implementation of the generalized Natário warp drive that has a divergenceless shift vector field.

The generalized class of warp drives developed by Natário (including both the Alcubierre and this zero-expansion warp drive) are shown to always have regions in the warp shell where the energy density is negative to at least some observers. You can't get away from it – to warp, you need negative energy density.

The Natário class of warp drives blueshift light coming in from the front and redshift light catching up from the back. For the case of a super-luminal warp bubble, there will of course be a horizon at the back of the bubble and light will not be able to get through from behind. For a bubble speed vs relative to the speed of light, light coming from straight ahead will be blueshifted up in frequency and photon energy by a factor of 1 + vs. For sub-luminal travel, light coming from behind will be redshifted by a factor 1 - vs. In general for n as the unit direction along which the light is propagating, the blueshift factor will be 1 + vs ⋅ n. If the warp shell is infinitesimally thin, light from outside that reaches the center of the bubble will not be distorted in direction although it will be frequency shifted, but observers still inside the bubble but displaced from the center will see the field of view distorted as well as frequency shifted.

Most of the difficulties that apply to the Alcubierre warp drive also apply to the generalized Natário warp drive (and thus to the zero-expansion Natário drive and Fell-Heisenberg drive as well). In particular, quantum energy inequalities, high magnitudes of the energy required, formation of time machines, instabilities related to the stress-energy making up the warp shell being unable to keep up with the bubble, the chicken-and-egg problem of needing FTL to achieve FTL, the formation of horizons making warp bubbles uncontrollable, and Hawking radiation at the horizons, are all still concerns for generic warp drives.

== Fell-Heisenberg warp drives ==

S. D. B. Fell and L. Heisenberg investigated what sort of warp structure is necessary to produce a warp drive with positive energy density everywhere<ref name="Fell-Heisenberg">S. D. B. Fell and L> Heisenberg, "Positive energy warp drive from hidden geometric structures", Classical and Quantum Gravity 38 155020 (2021) https://doi.org/10.1088/1361-6382/ac0e47 https://arxiv.org/abs/2104.06488</ref>. They found that warp drives of the generic Natário form with irrotational shift vectors had a simple expression for the energy density and vanishing momentum (irrotational means that the shift vector can be described as the gradient of a scalar potential field or, equivalently, that the curl of the shift vector field vanishes everywhere). Fell and Heisenberg proposed several potential fields that either had positive energy density everywhere except at discontinuities, or that were continuous but had regions that violated the weak energy condition (i. e. negative energy densities in some frames of reference). One specific example they provided of a warp field had an energy that is about 10,000 times less than the mass-energy of our sun. Or only about half the mass-energy of Jupiter. This is a significant improvement over many other proposed drives.

Despite the introduction discussing warp drive configurations that satisfy the various energy conditions, the configurations described in the paper are shown to locally violate the weak and strong energy conditions. Nonetheless, the energy density is still mostly positive. Santiago, Schuster, and Visser's<ref name="Santiago Schuster Visser 2022">J. Santiago, S. Schuster, and M. Visser, "Generic warp drives violate the null energy condition", Physical Review D 105, 064038 (2022) https://doi.org/10.1103/PhysRevD.105.064038</ref> work shows that the various energy conditions must still be violated by this warp drive; to not violate these, energy density must be positive in all reference frames not just those of the co-moving observer.

Any vector field can be described by its divergence, its curl, and a constant. The constant part of a warp drive's shift vector can be discarded because then the warp drive would not be localized. It is interesting, therefore, that the Natário zero-expansion drive has zero divergence and can be described only by its curl, while the Fell-Heisenberg drive has zero curl and can be described only by its divergence. These two classes of drives thus rely on opposite choices of setting one of the two different parts describing a vector field to zero. As might be expected, the Fell-Heisenberg drive is not zero expansion – space-time will change in volume as it passes through the warp bubble (although after going all the way through it will return to its original volume). However, objects in space-time will retain their original orientation as they pass through the warp bubble.

Any vector field with zero curl can be represented as the gradient of a suitable scalar function. Thus, the Fell-Heisenberg drive can be completely described by just scalar field (one number at all points of space and time) rather than a vector field.

Previous examples of warp drives, such as the Alcubierre and zero-expansion Natário drives, tend toward a flat space-time as you go far enough away faster than would occur if there were any mass present in the warp drive. That is, the ADM mass of such drives is zero. However, one of the Fell-Heisenberg examples described a spacetime where the geometry approaches that of what you would get for a massive object at the warp location, giving one of the first examples of a warp drive with mass.

In the Fell-Heisenberg drive the momentum vanishes everywhere. Yet the energy occupies regions where the shift vector is varying rapidly. The lack of momentum means that the energy will not move to keep up with the differential expansion and movement of the spacetime. As a consequence, at later times the energy will have a different distribution than what is necessary to maintain the given warp configuration. The exact time evolution is not solved but this likely leads to collapse of warp bubble. Note that this is not a problem unique to the Fell-Heisenberg drive. Assuming a geometry that describes a warp produces a specific distribution of the stress-energy tensor that satisfies Einstein's equation. But there is no guarantee that this stress energy tensor satisfies the continuity equation, which describes how the distribution of matter and momentum change with time. The issue of stability of any warp drive is not well understood, but most warp drive geometries will be unstable in the sense that their matter that generates the warp fields will not be able to keep up with the motion of the warp. The Fell-Heisenberg drive is unique only in that, with vanishing momentum, there is no configuration that allows stable propagation.

(n. b. The Heisenberg here is Lavinia Heisenberg, not the Werner Heisenberg of quantum physics and uncertainty principle fame.)

== Lentz warp drive ==

E. Lentz<ref name="Lentz">E. W. Lentz, "Breaking the warp barrier: hyper-fast solitons in Einstein–Maxwell-plasma theory", Classical and Quantum Gravity. 38 075015 (2021). arXiv:[https://arxiv.org/abs/2006.07125 2006.07125]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2021CQGra..38g5015L 2021CQGra..38g5015L]. doi:[https://doi.org/10.1088%2F1361-6382%2Fabe692 10.1088/1361-6382/abe692]. ISSN [https://search.worldcat.org/issn/0264-9381 0264-9381]. S2CID [https://api.semanticscholar.org/CorpusID:219635854 219635854].</ref> investigated a particular configuration of the Fell-Heisenberg class of warp drives, where the scalar field forms a selection of rhomboid regions, resulting in an energy density that is everywhere positive and a shift vector with vanishing tidal force at the center for a comfortable ride.

However, Santiago, Schuster, and Visser<ref name="Santiago Schuster Visser 2022"></ref> dispute claims that the Lentz drive satisfies the energy conditions, noting that everywhere positive energy density in one frame of reference is insufficient to establish that the energy density is positive in all reference frames; knowledge of the Cauchy stress tensor is also needed. They show that any generic warp drive will violate the strong energy condition, null energy condition, and weak energy condition.

== Conservation laws ==

(Discussion of asymptotic flatness: what is it? Where does it apply?)

(The "big four" conservation laws and how they relate to asymptotic flatness.)

(ADM mass -> 0; conservation of energy/mass issues.)<ref>Sebastian Schuster, Jessica Santiago,and Matt Visser, "ADM mass in warp drive spacetimes" General Relativity and Gravitation (2023) 55:14 https://doi.org/10.1007/s10714-022-03061-9</ref>

(Discuss issues of conservation of angular momentum.)

(If angular momentum conservation is ignored, discuss how momentum & energy are affected by outside forces. In a gravitational field equivalent to inertial frame ... like being in an accelerated elevator; will acquire momentum buildup while staying "at rest".)

== Warp Railroads ==

Van Den Broeck<ref name="VanDenBroeck_2000"></ref> suggested a warp railroad,

You can set up devices ahead of time along the path of the bubble that produce the negative energy regions for the warp bubble at the appropriate time without needing the negative energy to ever move with the bubble at all. At least for bubbles on a predictable schedule and route.

helps with:

-stability (elaborate)

-conservation laws (elaborate)

Mention Krasnikov<ref name="Krasnikov 1998">S. V. Krasnikov, "Hyperfast travel in general relativity" Phys. Rev. D 57 4760</ref>

Hyperwave<ref name="Pieri 2024">Lorenzo Pieri, "Hyperwave: hyper-Fast Communication within General Relativity", arXiv:2311.12069 [gr-qc] https://doi.org/10.48550/arXiv.2311.12069</ref> makes use of this warp railroad idea.

== Quantum effects ==

(quantum stuff here<ref name="Hiscock1997">W. A. Hiscock, "Quantum effects in the Alcubierre warp drive spacetime", Classical and Quantum Gravity 14 L183 https://doi.org/10.1088/0264-9381/14/11/002 https://arxiv.org/abs/gr-qc/9707024</ref><ref name="Finazzi et al 2009">S. Finazzi, S. Liberati, C. Barceló, "Semiclassical instability of dynamical warp drives", Physical Review D 79, 124017 (2009)https://doi.org/10.1103/PhysRevD.79.124017 https://arxiv.org/abs/0904.0141</ref>)

From<ref name="Pieri 2024"></ref>
<blockquote>
It should be emphasised that Quantum Inequalities have not been proven for realistic interacting quantum fields, where the they are expected to be weaker if they exist [Cadamuro, 2019], and they often require Minkowskian conditions at infinity, a condition which is not appropriate for fields inside bounded regions, as in a Casimir devices.
</blockquote>

== Warp drives and black holes ==

What happens if you drive your warp drive into a black hole?

The pop-sci reason that nothing can escape a black hole is that at the event horizon the escape velocity is faster than light and nothing can go faster than light. A warp drive can go faster than light.

The more detailed reason from general relativity is that at the event horizon space and time are rotated sufficiently that "inward" becomes "forward in time." You can no more go farther away from the singularity in a black hole once you are inside the event horizon than you can go backward in time. Any super-luminal travel can go backward in time, and warp drives can engage in super-luminal travel.

The Hawking area theorem that says that black holes can only grow, never shrink relies on the condition that the energy is everywhere positive. Warp drives have negative energy density.

So in principle, there's nothing that prevents a warp drive from taking a dip into a black hole and coming back out again.

Garattini and Zatrimaylov looked into the problem of a warp drive near (and in) a black hole<ref name="Garattini and Zatrimaylov">R. Garattini and K. Zatrimaylov, "Black holes, warp drives, and energy conditions", Physics Letters B 856 138910 (2024) https://doi.org/10.1016/j.physletb.2024.138910</ref>. The found that the black hole reduces the negative energy requirement for the warp bubble to move inward toward the black hole singularity, but increases the negative energy needed to move away from the center of the black hole. In addition, a warp drive parked at the event horizon would allow light from inside the horizon to pass through the bubble and escape back outside.

== Credit ==
Author: Luke Campbell

== References ==

[[Category:Physics]][[Category:Infrastructure]][[Category:Physics & Engineering‏‎]][[Category:Physics & Math & Engineering]][[Category:Transportation & Infrastructure‏‎]][[Category:Metric Engineering]]

Warp Drives

2026-07-30T02:43:27Z

Lwcamp: /* Van Den Broeck warp drive */

{{PageConstructionNotice}}

Science fiction often features spacecraft that can seemingly move across space and get between the place of departure and the destination much faster than light could have done. This appears to contradict the theory of relativity, which predicts unequivocally that nothing can move through space faster than light. Because relativity has been incredibly successful at describing nature, with its many other predictions regularly being confirmed to extraordinary accuracy and within the bounds of uncertainty of all the experiments that tested them, it gives confidence that relativity is a correct description of reality. Which seems to rather throw a wet towel on our hopes for rapid travel between stars.

However, while relativity does not allow things to move through space faster than light, it places no such restrictions on how fast space-time itself can expand, contract, or move around. This leads to the idea of a warp drive – the spacecraft remains stationary within a region of highly curved space-time, and that region moves at super-luminal speeds rather than the spacecraft.

== The Alcubierre warp drive ==

The first warp drive geometry that satisfied the Einstein field equations of relativity was proposed by Miguel Alcubierre<ref>M. Alcubierre, "The warp drive: hyper-fast travel within general relativity." Classical and Quantum Gravity. 11 (5): L73–L77 (1994). [https://arxiv.org/abs/gr-qc/0009013 arXiv:gr-qc/0009013]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1994CQGra..11L..73A 1994CQGra..11L..73A]. doi:[https://doi.org/10.1088%2F0264-9381%2F11%2F5%2F001 10.1088/0264-9381/11/5/001]. S2CID [https://api.semanticscholar.org/CorpusID:4797900 4797900].</ref>. In this geometry, a sphere of space-time moves at an arbitrary speed (potentially but not necessarily a speed much faster than light). Objects within the sphere are moved along with the sphere; an object at rest within the sphere would be moved along with the sphere indefinitely. Space is expanding at the rear boundary and contracting at the front boundary in order to keep the sphere moving. In order to satisfy the Einstein field equations, the boundary of the sphere must have a negative energy density. The challenges of space-time geometries with negative energy densities are described in our page on [[Wormholes#Exotic_energy_conditions|wormholes]], for our purposes it is enough to note that negative energy densities can pose problems if not handled carefully, there may be limits on how much negative energy you can have without nearby positive energy density, and it may not be possible to get enough negative energy to support a warp drive; although none of this is ruled out by physics – yet!

<table align=center>
<tr><td width=650>[[File:Energy_magnitude_of_Alcubierre_drive.png]]
<td width=650>[[File:Expansion_of_Alcubierre_drive.png]]
<tr><td>The magnitude of the energy of the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. (The energy density is negative everywhere in the shell, here only the magnitude is shown to aid visual comprehension.)
<td>The expansion of space in the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. Negative expansion means that space is contracting in that direction, positive expansion that it is expanding.
</table>

NOTE: What the? The energy distribution is symmetric! How does the drive know which way to go? There must be significant contributions of other components of the stress-energy tensor, and those have got to be asymmetric along the forward/backward axis. Check this when I get time.

(And what's with the ring in all the artwork, anyway?)

=== So what is going on? ===

What is it that makes a warp drive a warp drive, and how does it work?

A warp drive is a concept that comes out of the science of general relativity. General relativity collects space and time together into a single "structure" called spacetime. This structure is curved, and its curvature is determined by the material that is in it. Absent forces acting between matter, particles of matter move through spacetime on the shortest possible paths in this curved environment.
<blockquote>
"Spacetime tells matter how to move; matter tells spacetime how to curve." - John Archibald Wheeler
</blockquote>
All of this is encompassed in the Einstein field equation:
<div align=center>
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 8πG
<td rowspan="2" nowrap="nowrap" align="right"> T = R -  
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right"> R g
<tr>
<td style="border-top:solid 1px black;"> c4
<td style="border-top:solid 1px black;"> 2
</table>
</div>
Unfortunately, by convention, both the symbols G and R get used for a lot of different things. Here, symbols in bold are second rank tensors – constructs of arrays of 4×4 numbers where 4 is necessary because that is the number of dimensions of spacetime. Conventionally, these numbers are indexed by a value going from 0 to 3, with 0 being the part corresponding to the coordinate of time and 1 to 3 being the Cartesian coordinates that you might know as x, y, and z.

T is the stress-energy tensor: the (0,0) component represents the energy density at a given point; the (0,i) and (i, 0) components (with i going from 1 to 3) represent the momentum density; and the (i,j) components represent the stresses present, such as pressure or tension or shear.

The other term, R - R g/2, involves a partial description of the curvature. A full description of the curvature requires a construct of 4×4×4×4 components; too much for the stress energy tensor to fully constrain. Which is okay, because this allows curvature to exist away from where matter is, allowing gravity to have long range effects across empty space as well as for phenomena like gravitational waves. But it is enough that a given distribution of matter T gives enough constraints (along with specifying how the geometry behaves far away) that you can generally figure out what the actual gravity is.

This is the usual story that we are told. But there is an alternate way of looking at the same geometry such that the math works out the same, but it is described conceptually in a very different way. This divides spacetime up into "layers", called foliations, that represent slices of space and going from one foliation to the next is increasing in time. This is useful for computation because you can assign coordinates to your system. There are two additional constructs – the lapse function and the shift vector field. Starting at an event on one foliation, the lapse function at that position tells you that after a certain "proper time" (time perceived by an observer at that event) the time coordinate of the observer will be the proper time times the lapse function at that event. And the new spatial coordinates of the observer will be the proper time times the shift vector at that event. (Strictly speaking, this is only exactly true in the limit of the proper time approaching zero or when the lapse function and shift vectors are constant. But hopefully this gives a good enough insight to what these entities are so that we can go on to describe warp drives.)

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Lapse_shift.png]]
<tr><td>A folliation of space-time into slices of constant time coordinates (grey). At an event (black circle), the lapse function (yellow) shows how far into the future the point at that event will be projected in a unit amount of proper time experienced by an observer initially at the event. The shift vector (red) shows how that point changes coordinates after the unit amount of proper time. The cyan vector shows the new coordinates the point is projected onto. The lower black surface indicates a foliation of spacetime at the same time coordinate; after unit proper time, the events on the lower foliation get projected to the curved upper black foliation according to the shift vectors and lapse functions at each of their respective points.
</table>
</table>

Warp drives are generally restricted to spacetime geometries with a constant lapse function - all observers go forward by the same time coordinate after experiencing the same amount of proper time. Thus, all warp drives have no time dilation.
Far from the warp bubble, the shift vector vanishes – each spatial coordinate is mapped to the same spatial coordinate after any amount of proper time so long as the warp drive is not near. But the inside of the warp bubble is an extended region where the shift vector field is non-zero and constant. In this bubble, all spatial coordinates are mapped to a coordinate that shifts linearly in time with the proper time of the observer. Astute readers will see that this linear change in position with time is very similar to a velocity. This region of constant shift vector then moves with the shift vector velocity to keep stuff originally in the bubble so that it stays in the bubble. In between the inside of the bubble and the far away regions with zero shift vector, the shift vector field falls away to match one region to the other.

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Warp_shift.png]]
<tr><td>The shift vector of a one-dimensional slice through warp geometry, in red. At the edges there is no shift. At the center (inside the warp bubble) there is a region of uniform shift so that at a later time the inside of the warp bubble is rigidly translated to different spatial coordinates (in blue). The two foliations remain the same distance apart, because warp drives maintain a unit lapse function.
</table>
</table>

Traditionally, scientists would start with an interesting distribution of matter given by T and then find out what kind of curvature it created. This works well for planets and stars and black holes and cosmic strings and other things which are observed in the universe and you want to see what sort of effect they had. But starting in the 1980's, some physicists started looking at Einstein's field equations the other way around. They start with a desired geometry, find the curvature part of the field equation R - R g/2, and use that to find out what T has to be in order to get that curvature. This was originally done for [[Wormholes|wormholes]], but in 1994 Miguel Alcubierre used this same trick to figure out what could create a region of constant shift vector that moved with the shift vector - the warp drive. With this demand for a warp geometry, the required distribution of matter and energy and momentum and stress can be determined.

The original Alcubierre shift vector N→ produces a spherical warp bubble with an axi-symmetric distribution of, well, everything along the direction of motion with warp velocity vector v→.
<div align="center">
N→ = v→ f(rs)
</div>
Here, rs is the distance from the center of the warp bubble, and f(rs) can be any function that is 1 near rs = 0 and is 0 for rs well beyond the radius of the warp bubble.

=== Challenges and possible resolutions ===

Almost as soon as Alcubierre proposed his warp drive, others began picking it apart.
<ol>
<li>Pfenning and Ford<ref name="PfenningFord_2001">M. J. Pfenning and L. H. Ford, "The unphysical nature of 'Warp Drive'", Classical and Quantum Gravity. 14 (7): 1743–1751 (1997). arXiv:[https://arxiv.org/abs/gr-qc/9702026 gr-qc/9702026]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1997CQGra..14.1743P 1997CQGra..14.1743P]. doi:[https://doi.org/10.1088%2F0264-9381%2F14%2F7%2F011 10.1088/0264-9381/14/7/011]. S2CID [https://api.semanticscholar.org/CorpusID:15279207 15279207].</ref> found that if the negative energy density wall around the bubble obeys [[Wormholes#Quantum_energy_inequalities|quantum energy inequalities]], then for warp speeds of around light speed the shell thickness would be on the order of a hundred Planck lengths; a distance far smaller than any other known physical phenomenon or object.
<li>In the same work, Pfenning and Ford also found that for a bubble a hundred meters in radius, a warp speed of around the speed of light, and the required thickness of about 100 Planck lengths the energy in the bubble shell would be E ≈ -1063 kg c2. The magnitude of this latter value is ten orders of magnitude larger than the energy of the entire visible universe. They do note, however, that if quantum energy inequalities can be ignored then a warp drive with a hundred meters radius but a shell thickness of one meter would "only" have an energy magnitude of about a quarter solar mass.
<li>As with any method of faster than light travel, the warp drive could be used to make a time machine.
<li>Perhaps most seriously, if the warp drive is going faster than light speed, the negative energy regions on the outside of the shell won't be in the warp parts of the bubble. They will have to be moving through space at faster than light speed if the bubble is to maintain its integrity, which is the very problem that the warp drive was designed to avoid. The outside of the warp bubble shell that maintains the warp bubble would fall away and the warp bubble would quickly erode to sub-luminal speeds.<ref name="Van Den Broeck 2000">Chris Van Den Broeck, "On the (im)possibility of warp bubbles", arXiv:gr-qc/9906050 https://doi.org/10.48550/arXiv.gr-qc/9906050</ref>
<li>A more general problem that includes the previous one is that the geometry of the warp drive determines the distribution of energy, momentum, and stress (pressure, shear, and tension) by Einstein's field equation, which states that the stress energy tensor is directly proportional to a reduced order curvature tensor of space-time. However, the stress-energy tensor must also obey the continuity equation – basically, for the energy to change in a given volume of space that energy must come in or go out of the boundary of that volume and similarly for the momentum. So a warp drive can have the warp condition for one moment in time. But after that, the stress-energy that maintains the warp will start to move according to the forces on it and according to the natural paths that it takes through the warped space-time. There is no guarantee that the configuration of the stress-energy at later times will be sufficient to maintain the warp.
<blockquote>
From the perspective of simulating the warp drive dynamically, the key challenge is stability. One can straightforwardly generate the initial metric that describes the Alcubierre metric and infer the instantaneous matter configuration that must support it using the Einstein Equation, but a time-domain evolution of the coupled matter-gravity system necessitates that we specify the equation of state of the matter, which in turn determines how it evolves. In a perfect fluid, this involves specifying the relation between the fluid pressure and density, but in a more general case, the full relation between the
stress-energy components must be given. There is (to our knowledge) no known equation of state that would maintain the warp drive metric in a stable configuration over time - therefore, whilst one can require that initially, the warp bubble is constant, it will quickly evolve away from that state and, in most cases, the warp fluid and spacetime deformations will disperse or collapse into a central point.
 
– Clough, Dietrich, and Khan<ref>Katy Clough, Tim Dietrich, and Sebastian Khan, "What no one has seen before: gravitational waveforms from warp drive collapse", arXiv:2406.02466 [gr-qc] https://doi.org/10.48550/arXiv.2406.02466</ref>
</blockquote>
<li>In order to get the matter (or more generally, stress-energy) that forms the warp shell up to speeds that could allow the shell to exist, you need to be able to, well, accelerate that matter to warp speeds. Which gives a chicken-and-egg problem - in order to achieve superluminal travel you need to be able to get matter or energy moving superluminally in the first place<ref name="Bobrick and Martire">Alexey Bobrick and Gianni Martire, "Introducing Physical Warp Drives", arXiv:2102.06824 [gr-qc] https://doi.org/10.48550/arXiv.2102.06824</ref><ref>D. H. Coule, "No warp drive", Class. Quantum Grav. 15 (1998) 2523–2527. https://doi.org/10.1088/0264-9381/15/8/026</ref>, which rather makes warp drives unnecessary from the get-go.
<li>A warp bubble moving faster than light will not be able to get any signals in front of it as it will overtake its own signal. In fact, anyone in the interior of the bubble would not be able to send signals to the front surface of the bubble. In science-speak, a horizon forms at the front of the bubble<ref name="Natario">José Natário, "Warp drive with zero expansion", Classical and Quantum Gravity. 19 (6): 1157–1166 (2002). arXiv:[https://arxiv.org/abs/gr-qc/0110086 gr-qc/0110086]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2002CQGra..19.1157N 2002CQGra..19.1157N]. doi:[https://doi.org/10.1088%2F0264-9381%2F19%2F6%2F308 10.1088/0264-9381/19/6/308]. S2CID [https://api.semanticscholar.org/CorpusID:15859984 15859984].</ref>. As a consequence, the bubble cannot be controlled, steered, nor stopped from within the bubble itself.
<li>A Horizon will naturally emit Hawking radiation. It is estimated that the radiation will raise the temperature of the interior of the bubble to approximately 1032 K<ref name="Finazzi 2009">Stefano Finazzi, Stefano Liberati, and Carlos Barceló, "Semiclassical instability of dynamical warp drives", Phys. Rev. D 79, 124017 (12 June, 2009) https://doi.org/10.1103/PhysRevD.79.124017 , arXiv:0904.0141 [gr-qc] https://doi.org/10.48550/arXiv.0904.0141</ref> (at this level of accuracy, the result is the same if you replace temperatures in K by the same number in °C or °F). This temperature, however, may be lower if quantum inequalities do not have to be fulfilled.
<li>Semi-classical instabilities at the horizon are expected to lead to a divergence of the stress-energy tensor at the boundary in a short period of time, thus leading to the destruction of the warp bubble<ref name="Finazzi 2009"></ref>.
</ol>
Helpfully, Van Den Broeck<ref name="VanDenBroeck_2000">C. Van Den Broeck, "Alcubierre’s Warp Drive: Problems and Prospects" AIP Conference Proceedings. 504: 1105–1110 (2000). Bibcode:[https://ui.adsabs.harvard.edu/abs/2000AIPC..504.1105V 2000AIPC..504.1105V]. doi:[https://doi.org/10.1063%2F1.1290913 10.1063/1.1290913].</ref> suggested several solutions – or at least mitigations – for some of these problems
<ol>
<li>Quantum inequalities had not been shown to be true in general for highly curved space-times.
<li>A proposed warp drive geometry<ref name="VanDenBroeck_1999">C. Van Den Broeck, "A 'warp drive' with more reasonable total energy requirements". Classical and Quantum Gravity. 16 (12): 3973–3979 (1999). arXiv:[https://arxiv.org/abs/gr-qc/9905084 gr-qc/9905084]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1999CQGra..16.3973V 1999CQGra..16.3973V]. doi:[https://doi.org/10.1088%2F0264-9381%2F16%2F12%2F314 10.1088/0264-9381/16/12/314]. S2CID [https://api.semanticscholar.org/CorpusID:15466313 15466313]. </ref> (see the van Den Broeck drive, below) would be able to minimize the magnitude of the energy required to about that of the mass of our sun. While still large, it is not unphysically large.
<li>Time travel is always going to be a worry with faster than light travel. There's no neat solution to this one.
<li>It may be possible for the negative energy outside of the superluminal region to compress into the superluminal region to form a shock in space-time. The front surface of the bubble would then be a singularity. It is not clear if such a sharp jump in physical properties is possible, but neither is sure that it is impossible, either. However, while this might help with the negative energy in front of the bubble getting swept up because it cannot move faster than light, it does little to help with the material at the back of the bubble on the outside getting left behind because it cannot keep up.
</ol>
Other mitigating research has also shown
<ol start=9>
<li>Semi-classical divergence only occurs in warp drives with only one spatial dimensions. Those with two or more spatial dimensions (as in our universe) only contain points where divergences occur, and these divergences can be managed. In fact, their effects can be reduced by making the warp shape more elongated<ref>Carlos Barceló, Valentin Boyanov, Luis J. Garay, Eduardo Martín-Martínez,
and Jose M. Sánchez Velázquez, "Warp drive aerodynamics", JHEP08 (2022) 288 https://doi.org/10.1007/JHEP08(2022)288 , arXiv:2207.06458 [gr-qc] https://doi.org/10.48550/arXiv.2207.06458</ref>.
</ol>

=== Alcubierre warp interactions with light and matter ===

Space is not entirely empty. It is filled with a diffuse plasma in the form of the interstellar medium, as well as cosmic radiation, light from stars, and cosmic microwave background radiation. A warp drive propagating through space will encounter this stuff. What happens when this matter and radiation have a warp bubble pass across them?

The first analysis of matter encountering a warp bubble was performed by Pfenning and Ford<ref name="PfenningFord_2001"></ref>. They looked at the warp bubble interaction with a massive object at rest with the frame of reference of the warp drive (keep in mind that the rest frame is the same inside and outside the warp bubble; in this rest frame objects inside the bubble have no momentum even though they are, in some sense, changing location rapidly with time). When the warp bubble passes, the object experiences an acceleration in the direction of the warp bubble motion. When the warp shell passes and the object is inside the bubble, it will be moving with approximately the speed of the bubble. Pfenning and Ford analyzed this problem with a continuous distribution of shell energy that, strictly speaking, never falls to zero except at the bubble center and at spatial infinity, so unless the object passes through the center in Pfenning and Ford's description it will never quite get up to the bubble's speed. In this case, the object will move almost as fast as the object but will pass through the bubble in a finite time, after which it will again be at rest with respect to the reference frame but displaced along the direction of the warp bubble motion by some distance. With this description of the warp bubble, a spacecraft of finite size will always be moving a little slower than the warp bubble and would have to use rockets to keep up with it. in addition, the spacecraft would experience tidal forces that would cause stress on the spacecraft's structure.

Pfenning and Ford also examined cases where the shell is of a finite (possibly infinitesimal) width and falls to zero both inside and outside the bubble. In this case any matter encountering the bubble would be collected at the bow of the bubble and thereafter move along with it.

McMonigal et al. analyzed the situation for both massive particles and light moving along the axis of travel of the bubble<ref name="McMonigalEtAl_2012">B. McMonigal, G. F. Lewis, and P. O'Byrne, "Alcubierre warp drive: On the matter of matter". Physical Review D. 85 (6) 064024 (20 March 2012). arXiv:[https://arxiv.org/abs/1202.5708 1202.5708]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2012PhRvD..85f4024M 2012PhRvD..85f4024M]. doi:[https://doi.org/10.1103%2FPhysRevD.85.064024 10.1103/PhysRevD.85.064024]. S2CID [https://api.semanticscholar.org/CorpusID:3993148 3993148].</ref>. They found that light moving opposite the warp direction passed through the bubble without incident, being only somewhat delayed by passing through the bubble. A warp bubble that is warping at sub-luminal speeds can have light catch up from behind it. This light is able to pass through the bubble, and is somewhat advanced in its path by the speed of the bubble. For super-luminal warp bubbles, however, the situation is different. The bubble will catch up to light moving it its own direction that is originally in front of it. This light cannot escape forward, the bubble being too fast. Nor can it escape backward, as the light is propagating forward and the interior of the bubble is at rest. Thus, the light gets caught at the bow of the warp bubble, unable to escape for so long as the warp bubble is active. This light is strongly blue shifted to extremely energetic x-rays and gamma rays. The space behind the bubble is swept clear of forward-moving light.

The situation for matter over-run by a super-luminal warp bubble is similar. Matter moving backward passes through the bubble, being only swept a ways along the bubble's path as sign of its passing but otherwise continuing on their way. Matter at rest behaves the same way as Pfenning and Ford discovered. Meanwhile, matter moving in the same direction of the warp is overtaken and collects at the bow of the bubble, unable to leave for so long as the bubble is warping. This matter "surfing" on the bubble bow is highly accelerated to relativistic speeds, experiencing extreme time dilation.

For a sub-luminal bubble, there is a speed faster than the warp speed where particles coming up from behind overtake the bubble and pass through, out the front. Particles below this speed but still faster than the warp speed still overtake the bubble, but then bounce backwards out of the bubble from behind. They are still moving forward, but are now moving more slowly than the warp speed. Particles moving forward but slower than the bubble will be overtaken and then bounced out the front with a speed higher than the warp speed. Those particles moving backward will pass through the bubble and exit with their initial speed.

The spacecraft inside the bubble will observe that light moving backward compared to the warp direction is blue-shifted, and matter moving backward passes through with increased energy. Meanwhile, a sub-luminal bubble will see light that was moving forward as red-shifted and matter catching up to the bubble from behind will pass by with reduced energy (superluminal bubbles, of course, do not have any matter or radiation catching up to them from behind).

When a superluminal warp bubble that has been collecting matter and radiation for the duration of its journey and blue-shifting it to much higher energies is turned off, all that matter and energy is released as a blast of radiation in the direction the bubble was warping.

McMonigal et al. conclude by noting that any spacecraft in the warp bubble would need shielding to protect against the blue-shifted radiation and matter of increased energy. In addition the destination would be "blasted into oblivion" by the release of matter and radiation that had been caught in the bubble during the trip.

Clark et al.<ref>Chad Clark, William A. Hiscock, and Shane L. Larson, "Null geodesics in the Alcubierre warp drive spacetime: the view from the bridge", arXiv:gr-qc/9907019 (2018) https://doi.org/10.48550/arXiv.gr-qc/9907019</ref> examine what it would look like from within an operating warp drive. They find that light moving at 90 degrees to the direction of motion will look like it is still coming from 90 degrees. However, light from the front hemisphere is squished progressively toward the forward point the faster you go. For a warp bubble moving faster than the speed of light, there is a cone behind the warp bubble where no light can reach it. However, you don't see a black spot behind you – light from the regions in the backward hemisphere from outside that cone that can reach you is distorted toward the direction behind you to fill in that spot. Light from the forward direction is blue-shifted and light from behind is red-shifted. At speeds of around 200 times light speed, the cosmic microwave background would be blue shifted to energies similar to that of the solar photosphere, heating anything in the warp bubble as if they were close to the sun.

== Other warp geometries ==

The Alcubierre warp drive is the first and best known of the warp drives. It has been around long enough that there have been many studies done on it. The warp geometry is particularly simple, making analysis easier. But in the time that the Alcubierre warp was proposed, many other researchers have come up with other warp designs, and even generalized the idea to include a wide array of additional possibilities.

Most of the general statements about the challenges of the Alcubierre drive hold for these other drives. Many of the details about interactions with matter are also likely to hold, although these other styles of warp drive have not yet been subject to such rigorous investigation.

=== Van Den Broeck warp drive ===

If an Alcubierre warp bubble a hundred meters across requires a magnitude of energy greater than the entire energy in the observable universe, one option to reduce the magnitude of energy used is to make the warp bubble smaller. Much smaller. Pfenning and Ford<ref name="PfenningFord_2001"></ref> suggested making the warp bubble smaller than an atom. Of course, that brings up the problem of how to stuff a spacecraft in there. Van Den Broeck proposed a solution<ref name="VanDenBroeck_1999"></ref>: make the warp bubble only about the size of an atomic nucleus but expand the space inside the bubble enormously so that a spacecraft could fit in. This is somewhat like a nuclear sized wormhole that leads to a pocket universe that holds the spacecraft. Unfortunately, the metric doesn't fit neatly into any [[Wormholes#Three_dimensions|embedding diagram]] that I can figure out, but the math works out so that spatial coordinates inside the bubble are expanded by a factor of 1017 and a spacecraft would have a few hundred meters of bubble interior to putz around in. This trick manages to reduce the magnitude of energy to only about the mass energy of a few stars similar to our own. Other than that, it is otherwise a normal Alcubierre drive.

Further work<ref>S. Krasnikov, "The quantum inequalities do not forbid spacetime shortcuts", Physical Review D. 67 (10) 104013. https://doi.org/10.1103%2FPhysRevD.67.104013, arXiv:gr-qc/0207057 (2003) https://doi.org/10.48550/arXiv.gr-qc/0207057 </ref> has managed to use Van Den Broeck's trick to reduce the magnitude of the negative energy to only about a milligram.

== Natário warp drive ==

The Alcubierre drive is not the only way to construct a warp drive. José Natário<ref name="Natario"></ref> showed that it was just one example of an entire class of warp drives. These are space-time geometries with a smooth and localized shift vector field and a unit lapse function.

Natário then went on to find in that class a set of warp drives that do not have any expansion or contraction of space at all. Instead, space encountering the front boundary of the warp bubble instead "slides" around the outside of the bubble until it gets to the corresponding place on the back and is left behind there. Another way to think of it is that space entering the bubble shell is compressed in the radial direction of the shell but is simultaneously expanded in the tangential direction so that there is no net change in volume. When it gets to the back, the opposite occurs. In the language of mathematics, the shift vector field is divergenceless.

<table align=center>
<tr><td width=360>[[File:Natario_drive_flow_of_space.png]]
<td width=445>[[File:Natario_warp_drive_horizons.png]]
<tr><td style="vertical-align: top;">A representation of the motion of space in a zero-expansion Natário drive with a vanishingly thin warp shell around the bubble.
<td>The horizons of the generalized Natário class of warp drives (including the Alcubierre warp drive and all fell-Heisenberg drives) at superluminal speed in the direction of the green arrow. No event inside the bubble can affect events outside the bubble in front of the blue line. No events outside the bubble behind the red line can affect events inside the bubble. The lines have an angle α with respect to the direction of motion with sin(α) = 1/vs for vs the speed of the bubble relative to light speed. This is analogous to the Mach cone of supersonic objects.
</table>

NOTE: There are two different things that can be meant by the term "Natário warp drive". The generalized Natário warp drive is the general theory of all warp drives that includes within it the Alcubierre, Fell-Heisenberg, and Lentz warp drives, among infinitely many others. The zero-expansion Natário warp drive is the specific implementation of the generalized Natário warp drive that has a divergenceless shift vector field.

The generalized class of warp drives developed by Natário (including both the Alcubierre and this zero-expansion warp drive) are shown to always have regions in the warp shell where the energy density is negative to at least some observers. You can't get away from it – to warp, you need negative energy density.

The Natário class of warp drives blueshift light coming in from the front and redshift light catching up from the back. For the case of a super-luminal warp bubble, there will of course be a horizon at the back of the bubble and light will not be able to get through from behind. For a bubble speed vs relative to the speed of light, light coming from straight ahead will be blueshifted up in frequency and photon energy by a factor of 1 + vs. For sub-luminal travel, light coming from behind will be redshifted by a factor 1 - vs. In general for n as the unit direction along which the light is propagating, the blueshift factor will be 1 + vs ⋅ n. If the warp shell is infinitesimally thin, light from outside that reaches the center of the bubble will not be distorted in direction although it will be frequency shifted, but observers still inside the bubble but displaced from the center will see the field of view distorted as well as frequency shifted.

Most of the difficulties that apply to the Alcubierre warp drive also apply to the generalized Natário warp drive (and thus to the zero-expansion Natário drive and Fell-Heisenberg drive as well). In particular, quantum energy inequalities, high magnitudes of the energy required, formation of time machines, instabilities related to the stress-energy making up the warp shell being unable to keep up with the bubble, the chicken-and-egg problem of needing FTL to achieve FTL, the formation of horizons making warp bubbles uncontrollable, and Hawking radiation at the horizons, are all still concerns for generic warp drives.

== Fell-Heisenberg warp drives ==

S. D. B. Fell and L. Heisenberg investigated what sort of warp structure is necessary to produce a warp drive with positive energy density everywhere<ref name="Fell-Heisenberg">S. D. B. Fell and L> Heisenberg, "Positive energy warp drive from hidden geometric structures", Classical and Quantum Gravity 38 155020 (2021) https://doi.org/10.1088/1361-6382/ac0e47 https://arxiv.org/abs/2104.06488</ref>. They found that warp drives of the generic Natário form with irrotational shift vectors had a simple expression for the energy density and vanishing momentum (irrotational means that the shift vector can be described as the gradient of a scalar potential field or, equivalently, that the curl of the shift vector field vanishes everywhere). Fell and Heisenberg proposed several potential fields that either had positive energy density everywhere except at discontinuities, or that were continuous but had regions that violated the weak energy condition (i. e. negative energy densities in some frames of reference). One specific example they provided of a warp field had an energy that is about 10,000 times less than the mass-energy of our sun. Or only about half the mass-energy of Jupiter. This is a significant improvement over many other proposed drives.

Despite the introduction discussing warp drive configurations that satisfy the various energy conditions, the configurations described in the paper are shown to locally violate the weak and strong energy conditions. Nonetheless, the energy density is still mostly positive. Santiago, Schuster, and Visser's<ref name="Santiago Schuster Visser 2022">J. Santiago, S. Schuster, and M. Visser, "Generic warp drives violate the null energy condition", Physical Review D 105, 064038 (2022) https://doi.org/10.1103/PhysRevD.105.064038</ref> work shows that the various energy conditions must still be violated by this warp drive; to not violate these, energy density must be positive in all reference frames not just those of the co-moving observer.

Any vector field can be described by its divergence, its curl, and a constant. The constant part of a warp drive's shift vector can be discarded because then the warp drive would not be localized. It is interesting, therefore, that the Natário zero-expansion drive has zero divergence and can be described only by its curl, while the Fell-Heisenberg drive has zero curl and can be described only by its divergence. These two classes of drives thus rely on opposite choices of setting one of the two different parts describing a vector field to zero. As might be expected, the Fell-Heisenberg drive is not zero expansion – space-time will change in volume as it passes through the warp bubble (although after going all the way through it will return to its original volume). However, objects in space-time will retain their original orientation as they pass through the warp bubble.

Any vector field with zero curl can be represented as the gradient of a suitable scalar function. Thus, the Fell-Heisenberg drive can be completely described by just scalar field (one number at all points of space and time) rather than a vector field.

Previous examples of warp drives, such as the Alcubierre and zero-expansion Natário drives, tend toward a flat space-time as you go far enough away faster than would occur if there were any mass present in the warp drive. That is, the ADM mass of such drives is zero. However, one of the Fell-Heisenberg examples described a spacetime where the geometry approaches that of what you would get for a massive object at the warp location, giving one of the first examples of a warp drive with mass.

In the Fell-Heisenberg drive the momentum vanishes everywhere. Yet the energy occupies regions where the shift vector is varying rapidly. The lack of momentum means that the energy will not move to keep up with the differential expansion and movement of the spacetime. As a consequence, at later times the energy will have a different distribution than what is necessary to maintain the given warp configuration. The exact time evolution is not solved but this likely leads to collapse of warp bubble. Note that this is not a problem unique to the Fell-Heisenberg drive. Assuming a geometry that describes a warp produces a specific distribution of the stress-energy tensor that satisfies Einstein's equation. But there is no guarantee that this stress energy tensor satisfies the continuity equation, which describes how the distribution of matter and momentum change with time. The issue of stability of any warp drive is not well understood, but most warp drive geometries will be unstable in the sense that their matter that generates the warp fields will not be able to keep up with the motion of the warp. The Fell-Heisenberg drive is unique only in that, with vanishing momentum, there is no configuration that allows stable propagation.

(n. b. The Heisenberg here is Lavinia Heisenberg, not the Werner Heisenberg of quantum physics and uncertainty principle fame.)

== Lentz warp drive ==

E. Lentz<ref name="Lentz">E. W. Lentz, "Breaking the warp barrier: hyper-fast solitons in Einstein–Maxwell-plasma theory", Classical and Quantum Gravity. 38 075015 (2021). arXiv:[https://arxiv.org/abs/2006.07125 2006.07125]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2021CQGra..38g5015L 2021CQGra..38g5015L]. doi:[https://doi.org/10.1088%2F1361-6382%2Fabe692 10.1088/1361-6382/abe692]. ISSN [https://search.worldcat.org/issn/0264-9381 0264-9381]. S2CID [https://api.semanticscholar.org/CorpusID:219635854 219635854].</ref> investigated a particular configuration of the Fell-Heisenberg class of warp drives, where the scalar field forms a selection of rhomboid regions, resulting in an energy density that is everywhere positive and a shift vector with vanishing tidal force at the center for a comfortable ride.

However, Santiago, Schuster, and Visser<ref name="Santiago Schuster Visser 2022"></ref> dispute claims that the Lentz drive satisfies the energy conditions, noting that everywhere positive energy density in one frame of reference is insufficient to establish that the energy density is positive in all reference frames; knowledge of the Cauchy stress tensor is also needed. They show that any generic warp drive will violate the strong energy condition, null energy condition, and weak energy condition.

== Conservation laws ==

(Discussion of asymptotic flatness: what is it? Where does it apply?)

(The "big four" conservation laws and how they relate to asymptotic flatness.)

(ADM mass -> 0; conservation of energy/mass issues.)<ref>Sebastian Schuster, Jessica Santiago,and Matt Visser, "ADM mass in warp drive spacetimes" General Relativity and Gravitation (2023) 55:14 https://doi.org/10.1007/s10714-022-03061-9</ref>

(Discuss issues of conservation of angular momentum.)

(If angular momentum conservation is ignored, discuss how momentum & energy are affected by outside forces. In a gravitational field equivalent to inertial frame ... like being in an accelerated elevator; will acquire momentum buildup while staying "at rest".)

== Warp Railroads ==

Van Den Broeck<ref name="VanDenBroeck_2000"></ref> suggested a warp railroad,

You can set up devices ahead of time along the path of the bubble that produce the negative energy regions for the warp bubble at the appropriate time without needing the negative energy to ever move with the bubble at all. At least for bubbles on a predictable schedule and route.

helps with:

-stability (elaborate)

-conservation laws (elaborate)

Mention Krasnikov<ref name="Krasnikov 1998">S. V. Krasnikov, "Hyperfast travel in general relativity" Phys. Rev. D 57 4760</ref>

Hyperwave<ref name="Pieri 2024">Lorenzo Pieri, "Hyperwave: hyper-Fast Communication within General Relativity", arXiv:2311.12069 [gr-qc] https://doi.org/10.48550/arXiv.2311.12069</ref> makes use of this warp railroad idea.

== Quantum effects ==

(quantum stuff here<ref name="Hiscock1997">W. A. Hiscock, "Quantum effects in the Alcubierre warp drive spacetime", Classical and Quantum Gravity 14 L183 https://doi.org/10.1088/0264-9381/14/11/002 https://arxiv.org/abs/gr-qc/9707024</ref><ref name="Finazzi et al 2009">S. Finazzi, S. Liberati, C. Barceló, "Semiclassical instability of dynamical warp drives", Physical Review D 79, 124017 (2009)https://doi.org/10.1103/PhysRevD.79.124017 https://arxiv.org/abs/0904.0141</ref>)

From<ref name="Pieri 2024"></ref>
<blockquote>
It should be emphasised that Quantum Inequalities have not been proven for realistic interacting quantum fields, where the they are expected to be weaker if they exist [Cadamuro, 2019], and they often require Minkowskian conditions at infinity, a condition which is not appropriate for fields inside bounded regions, as in a Casimir devices.
</blockquote>

== Warp drives and black holes ==

What happens if you drive your warp drive into a black hole?

The pop-sci reason that nothing can escape a black hole is that at the event horizon the escape velocity is faster than light and nothing can go faster than light. A warp drive can go faster than light.

The more detailed reason from general relativity is that at the event horizon space and time are rotated sufficiently that "inward" becomes "forward in time." You can no more go farther away from the singularity in a black hole once you are inside the event horizon than you can go backward in time. Any super-luminal travel can go backward in time, and warp drives can engage in super-luminal travel.

The Hawking area theorem that says that black holes can only grow, never shrink relies on the condition that the energy is everywhere positive. Warp drives have negative energy density.

So in principle, there's nothing that prevents a warp drive from taking a dip into a black hole and coming back out again.

Garattini and Zatrimaylov looked into the problem of a warp drive near (and in) a black hole<ref name="Garattini and Zatrimaylov">R. Garattini and K. Zatrimaylov, "Black holes, warp drives, and energy conditions", Physics Letters B 856 138910 (2024) https://doi.org/10.1016/j.physletb.2024.138910</ref>. The found that the black hole reduces the negative energy requirement for the warp bubble to move inward toward the black hole singularity, but increases the negative energy needed to move away from the center of the black hole. In addition, a warp drive parked at the event horizon would allow light from inside the horizon to pass through the bubble and escape back outside.

== Credit ==
Author: Luke Campbell

== References ==

[[Category:Physics]][[Category:Infrastructure]][[Category:Physics & Engineering‏‎]][[Category:Physics & Math & Engineering]][[Category:Transportation & Infrastructure‏‎]][[Category:Metric Engineering]]

Wormholes

2026-07-28T04:04:51Z

Lwcamp: /* Exotic energy conditions */

[[File:ltwormhole.jpg|link=https://www.youtube.com/watch?v=SuJ-2nTvAWo|thumb|600px|Still from a raytraced simulation of a long-throated wormhole, by Pablo Antonio Cano (YT). [https://www.youtube.com/watch?v=V7e-1bRpweo Check out Scott Manley's 360° video as well.]]]

Wormholes are hypothetical structures in space-time allowed by the general theory of relativity.
They provide short cuts through space-time that connect one region of space and time to another; potentially, these regions can be very distant from each other.
The symmetries in space-time of our universe mean that wormholes move like physical objects, and acquire the conserved (or in some cases approximately conserved) quantities of things that go into them and lose those quantities of things that come out – properties like energy, mass, momentum, angular momentum, and electric charge.

==What is a wormhole? The geometry of space-time and how to twist it into a pretzel==

We're going to have to start out with some pretty heady stuff. Like the very nature of existence. And what is reality, anyway?
Usually, space and time are so much an integral part of our existence that we don't even think of them at all. They simply form a backdrop on which all the interesting stuff plays out. But when we do think about them, we're so accustomed to living in a world where they exist and where distances are well defined and events can be uniquely identified based on where and when they happened (or are happening or will happen) that a lot of stuff seems so obvious that we don't stop to think about if it really has to be this way or if there are other possible ways of doing things.

===A toy model===

So lets consider a hypothetical scenario, where an ant can be in any of several states, represented by the squares in the pictures below. The ant can transition to any other state that has a numbered side with the same number as the state it is in. We will start on the left. If the ant chooses the "2" transition, we end up with the picture in the middle. If it then chooses the "10" transition, we end up with the state on the right.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Space_elements_1.png|frameless]]
<td width=400>[[File:Space_elements_2.png|frameless]]
<td width=400>[[File:Space_elements_3.png|frameless]]
</table>
But now notice that the behavior of this system is exactly the same as if we re-arrange the states into a grid, as shown below on the left. The disconnected states now become a space; an expanse of coordinates on which the ant can exist. Now, the ant's transition between states can be described as a trajectory as shown in the right picture below.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:Space_elements_grid.png|frameless]]
<td width=255>[[File:Space_elements_grid_path.png|frameless]]
</table>

This gives us an inkling of how something like space-time might end up being assembled from individual events. But what if we don't have things match up into quite such a perfect grid? We
ll take our space elements and tile them into an infinite grid. Two sections of the grid are shown below. But now we remove two of those space-tiles, and we connect the edges adjacent to the removed tiles with a corresponding edge of the other removed tile (although, as shown in the figure, we don't have to keep the same orientation). Now the ant, entering side 1 (which we have color-coded red for convenience) comes out of side 1 of the other missing square. This is perhaps the most basic wormhole.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Space_elements_wormhole_grid_1.png|frameless]]
<td width=400>[[File:Space_elements_wormhole_grid_2.png|frameless]]
</table>

Of course, this simple assembly only stitches together a space. But the world we live in is one of space-time. As the ant transitions from one square to another, it is also transitioning further ahead in time. But when the ant passes through the wormhole, while it locally experiences no discontinuity that wormhole might connect to a different time as well as a different region of space. So the ant might emerge far in the past or future of when it stepped through.

===Manifolds and coordinate patches===

The toy model used above hopefully prepares you for some of the ideas to follow - one of how we can piece together a complete space and time by connecting together different pieces. But before we go further, lets define an important term: a manifold is a "shape" of a given number of dimensions where, if you zoom in close enough, you can always get to a scale where it looks like the shape is flat. We'll use this definition to help us describe how we can put space and time together to get a wormhole. But first, let's look at a few illustrative examples.

====One dimension====

The simplest one dimensional shape is a line. And a line is everywhere flat. So a line is a one dimensional manifold.

A one dimensional shape might seem to be a wiggly path if inscribed in two or more dimensions, but it has no intrinsic curvature. That is, if everything about your world is confined to that path, you can't tell that it is curved – all you can do is measure the distance you go along the path. This greatly limits the kinds of shapes that we can consider. So any curved path of infinite extent looks from the inside like a line, and so any infinite length path is also a manifold.

But there's another class of one dimensional shapes that are manifolds – loops. A loop is a path of finite length that connects back with itself. You can make one by drawing a closed curve in a higher dimensional space. Or you can take a straight line but then say that a given point here maps on to another point there, and all the points between here and there form a loop. When you are marching along the line starting at there and you reach here you are also at there, so if you go further you just end up back where you were. This last bit may seem like a pointless bit of pedantry that just confuses a simple problem, but it will be a very useful way of looking at things as we go on.

====Two dimensions====

The simplest two dimensional manifold is a plane. A plane is everywhere flat, and extends off to infinity. A small region around any point on a plane (naturally) locally looks like a plane, so it meets our definition of a manifold.

=====Curved manifolds=====

But in two dimensions, we can have something that we don't have in one dimension – intrinsic curvature. Consider a sphere. If you get close enough to the sphere, it will look like you are on a plane; we live on a world that is (approximately) spherical (if you neglect mountains and ocean basins and all that other stuff), but when you stand up and look around you it mostly looks flat. So a sphere is a manifold. But take a line on the surface of your sphere. Extend it out along the surface, drawing it as straight as possible. Eventually, it will go all the way around the sphere, dividing it in two, and meet itself. This path is called a great circle. The Earth's equator is (approximately) a great circle, as are the lines of longitude (but not lines of latitude, because they do not divide the Earth into two equal-sized halves). The straightest-possible path on a curved surface (like a great circle on a sphere) is called a geodesic. But to keep the jargon down, we'll just call them lines for this non-technical presentation, and hopefully this will keep things more intuitive and understandable to the layman.

In flat space, two lines that are initially parallel remain parallel, always remaining the same distance apart. But in a curved space, initially parallel lines do not remain the same distance apart. On a sphere, for example, any two great circles will intersect at two points. A sphere thus has intrinsic curvature.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:Sphere_and_great_circles.png|frameless]]
<td>[[File:Hyperbolic_surface_and_lines.png|frameless]]
<td>[[File:Cylinder_and_lines.png|frameless]]
<tr>
<td width=400>Two great circles going around a sphere. Although initially parallel, they meet at two points.
<td width=400>Two initially parallel lines on a hyperbolic surface. As you get farther from the place where they are parallel, the lines diverge.
<td width=400>Lines that are initially parallel on a cylinder remain parallel everywhere.
</table>

On a sphere, parallel lines always end up converging. One example of a curved surface where parallel lines diverge is a hyperbolic surface. Like a sphere, a hyperbolic surface is a manifold. But it has the opposite sense of curvature.

Now consider a cylinder. Again, a cylinder is a manifold. A cylinder can be made by rolling up a plane. Like our construction of a loop, this can be done by simply taking a long rectangular section of a plane and saying that two of the parallel edges are actually the same (this is essentially what you do if you roll up a sheet of paper so that two of its parallel edges touch, and then pretend that they are merged together where they meet). This "rolling up" preserves distances, so any two parallel lines on the plane remain the same distance apart on the cylinder (think about that sheet of paper, and if you drew two parallel lines on it - they will stay the same distance apart when the paper is rolled up). A cylinder has extrinsic curvature – you can tell that it is curved when looking at it from three dimensions. But it has no intrinsic curvature – if you are confined to the surface of the cylinder it behaves locally as if it is flat.

A manifold that is flat obeys all the rules of Euclidean geometry (although you'll need to go to the three dimensional case for dealing with volumes), and the standard Cartesian coordinate system (representing position by two orthogonal coordinates, commonly called <math>x</math> and <math>y</math>) can be used to specify any point. A manifold that is not flat violates Euclid's fifth postulate; the geometry of the manifold will be non-Euclidean (somewhere, H. P. Lovecraft is quietly having a mental breakdown). It cannot be represented by Cartesian coordinates, but different sets of coordinates – or perhaps patches of multiple coordinate systems that apply in different places – can be defined instead to locate points.

=====Topology: simply and multiply connected manifolds=====
If you take a plane or a sphere, any closed loop on the manifold can be shrunk in a continuous manner until it vanishes. This is called simply connected. But not all manifolds are simply connected. Those which are not simply connected are called multiply connected. The most basic multiply connected manifold is a torus, like a bagel or doughnut or bicycle inner tube. While there are closed loops on a torus that can be shrunk to a point and made to vanish, those that go all the way around the torus tube cannot – they'll get caught up circling the tube. Likewise, loops that go all the way along the tube will get caught up on the central hole if they are shrunk and again cannot be made to go to a point.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:torus_and_loops.png|frameless]]
<td width=400>[[File:Topological_torus_and_loops.png|frameless]]
<tr>
<td colspan=2>Three classes of loops on a torus. On the left is a standard geometrical torus; on the right is a topological torus that is flat but has opposite edges identified so that they are connected. The blue loop doesn't go around the torus, it can be continuously deformed to shrink to a point and vanish. The red line goes around the torus the long way. If you shrink it, it will get caught up going around the hole in the middle and get stuck there; it cannot be continuously deformed to make it any smaller. The green line loops around the torus from the outside to the inside and back. Again, if you shrink it the line gets caught up circling the torus tube and can't be made any smaller.
</table>

And this brings us to another interesting kind of manifold. You can get a manifold that is flat but has the topology of a torus. You can do this by taking a rectangle and saying that the opposite sides are actually the same. This gives you the effect of a "wrap-around screen" like on that old arcade game Asteroids – if you go off the edge on one side, you just come back on the opposite side at the same height. in physics, this is known as "periodic boundary conditions". Closed loops that go all the way around one way or the other cannot be shrunk to a point. But this kind of surface cannot be embedded in three-dimensional space! We might have grown used to surfaces that we can envision existing in the space we live in, but there are manifolds, even fairly simple two-dimensional manifolds, that cannot be embedded in higher-dimensional spaces like that. Remember this when we get to actual wormholes.

You can have multiply connected manifolds that are not toruses, even though they might have the same topology; any manifold with one or more "loops" or "handles" is multiply connected.

=====Non-orientable manifolds=====

And there is one other interesting kind of manifold that is relevant to wormholes – manifolds that are non-orientable. An orientable manifold is one where you can define a consistent sense of clockwise and counter-clockwise everywhere on the manifold. If you start with a clock spinning clockwise, or a right-handed glove, or a copy of Lewis Carrol's "Through the Looking-Glass, and What Alice Found There", you can move it anywhere on the surface and if it meets back up with itself it will still be spinning clockwise, or right handed, or normally readable. But in a non-orientable surface, there are ways you can move the object so that when you get it back to where it originally was the clock is spinning counter-clockwise, or the glove is left-handed, or the book is written in mirror writing. The simplest example of such a surface is a Mobius strip. As shown below, if you move an oriented circle around the strip, it comes back with the opposite orientation.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Topological_Mobius_strip.png|frameless]]
<td width=400>[[File:Mobius_twist.png|frameless]]
<td width=400>[[File:Mobius_strip.png|frameless]]
<tr>
<td colspan=3>A Mobius strip … a strip of surface with a twist so the edges where it connects meet up "the wrong way around". On the right is a 2D strip where you connect along the short edge matching the 1's together and the 2's together. The center shows the same strip with a twist in it so you can line the edges up the way you are supposed to, demonstrating how the twist reverses the orientation of the spinning circles. An embedded image in 3D space with the edges connected together is shown on the left. You cannot define a consistent orientation to this surface – starting at the red counter-clockwise circle, if you move the circle continuously along the surface when you get back where you started you will find it is going clockwise!
</table>

====Three dimensions====

The most basic three-dimensional manifold is a flat space. This is, well, just a normal space. You can go off in any combination of three independent directions (left-right, up-down, front-back), and because it is flat, initially parallel lines stay parallel. It all behaves much as you would expect three dimensional spaces to behave.

But you can get variations on this. They are harder to visualize than two-dimensional cases, because we don't have the brainspace for processing curved spaces inside of a four-dimensional hyperspace. But you can do things like identifying two planes to be the same, such that if you go far enough in one direction you can end back up where you started again (akin to a cylinder), or make them curved such that distances are distorted and lines don't remain parallel and maybe you can put larger things inside of smaller surfaces. There are 3D equivalents of spherical and hyperbolic surfaces (called, naturally enough, spherical and hyperbolic spaces). There are even non-orientable spaces.

One way of trying to envision a curved space, and to gain some intuition about them, is to take a two-dimensional slice of that curved space and embed that in a flat three-dimensional space. This is called an embedding diagram. Remember that this is a reduced representation that throws away a lot of the information (unless there are special symmetries you can exploit) and is not actually the thing being described.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Space_with_embedded_plane.png|frameless]]
<td width=400>[[File:Embedded_plane.png|frameless]]
<td width=400>[[File:Embedding_diagram_example.png|frameless]]
<td width=400>[[File:Embedding_diagram_example_very_curved.png|frameless]]
<tr>
<td>A space with an embedded plane. We show a sphere in this space, with the plane bisecting the sphere.
<td>Just the embedded plane. Note how in a spherically symmetric manifold (like this space), a circle on the embedding diagram represents a complete 2-dimensional spherical surface.
<td>An embedding diagram of a "dimpled" space. Just as before, the closed circles represent closed spherical surfaces. It is just that in this case, the distance between two concentric spheres of radius <math>R_1</math> and <math>R_2</math> is longer than the usual <math>R_1 - R_2</math> because of the spatial curvature, as shown by the increasing distance along the dimple's surface.
<td>An embedding diagram of an extremely curved section of space. In this example, there are several places where we have a bigger sphere inside of a smaller one! Curved space is weird like that.
</table>

Remember that not every manifold can be embedded, so not every curved space can be represented by an embedding diagram. But you can take patches of the curved space and show them by embedding diagrams, and then describe how these patches stitch together to create a contiguous surface.

====Space-time: three space dimensions and one time dimension====

This is the world we live in. So this is the class of manifolds that we really need to use when making a wormhole. Sometimes it's abbreviated as 3+1D (as opposed to 1D, 2D, 3D, etc.).

The basic, infinite, flat space-time manifold is often called Minkowski space. I know, it's a big word using scary foreign-sounding names. But all it means is that its a 3+1D manifold that goes on forever and is flat. In flat space-time, everything obeys the rules of special relativity. Which means it really doesn't work when masses become significant enough to make gravity.

Our best model of how gravity works is general relativity. In this theory, the curvature of space-time is created by mass. At least, that's what they always say in the over-simplified pop-sci summaries of general relativity. Actually, space-time curvature is created by all the components of the stress-energy tensor. It's just that in almost all cases, mass is the most significant part of this. But momentum, stresses, pressures, and shear forces all have contributions as well. In general relativity, physical objects move along the straightest possible lines in this curved space-time. The way the curvature causes the lines to bend and converge is what leads to the appearance of gravity as a force if the curvature is sufficiently weak.<ref name="Misner_Thorne_and_Wheeler">Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler, "Gravitation", W. H. Freemann and Company, New York (1973)</ref>

The most basic curved space-time manifold is the Schwarzschild geometry, which is what you get when you have an uncharged, non-spinning spherical mass doing its distortion thing on space-time. The Schwarzschild geometry is static meaning it does not change in time, and is spherically symmetric, meaning that it has a definite center and it doesn't matter which direction you are relative to that center – it is independent of angle. If the mass becomes dense enough, space-time gets so distorted that an event horizon forms. Nothing that goes into the event horizon can ever come back out, and the inevitability of the geometry crushes the mass down to an infinitely dense point at the center called a singularity. Nothing on the inside of the horizon can ever escape. It forms a black hole, that forever traps any light or matter that enters it. For masses that don't exceed this critical density, the gravitational distortion outside of the object is the same as for a black hole of equal mass, but there is no event horizon. The object can still be seen and interacted with.<ref name="Misner_Thorne_and_Wheeler"></ref>

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Schwarzschild_star.png|frameless]]
<td width=400>[[File:Schwarzschild_black_hole.png|frameless]]
<tr>
<td>An embedding diagram of the Schwarzschild geometry of a dense, massive star. The physical extent of the star is shown in blue.
<td>An embedding diagram of the Schwarzschild geometry of a black hole of the same mass as the star to the left. The event horizon is shown in red.
</table>

There is one distinction between space-time manifolds that will become important later. Some manifolds have an isolated region of high curvature (corresponding to a concentration of mass, or at least of stress-energy) but as you go farther away from the strongly curved region the space-time becomes increasingly flat. If you can always go out far enough away from a curved region that space-time becomes flatter than whatever flatness criterion you choose, no matter what direction you choose to leave the curved region, the manifold is called asymptotically flat. For our purposes, if you can go far enough away that the Newtonian approximation to general relativity is accurate, and gravity can be described as a force rather than requiring a geometric description, then the geometry can be considered asymptotically flat. You can also allow gravitational waves in your asymptotically flat space-time – their very small space-time curvature is not enough to cause problems and they can still be described by a linearized version of gravity; linear deviations from Newtonian gravity such as gravito-magnetic effects can also fall into asymptitcally flat region of space-time. Both the Schwarzschild and the Minkowski geometries are asymptotically flat. Spherical or hyperbolic geometries are not asymptotically flat. Planets, stars, galaxies, neutron stars, black holes, and galactic clusters are all approximately asymptotically flat; but the universe as a whole is not. As long as we confine our attention to merely galactic clusters, we can make the approximation of asymptotic flatness.

===Making a wormhole===

Now that we have covered some of the introductory material, we can get down to the main course. To make a wormhole, take two regions of space-time, usually far apart and usually asymptotically flat (although neither of these is strictly required) and connect them together. That's it. You're done. Sounds easy, doesn't it?

Note that if you are connecting two regions in the same universe, your new geometry will be multiply connected. A loop that goes through the wormhole and then connects back up outside the wormhole cannot be shrunk to a point. There will now be two equivalent ways to get between any two points – the path that goes through your usual space-time, and the path that goes through the wormhole. If you use a wormhole to connect two separate universes, the combination of the two universes and the wormhole can still be simply connected.

In general relativity, you can distort the geometry of space-time with stress-energy, but there is no way to change its topology; that is, no way to turn a simply connected universe into one that is multiply connected, or to add or remove more "loops" or "handles" to an already multiply connected universe. Such changes, if they are possible at all, would only be possible in the realm of quantum gravity and we do not know enough about quantum gravity to know if it would be possible or not. So, it might not be possible to make intra-universal wormholes at all. Or maybe quantum gravity does allow topology changes in space-time and there are ways to connect different parts of our universe together with wormholes. On the plus side, if you can't change the topology of space-time, you can always bud out a new universe from our own – perhaps by triggering the inflationary field that started our universe. It will be connected to ours by a wormhole. And because the topology can't change, that wormhole would never be able to break. We would have an eternal bridge to the new universe. So a system of physics that does not allow topology changes still allows wormholes to other universes.

==Common kinds of wormholes==

===Spherically symmetric wormholes===

General relativity involves a lot of non-trivial math. You need to deal with four-dimensional curvature tensors with up to 20 independent components, and figure out how they relate to a given distribution of matter, fields, energy, stresses, and material flows. If you can reduce the number of dimensions you need to keep track of, it can simplify things a lot. One way of doing this is assuming your space-time geometry does not change with time (it is static), and looks the same in all directions when viewed from a central point (it is spherically symmetric). This makes things a lot easier to analyze.

Astute readers will have noticed that these conditions are satisfied by the Schwarzschild geometry. In fact, if you play around with the Schwarzschild geometry enough, you will find that you can even get a wormhole-like solution. At the event horizon, you can connect your black hole solution to a time-reversed structure called a white hole. This is called an Einstein-Rosen bridge. Unfortunately, it has some issues. For one, it can't form naturally from infalling matter creating a black hole. For another, going through an Einstein-Rosen bridge involves passing through an event horizon, so the trip is necessarily one way. And finally, you can show that even the smallest amount of matter or radiation falling into the Einstein-Rosen bridge leads to it collapsing before the matter or radiation can pass through, trapping it inside the black hole. So this is not looking particularly useful.

So what are we going to do? Well, the Einstein-Rosen bridge, like a lot of early work in general relativity, was made by assuming a distribution of stress-energy (in this case, matter) that was assumed to be physically plausible, and finding out what the resulting geometry would be. But we want a wormhole, dangit! So what happens if we just demand a static tunnel connecting two asymptotically flat regions of space-time? If you do this, you get a Morris-Thorne wormhole<ref name="MorrisThorne">Michael S. Morris and Kip S. Thorne, "Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity", American Journal of Physics 56(5), 395-412, May 1988</ref>. You still have the relation between stress-energy and geometry, but now it is your assumed geometry dictating the stress-energy rather than the other way around. One difficulty with this method is that the distribution of stress-energy you end up with might not end up seeming physically plausible. In particular, it can be shown that any wormhole must have regions in it with negative energy density<ref name="Visser_Lorentzian_wormholes">Matt Visser, "Lorentzian Wormholes: From Einstein to Hawking", Springer/AIP Press (1996)</ref>, and these spherically symmetric wormholes are no different. We will discuss the issues around and solutions to the negative energy density problem later.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Morris_Thorne_wormhole.png|frameless]]
<tr>
<td>An embedding diagram of a Morris-Thorne wormhole. For this wormhole, we just took two copies of the Schwarzschild geometry from above and connected the middle "dimples" with a cylindrical patch of space-time.
</table>

The places where you enter a wormhole are called its mouths, and most wormholes have two. The space-time tunnel between the mouths is called the wormhole throat.

====Some common problems with spherically symmetric wormholes====

if you don't have your distribution of matter and energy going off to infinity, then eventually if you go far enough away your geometry will reduce to the Schwarzschild geometry. A good many published examples of such wormholes are actually Schwarzschild-like up until pretty close to where the event horizon would be. The first problem you might encounter, then, is the formation of a horizon. A single horizon makes any trip one-way. Going into it on the side with the horizon would be indistinguishable from entering a black hole - except that you pop out somewhere else instead of getting removed from existence at the singularity. And because of the horizon, you couldn't ever go back. If both sides have a horizon, you simply can't get out of it. Why do you even have this thing, anyway?

The strong curvature as you get close to the throat can also produce very strong tidal forces. Due to the gravitational forces, if you go in feet-first your feet could be pulled much harder than your head as you go in and you could end up getting pulled apart – spaghettified is the technical term. In addition to these pure spatial tides, it turns out that you also have velocity-dependent tides. These tides are transverse to the radial direction - if you dive in really fast, you will be squeezed even harder from side to size.

Early attempts to limit the tidal forces while still keeping a throat large enough to fit a person through resulted in very large masses. Like planetary scale masses or even solar scale masses. This would mean you would need to spend a lot of propellant de-orbiting into the wormhole and then rocketing your way back out. Some of these early wormholes also had issues like the extreme gravity blue-shifting the cosmic microwave background to blow-torch intensity x-rays. Sometimes, attempts to keep the tides down also lead to really large throat sizes. Like the distance from the Sun to the Earth sizes. Or even light year sizes.
<ref name="Garattini_arXiv">[https://arxiv.org/abs/2008.05901|Remo Garattini, "Generalized Absurdly Benign Traversable Wormholes powered by Casimir Energy", arXic:2008.05901v1 [gr-qc] 12 Aug 2020]</ref> These gigantic wormholes would be rather inconvenient.

None of these are necessarily intrinsic limitations to spherical wormholes. But it does indicate that a perfectly reasonable assumption for the wormhole tech in your science fiction universe could involve wormholes with Jupiter-level masses, extreme tides, and other fun details.

===Thin shell wormholes===

Take a shape located somewhere in flat (Minkowski) space-time. The important part of this shape is that it must be a closed surface that completely encloses a given volume. Now go someplace else in space-time and take an identical shape. Remove all of the space-time from inside both shapes, and then say that the corresponding parts of the two surfaces of the shapes are actually the same points. Now you have a thin shell wormhole. The throat region is infinitesimally short – the section that corresponds to the wormhole is just that surface of your shape, hence the name "thin shell".

The first thin shell wormholes proposed were simple polyhedra <ref name="Polyhedral_wormholes">Matt Visser, "Traversable wormholes: Some simple examples", Physical Review D vol. 29 no. 10 pages 3182-3184 (1989)</ref>. These are attractive choices for a potential wormhole-user because the flat faces of a polyhedral thin-shell wormhole have no tidal effects whatsoever. Further, the flat faces do not have any stress-energy either. It is simply flat space-time, no different from any other area of flat space-time. Going through the face of a polyhedral wormhole is no different from going through the flat space-time of the doorway separating your hallway from your bedroom. The negative energy stuff is concentrated entirely in the edges and corners of the polyhedron. And hoo boy, do you need a lot of it. For a cubic wormhole, each edge requires approximately <math>-1.52 \times 10^{43}</math> J/m of length. This is <math>-1.69 \times 10^{26}</math> kg/m, or close to a tenth of a Jupiter mass per meter of edge length. It has been proposed that perhaps cosmic strings – defects of the primordial universe where things couldn't line up the right way and left a linear flaw in reality – with really high tension might be able to meet this requirement and hold open a polyhedral wormhole.

But there's a strange thing when you consider the mass that these wormholes require. They are embedded in flat space-time. Which means that outside of that thin shell they have, they induce no curvature on the space-time around them. Remember that mass causes space-time to curve, dimpling up as in the Schwarzschild geometry shown earlier. Because these wormholes induce no far-flung curvature to imprint on the surrounding space-time, they have zero mass (technically ADM mass). And because mass is equivalent to energy, it should technically be possible to make one of these for no energy input at all from those of us in flat space-time. Forces are the rate of change of energy with distance, pressures are the rate of change of energy with volume. But because the energy is always zero no matter what size they are (as measured from out in flat space-time), there would not be any force or pressure needed to hold them open. The extreme conditions of their interior contrasts with the relatively benign conditions once away from the thin shell (or, indeed, within any flat area within the shell).

A curious feature of polyhedral wormholes is that each face of the polyhedron might connect to a different polyhedron. You might, for example, have a collection of seven cubical wormhole mouths, and each of the six faces of the wormhole you are facing connect to a different one of the six remaining mouths.

For more general thin-shell wormholes, you just need to smear that negative mass out a bit. As long as you keep a flat area to go through, travelers will not experience any strange tides or encounter space-warping exotic energies. You could, for example, have your standard circular portal beloved of fiction. Or you could have any other arbitrary shape you wanted.

Thin shell wormholes are what you get when you take a mathematical limit; they would be an approximation of a more physical distribution of stress and energy. Any realizable version would have a finite, although possible small, thickness to its shell and hence a finite, although possible small, throat length. As we will discuss later, they would likely also have a positive, although possible small, mass. But as an approximation, and as a valid solution to Einstein's field equations in general relativity, they allow us to explore many aspects of wormhole physics using a relatively simple model that is easy to calculate, and can lead to a lot of interesting inspiration for fictional ideas of wormholes.

====Tidal distortion during transit====

Consider a thin shell wormhole whose shell-shape is everywhere convex - no flat areas this time. For convenience, let's make it a sphere. As a traveler passes through the shell, the sections of his body that pass the shell must suddenly "pop" from curving one way to curving the other way, as illustrated below:

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Velocity_dependent_tides_pre_transit.png|frameless]]
<td width=400>[[File:Velocity_dependent_tides_in_transit.png|frameless]]
<tr>
<td>A traveler (left rectangular shape) about to go through a spherical thin-shell wormhole (here represented as circles, with the interior volume blacked out to indicate that there is no space-time there and nothing can exist in that region).
<td>The traveler in the process of passing through the wormhole. Note that on the wormhole shell, the portion of the traveler to the left has a leftward-curvature, while the portion on the right has a rightward curvature. As the traveler passes through, each section that goes through the shell immediately jumps from left-curving to right-curving. This will induce stresses in the traveler as his interior is deformed.
</table>

it is clear that as the curvature of the wormhole becomes large compared to the size of the traveler, the distortion will decrease. In the limit of a flat section of the surface, no distortion is experienced at all.

A consequence of this is that something going through a curved section of a thin shell wormhole will experience stresses (forces) and strains (distortions). The elastic energy stored by a strained object is proportional to the product of the stress and the strain, added up over all of its volume. Assuming that the object going in to the wormhole is initially in its relaxed state, it will require energy to push it into the wormhole shell. If it enters with insufficient initial kinetic energy to get it through or external forces pushing on it, it will just bounce off.

===Cylindrical and toroidal wormholes===

In principle, you can have a wormhole with cylindrical symmetry
<ref>Kirill A. Bronnikov and José P. S. Lemos, "Cylindrical wormholes", arXiv:0902.2360 [gr-qc] 24 Feb 2009</ref>
<ref>K. A. Bronnikov and V. G. Krechet, "Potentially observable cylindrical wormholes without exotic matter in general relativity", arXiv:1807.03641v4 [gr-qc] 10 May 2019 https://arxiv.org/abs/1807.03641</ref>
. This forms a valid solution to Einsteins field equations in general relativity. However, a true cylindrical wormhole would be infinite in length. So they probably don't exist naturally, and you won't be able to build one. Unlike all the other wormholes discussed here, they are not asymptotically flat. Unlike wormholes of finite extent, not all cylindrical wormholes require negative energy regions.

So the thought is that maybe you can get close enough to a cylindrical wormhole by making a really long wormhole that bends ever so slightly and comes back and meets itself, forming an extremely stringy torus. The general case for this has not yet been worked out, but a thin shell approximation of a toroidal wormhole has been developed
<ref>Vladimir Dzhunushaliev, Vladimir Folomeev, Burkhard Kleihaus, and Jutta Kunz, "Thin-shell toroidal wormhole", arXiv:1901.07545v2 [gr-qc] 28 Jan 2019</ref>
.
These thin shell toriodal wormholes have been shown to be stable to perturbations, one nice benefit that has not been established for a lot of other kinds of wormholes. They do require some negative energy regions on their surface, but there will also be some regions that do not contain negative energy.

===Non-orientable wormholes===

In the discussion on non-orientable manifolds, we showed how in these manifolds you can't properly determine the clockwise - counter-clockwise orientation of objects or motions, or the orientation of objects with respect to their mirror reflections. If you move an object along certain paths, when it comes back it might come back the wrong way around. So it is probably no surprise that a non-orientable wormhole turns the universe into a non-orientable manifold. Something going through a non-orientable wormhole comes out looking like it has been mirror reflected.

Normally this seems like it would be rather boring and useless, but for an obscure constraint from quantum physics called the CPT theorem. This theorem, which holds for all physical phenomena, means that the product of the discrete symmetries of charge conjugation (C), parity transformation (P), and time reversal (T) always returns you to your original state. Huh? What does that mean? Well, parity transformation is equivalent to a mirror reflection; so passing through a non-orientable wormhole means parity is inverted. This means that exactly one of the other two symmetries must also be inverted for the CPT theorem to hold. The thing coming out on the other end is still manifestly going forward in time, so it must be charge conjugated, whatever that is.

Charge conjugation means you turn all particles into their antiparticles. So anything passing through a non-orientable wormhole emerges on the other end made entirely out of antimatter.

While this particular turn of events might discourage people from passing through, it does offer obvious application for energy generation, space propulsion, and weaponry.

==Exotic energy conditions==

It is fairly easy to see that any wormhole requires negative energy densities (or more generally, regions where the tension is higher than the energy in natural units) to exist. Take a look at any of the wormhole diagrams above – the spherically symmetric ones will probably make it the most clear. If you send a parallel ray of stuff – maybe particles or light or golf balls – through the wormhole, and trace out the straightest possible path of all of those parallel lines, you will see that after they go through the wormhole all those initially parallel trajectories will now be diverging. Positive energy (such as mass) causes attractive forces, drawing trajectories toward it and making the paths converging. To get diverging paths you need the opposite – repulsive gravity – which you can get from negative energy densities or very high tensions. Having the tension be higher than the energy density turns out to be equivalent to a negative energy density – in some frames of reference observers will measure a negative energy density there.

A lot of physicists used to be reflexively dismissive of negative energy densities. After all, they can't really exist; all energy in the real world is positive.

Except that it is not. There are certain odd cases that allow for negative energy densities. One of these is the Casimir vacuum. A conducting boundary cannot support an electric field parallel to that boundary (if there was such a field, charge would flow under the force of the field until the parallel component of the field is screened out). So if you have two parallel conductive plates, the only electromagnetic waves that you can get between them require their fields to vanish at the plates. This restricts the number of ways the electromagnetic field can vibrate, called modes. In quantum mechanics, merely the potential for a vibration is associated with a certain energy (called zero point energy) even if no quanta of vibration are actually present. By making these modes impossible, they cannot contribute their zero point energy to the region between the plates. This lowers the total energy of the space between the plates compared to the region outside the plates (or with the plates absent). Because the energy of empty space is zero, this means that the region between these Casimir plates has negative energy. The Casimir effect has been measured in the laboratory. It is fairly weak, but it is present. In fact, not long after traversable wormholes were first described, one proposal suggested holding open the wormhole using the Casimir effect<ref name=MirrosThorneYurtsever">Michael S. Morris, Kip S. Thorne, and Ulvi Turtsever, "Wormholes, Time Machines, and the Weak Energy Condition", Physical Review Letters, Volume 61, Number 13, 26 September 1988, pages 1446-1449</ref>

Another situation that gives rise to a negative energy density is a squeezed vacuum. A squeezed state is a way of manipulating a quantum oscillator (like a pulse of light) to get around the uncertainty relations of quantum mechanics to increase the precision in one variable by increasing the uncertainty in its conjugate variable. Applying this operator reduces the energy of the oscillator, and if this is applied to the vacuum state, which already has zero energy, you can end up with regions of negative energy.

A third way to get negative energy densities is using quantum energy teleportation<ref name=FunaiMartinMartinez">Nicholas Funai and Eduardo Martín-Martínez, "Engineering negative stress-energy densities with quantum energy teleportation", Physical Review D 96, 025014 DOI:https://doi.org/10.1103/PhysRevD.96.025014</ref>. This method uses an observer that communicates observations about the state of vacuum fluctuations in her vicinity to a second agent, who uses that information to select correlated vacuum fluctuations for extracting energy for himself. In the process, the energy of the vacuum will go negative.

It is worth noting that not just any old negative energy region of exotic space-time will work. In particular, dark energy is a condition of the universe we live in where even bare vacuum itself has a small amount of negative energy. The current best model we have for this is as a cosmological constant - where every part of the universe has an equal density of negative energy and a related amount of pressure. But the wormhole allowing contributions to space-time curvature of that negative energy are exactly cancelled by the wormhole-denying contributions of the pressure. So according to our best current models, dark energy will not support a wormhole.

Also, the Casimir vacuum described above is also problematic. The exotic vacuum between the plates does have the necessary properties (if not magnitudes) to support a wormhole, but this is generally far overwhelmed by the immediately adjacent positive mass matter making up the conductive plates. The technical term for the criterion that determines whether you can have a wormhole or not is the Averaged Null Energy Condition (ANEC). If you project a ray through space-time that follows the path that light (in a vacuum) could take, and do some math along that ray and add up all the contributions from all the places along that ray, then if your result is zero or positive then the ANEC is said to be satisfied. If this is the case, you can't get a wormhole. If the ANEC is less than zero, it is said to be violated and in principle you could use that weird exotic energy stuff to prop open a wormhole's throat. There are other so-called energy conditions, and if you have regions of space-time that also violate those energy conditions they can be used to hold open wormholes as well - but all of these also violate the ANEC, so the ANEC is all you need to consider.

===So just how do I get the ANEC-violating stuff I need?===

The short answer is, we don't know.

As we already said, a cosmological constant form of dark energy won't do it (although different ideas about dark energy, such as phantom energy, could conceivably work). The inflationary vacuum thought to have existed very briefly during the first instances of our universe's existence is also usually thought to be a cosmological constant (although one that stopped existing after a very brief period of time, so I guess it is less than constant), so the inflationary vacuum probably won't work either.

If you can get the Casimir vacuum without the massive conducting plates that bound it, you might be able to do something with that. For example, the twisted geometry of space-time itself in a wormhole throat imposes boundary conditions that restrict the modes of vibration inside of it similar to the Casimir effect. This is called the topological Casimir effect, potentially it could be used to help support a wormhole. Can you use this to hold open a wormhole? Maybe a sufficiently clever person will eventually figure out how to do so.

Even more serendipitous, it turns out that gravity itself naturally squeezes the vacuum
<ref name="gravitationally_squeezed_vacuum">David Hochberg and Thomas W. Kephart, "Lorentzian wormholes from the gravitationally squeezed vacuum", Physics Letters B 268, 377-383 (1991)</ref>
. If you have the curved space-time of gravity, all of the modes of oscillation in it become squeezed states. So it might be possible that the curved space-time of a wormhole can squeeze the vacuum enough to support itself. Or a combination of gravitational vacuum squeezing and the topological Casimir vacuum from the wormhole's geometry constraints might be all that is needed. At least one proposal has suggested a class of wormholes that can be self supporting in such a way
<ref>S. V. Krasnikov, "Toward a Traversable Wormhole", arXiv:gr-qc/0003092v1 22 Mar 2000</ref>
<ref>S. Krasnikov, "Traversible Wormhole", Physical Review D, volume 62, article 084028 (2000)</ref>,
although the throat of these wormholes is very "wrinkly" or "crumpled up" and it is unclear what the effects of that would be on anyone passing through.

====Quantum energy inequalities====

When you apply quantum field theory to what happens when you have a region where the quantum fields produce local negative energy densities, you end up with a limit of how much negative energy stuff you can have before it has to be balanced out by an even greater amount of nearby positive energy stuff
<ref>[https://web.archive.org/web/20060206055950/http://maths.york.ac.uk/www/PhysicsQIneq.htm|Quantum Energy Inequalities]</ref>
<ref>[http://www.phys.lsu.edu/mog/mog20/node16.html|Quantum field theory on curved spacetime at the Erwin Schrödinger Institute]</ref>
<ref name="Fewster_Roman_2005">Christopher J. Fewster and Thomas A. Roman, "On wormholes with arbitrarily small quantities of exotic matter", Physical Review D 72, 044023 (2005)</ref>
.
In other words, you can have regions with negative energy densities. But only if it is right next to places with even more positive energy so that the net energy averaged over the whole region is positive.

It has been shown<ref name=FunaiMartinMartinez"></ref> that the method of quantum energy teleportation can saturate these inequalities – driving them to their physical limits – even if they cannot exceed them.

====Matter with negative mass====

It might be tempting to try to solve the negative energy density requirement for wormholes by introducing some sort of matter that naturally has negative mass. Negative mass would behave in unusual and non-intuitive ways. The force on an object is its acceleration times its mass. Most things accelerate in the direction you push them. But when the mass is negative, the acceleration will be in the opposite direction to the force on the object. If you try to push negative mass matter with your hand, then the negative matter is constrained to be accelerating in the direction of your hand ... so instead of pushing on it you must be pulling on it, and by Newton's third law of motion the negative mass will pull back on your hand. So as you try to push it it will tug you forward<ref name="ScienceMeetsFiction">[https://www.youtube.com/watch?v=zEGsq7H5egE| Science Meets Fiction, "What Does Negative Mass Mean? Part 1"], [https://www.youtube.com/watch?v=1Xr4dTCZc7g| Science Meets Fiction, "What Does Negative Mass Mean? Part 2"], [https://www.youtube.com/watch?v=2YFyvR7M_LI| Science Meets Fiction, "Negative Mass Part 3: Energy, Friction, Gravity, and More"], [https://www.youtube.com/watch?v=Pr3j00DIrvM&t=1498s| Science Meets Fiction, "Negative Mass Part 4: Life, the Universe, and Everything(-ish)"]</ref>.

Gravitationally, the negative mass will exert a repulsive force on positive mass things around it, so large amounts of negative mass will push people away. Large nearby positive mass bodies normally extert attractive forces on things, but because the gravitational force is proportional to mass negative mass things will experience a force away from the positive mass … but remember that the acceleration of the negative mass is in the opposite direction to the force. The negative mass will still fall toward the positive mass object, and will be gravitationally repelled away from negative mass objects. This gets even weirder if a negative mass thing is next to a positive mass thing with the same magnitude to both their masses. The positive mass thing will accelerate away from the negative mass thing by gravity, but the negative mass thing also falls toward the positive mass thing. And because they have equal mass magnitudes, the acceleration will be the same. Both objects will continually accelerate, the negative mass chasing the positive mass, forever. The motion of the positive mass will give it positive kinetic energy and momentum in the direction it is going; the negative mass with the same motion will have negative kinetic energy to exactly balance the positive kinetic energy of the positive mass, and momentum opposite its direction of motion to exactly cancel the momentum of the positive mass in the other direction – conservation laws are still upheld, even if you have perpetual motion and reactionless acceleration. Unfortunately for this idea, however, you can't just hold the two objects apart with any sort of braces or connections. Any external force, no matter how small, will be amplified across the connections to infinity and break them. So that any slight imbalance in the initial masses will lead to them eventually drifting apart or colliding.

However, negative mass would introduce all kinds of problems to the smooth operation of how the world works. For one thing, if negative mass (or equivalently energy) can exist, why doesn't it just spontaneously pop out of empty space accompanied by an equal magnitude of positive mass (or energy)? For another, negative masses lead to all kinds of runaway instabilities. For example, consider a negative mass thing in air. If it moves through the air, the drag force is in the opposite direction to its motion. But the negative mass accelerates opposite the force, so it accelerates in the direction of its motion. Unlike positive mass things that slow down from drag, negative mass things go faster! And the faster they go the more drag they experience, leading to a runaway exponential increase in their speed. Eventually they will be going so fast that they will be heating up the air, driving shock waves, and even producing radiant fireballs. All the energy for those phenomena come from the negative mass gaining negative energy the faster it goes. Any initial motion, no matter how small, will get amplified without bound. And because on the molecular scale the forces from atoms colliding with the negative mass will be subject to statistical fluctuations, even if initially exactly at rest the negative mass will soon start accelerating and run away off to infinity.

Or consider a negative mass thing with an electric charge. If you put something with an electric charge in an electric field the field will exert a force on the charge. For a normal positive mass charged thing, it will move in the direction of the force. The work done is the force in the direction of the displacement in position, so because the force and motion of the object are in the same direction, the field does work on the charged object and energy flows from the field into the kinetic energy of the charged object. But if the charged object has negative mass it will begin to move opposite to the direction of the force. This will do work on the field instead of the object, increasing the energy of the field. The energy comes from the kinetic energy of the negative mass charged object – but because kinetic energy is proportional to mass and increases with increasing speed, as the negative mass object loses kinetic energy it gains speed, going ever faster. Both the field strength and the speed of the object increase without bound. If the electric field originates from the electric part of an electromagnetic wave, it will vary sinusoidally with time and space. As the negative mass object is forced into a sinusoidal trajectory opposing the electric force on it, it will amplify that wave making it ever more intense. A single, isolated charge exposed to an electromagnetic wave will eventually have its inertia keep it moving from the previously imposed force even after the force has switched direction, so if the negative mass is free with no other forces acting on it there will be no net amplification. But if there are any other damping forces, or internal arrangements of bound charge that can resonate with the wave, then instead of being absorbed (as would happen with a positive mass), the wave will be amplified indefinitely and the ever-increasing energy of the wave will come from the ever increasing temperature of the negative mass thing (because the mass is negative, the thermal motion of temperature gives it a negative energy whose magnitude only increases as the temperature rises).

Elastic forces, such as a spring, exert a force on an object proportional to how far the force-giving thing is displaced from equilibrium, with the force directed so as to bring the elastic material back into equilibrium. But a negative mass moves opposite the force, so it keeps moving farther and farther away from equilibrium, stretching the elastic medium (or spring) more and more in another exponential runaway process. Again, even the slightest initial displacement causes runaway instability, and thermal or quantum fluctuations ensure that there will be some slight initial displacement.

And, of course, the quantum energy inequalities indicate that you can't just have isolated lumps of negative mass stuff floating around.

So just having lumps of stuff that have a negative mass is probably not realistic, and it were possible it would lead to all kinds of universe-shattering cataclysms. Hopefully, we can get wormholes to stay open just using the exotic vacuum states from the Casimir effect and squeezed vacuums.

It is worth mentioning that there is still one loophole. If you can find stuff that has highly localized negative mass intrinsically surrounded in close proximity by a greater amount of positive mass, you may be able to use those regions of negative mass to prop open a wormhole if you can sufficiently isolate it from the necessary positive mass stuff it generates around it.<ref name="Woodward_2011">[https://doi.org/10.1016/j.phpro.2011.08.003|J. F. Woodward, "Making Stargates: The Physics of Traversable Absurdly Benign Wormholes", Physics Procedia Volume 20, 2011, Pages 24-46]</ref>

===Reducing or eliminating ANEC violating stuff===

As soon as people figured out that the wormholes they want needed stuff they couldn't get, they set about trying to find ways to use as little of the stuff they couldn't get as possible. A number of attempts have been made to hammer out a geometry for spherical wormholes that are stable, meet the quantum energy inequalities, and allow a person to pass through them in a reasonable length of time without being shredded by tides
<ref name="Fewster_Roman_2005"></ref>
<ref name="Visser_Kar_Dadhich">Matt Visser, Sayan Kar, and Naresh Dadhich, "Traversable Wormholes with Arbitrarily Small Energy Condition Violations", Physical Review Letters Vol. 90 No. 20 article 201102 (2003), https://arxiv.org/abs/gr-qc/0301003v2</ref>
<ref name="Kuhfittig">Peter K. F. Kuhfittig, "More on wormholes supported by small amounts of exotic matter", Physical Review D 73, 084014 (2006); Peter K. F. Kuhfittig, "Wormholes supported by small amounts of exotic matter: some corrections", arXiv:gr-qc/0508060v1 15 Aug 2005</ref>
.
One of these studies<ref name="Visser_Kar_Dadhich"></ref> showed that there exist classes of wormhole geometries that can be constructed with arbitrarily small amounts of ANEC violating matter; however, followup studies<ref name="Zaslavskii>O. B. Zaslavskii, "Traversable wormholes: Minimum violation of the null energy condition revisited" Physical Review D 76, 044017 (2007), DOI: [http://dx.doi.org/10.1103/PhysRevD.76.044017 10.1103/PhysRevD.76.044017] </ref> indicate that as the amount of exotic (ANEC-violating) matter is decreased, either a horizon forms (meaning you can't go through it) or the length of the throat diverges such that it becomes infinitely long as the amount of ANEC-violating stuff approaches zero.
Often, these resulting structures end up having extreme dimensions, such as a throat radius on the order of light years
<ref name="Garattini_arXiv"></ref><ref name="Fewster_Roman_2005"></ref>.

Thin shell wormholes have also been the subject of investigation. One study <ref name="Mazharimousavi_Halilsoy">S. Habib Mazharimousavi, M. Halilsoy, "3 + 1-dimensional thin shell wormhole with deformed throat can be supported by normal matter", Eur. Phys. J. C (2015) 75:271 DOI 10.1140/epjc/s10052-015-3506-6</ref> found a number of thin shell wormholes with sharp-edged shapes could be supported entirely without any exotic energy. However, this only applies if the edges are infinitely sharp. A finite radius of curvature at the edges would require negative energy at those edges.

If the wormhole is taken to have the properties of a class of subatomic particles called a fermion, then if the ratio of the electric charge to the mass is sufficiently large you can get a wormhole that can remain open without any additional negative energy regions at all
<ref name="Blazquez-Salcedo_Knoll_Radu">Jose Luis Blázquez-Salcedo, Christian Knoll, and Eugen Radu, “Transversable wormholes in Einstein-Dirac-Maxwell theory”, Physical Review Letters 126, 101102 (2021); ArXiv: 2010.07317v2 [gr-qc].</ref>
.
However, an analysis<ref>Ben Kain, "Are Einstein-Dirac-Maxwell wormholes traversable", [https://arxiv.org/abs/2305.11217 arXiv:2305.11217 [gr-qc]], Phys. Rev. D 108, 044019 (2023) https://doi.org/10.1103/PhysRevD.108.044019</ref> of this class of wormholes was unable to find any solutions which did not collapse into a black hole before any signal could propagate through them and concluded that they were not traversable.

==Fields and conserved properties==

Our best theories of how the world works – quantum mechanics and general relativity – predict that certain things are [[Conservation_Laws:_Limits_to_Cheating|conserved]]. For asymptotically flat space-times, these quantities are energy, linear momentum (or just momentum), angular momentum, and electric charge. Furthermore, in the non-relativistic limit mass is conserved independently of non-mass energy.

And not only are these things conserved, but they are always conserved locally. This means you can't just get rid of, say, some energy in one place and expect things to balance out by having that energy appear somewhere else.

What does it mean for something to be conserved? Mathematically, it means that the thing obeys the continuity equation. There are various ways of writing the continuity equation, all equivalent, but basically it boils down to the following: if you have a region of space surrounded by a closed surface, then the amount of conserved stuff inside that surface can only increase if some of the stuff enters through the surface and it can only decrease if some of the stuff leaves through the surface. Makes sense, right? If you have three loaves of bread in a room, then if the deliveryman brings in another two loaves you will have five loaves. But unlike loaves of bread, which can be baked or eaten or grow moldy or burn up or be broken into crumbs, the only way to get energy or electric charge or the other conserved things into or out of the room is to have them come into or go out of the room.

Each of these conserved quantities has an associated field. Electric charges create an electric field. Energy and momentum and angular momentum each create a distribution of the curvature tensor of space-time. In the non-relativistic limit, this is the mass part of the energy creating the gravitational field.

There is a relationship between the amount of a conserved quantity in our hypothetical closed surface and the net amount of field that penetrates that surface. This is called [https://en.wikipedia.org/wiki/Gauss%27s_law Gauss's law]. Basically, it says that if you add up all of the amount of field that goes out of the surface, and subtract off all of the field that goes back into the surface, the resulting amount will be directly proportional to the amount of the conserved stuff in there. Do you start off with a kilogram of mass inside your room? Then you have a certain amount of gravitational field from that mass that leaks out of the room. Add another kilo for two kilos total, and the amount of gravitational field leaving is also doubled. The distribution of where that field comes out may change, but as long as you don't take that mass out of the room or add more mass in the room, the total amount of field leaking out won't change. And any gravitational field from masses outside of the room that leaks into the room must also leak back out again somewhere. Those masses outside the room will not affect the net amount of field leaking through the surface. What this means is that you can determine the amount of conserved stuff inside a surface if you know the amount of field going through that surface.

This can be visualized in the field line approximation. The field around a positive charge can be approximated as a series of directed lines that radiate away from the charge. The field around a negative charge in this approximation is a series of lines that converge on the charge (for non-relativistic gravitation, you only have positive charges because mass is always positive. We'll just ignore the fact that wormholes require negative energy density stuff in order to exist which will lead to negative mass – we're just trying to get across the basic idea here). Field lines can only ever start on a positive charge and only ever end on a negative charge. The number of lines radiating away or converging on the charge is proportional to the size of that charge (so if you double the charge, you double the number of lines that connect to it). Field lines propagating through space bend away from a positive charge and bend toward a negative charge.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=322>[[File:charges_positive.png|frameless]]
<td width=322>[[File:charges_negative.png|frameless]]
<td width=322>[[File:charges_dipole.png|frameless]]
<tr>
<td>The field lines radiating away from a positive electric charge.
<td>The field lines converging on a negative electric charge.
<td>The field lines emerging from a positive charge and curving around to go into a negative charge of the same magnitude. This configuration, with a positive and negative charge of the same magnitude, is called a dipole.
</table>

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=322>[[File:charges_positive_Gauss_law.png|frameless]]
<td width=322>[[File:charges_negative_Gauss_law.png|frameless]]
<td width=322>[[File:charges_dipole_Gauss_law.png|frameless]]
<td width=322>[[File:charges_exterior_dipole_Gauss_law.png|frameless]]
<tr>
<td>A closed surface encloses a positive charge. A net amount of field lines exit the surface.
<td>A closed surface encloses a negative charge. A net amount of field lines enter the surface.
<td>A closed surface encloses a dipole. The total amount of charge inside the surface is zero because the negative charge cancels the positive charge. No net amount of field goes through the surface - just as many field lines enter as leave.
<td>A closed surface that has no charge inside of it, but various charges outside of it (in this case a dipole). Again, no net amount of field goes through the surface - every field line that enters comes back out somewhere else.
</table>

Now consider the case of an isolated positive charge, like one one shown above, that is far away from anything else. We will bring up one mouth of a wormhole close to the charge. Because the field lines going out of the charge cannot break, they cannot go through the wormhole. Instead, they will curve around the wormhole mouth. As the charge enters the wormhole mouth and travels into its throat, it drags its field lines along with it. With all of its field lines entering the mouth, that mouth looks like it has a net positive charge. As the charge leaves the other mouth, it continues to drag its field line out of the wormhole - the field lines that start on the charge now curve around to enter the wormhole mouth. The mouth that the charge exited from now looks like it has a negative charge.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Wormhole_with_charges_initial.png|frameless]]
<td width=400>[[File:Wormhole_with_charges_inside.png|frameless]]
<td width=400>[[File:Wormhole_with_charges_passed_through.png|frameless]]
<td width=400>[[File:Wormhole_with_charges_long_gone.png|frameless]]
<tr>
<td>The charge approaches the wormhole.
<td>The charge is inside the wormhole. The mouth of the wormhole that the charge entered looks like it has a positive charge.
<td>The charge has passed through the wormhole. The mouth that the charge entered still looks like it has a positive charge, but now the mouth that the charge came out of looks like it has a negative charge.
<td>The positive charge has now moved very far away. Both mouths of the wormhole still appear to be charged.
</table>

<table align=right border=1 width=600>
<tr><td> Technical details:
If you work through the actual math, there is a bit of additional nuance to this. The divergence theorem strictly requires a surface that is the boundary of the volume you are integrating over, so applying the divergence theorem to Gauss's law when the volume contains a wormhole only tells you the total electric flux coming out both sides of the wormhole without information as to how much flux is coming out of each one or how the flux changes over time.
To make the claims made here, you also need to use the Maxwell-Ampére law (the fourth of Maxwell's four equations) which, when integrated over any closed surface (not necessarily one that is the boundary to a volume) tells you that the electric flux changes only in proportion to the charge that goes through and nothing else. This establishes the persistent electric flux of wormhole mouths and local conservation of charge at the mouths. The other commonly considered conserved quantities also have an associated field with an analogue of both Gauss's law and the Maxwell-Ampére law, with similar consequences<ref name=ConservedQuantitiesAtWormholeMouths>[http://panoptesv.com/SciFi/WormholeConserved/conserved_quantities_through_wormholes.pdf Luke Campbell, "On the local conservation of conserved quantites at wormhole mouths", self published (2024)]</ref>.
</table>

This can also be understood by the continuity equation. Picture an imaginary closed surface around the mouth of the wormhole. This is like the case of the room, before, except that now there is a way out of the enclosed volume that does not pass through the surface – by going through the wormhole! Any conserved quantity that goes into the surface adds its quantity to the stuff in the surface. If it goes through the wormhole, it does not come out through the surface. The conserved quantity inside that surface stays the same! To an outside observer, it looks like the wormhole mouth has that quantity.

Either way you look at it, the results are the same. A wormhole mouth acquires any conserved quantity of the things that enter it, and loses any conserved quantity of the stuff that exits from it.<ref>C. W. Misner and J. A. Wheeler, “Classical Physics as Geometry: Gravitation, Electromagnetism, Unquantized Charge, and Mass as Properties of Curved Empty Space”, Annals of Physics 2, 525-603 (1957)</ref><ref>Valery P. Frolov and Igor D. Novikov, ”Physical effects in wormholes and time machines”, Physical Review D, Volume 42, Number 4, Pages 1057-1065, (1990) DOI:https://doi.org/10.1103/PhysRevD.42.1057</ref><ref>John G. Cramer, Robert L. Forward, Michael S. Morris, Matt Visser, Gregory Benford, and Geoffrey A. Landis, "Natural wormholes as gravitational lenses", Physical Review D, Volume 51, Number 6, Pages 3117-3120 (1995) DOI:https://doi.org/10.1103/PhysRevD.51.3117</ref>
<ul>
<li> A wormhole mouth acquires the energy of anything that goes into it. Its energy decreases by the energy of anything that goes out of it.
<li> In the non-relativistic limit, this means that a wormhole mouth acquires the mass of anything that goes into it. Its mass decreases by the mass of anything that goes out of it.
<li> A wormhole mouth acquires the electric charge of anything that goes into it. It gains negative the electric charge of anything that goes out of it.
<li> A wormhole mouth acquires the momentum vector of anything that goes into it. It gains the negative of the momentum vector of anything that goes out of it.
<li> A wormhole mouth acquires the angular momentum vector of anything that goes into it. It gains the negative of the angular momentum vector of anything that goes out of it.
</ul>

This perhaps should not be a surprising result – other general relativistic space-time distortions also exhibit this behavior. A black hole, for example, gains the mass, charge, momentum, and angular momentum of anything that goes into it.

You can use these same arguments to find that the same results apply to any asymptotically flat manifolds of space-time. They do not necessarily hold for space-time geometries that are not asymptotically flat. If the gravitational distortion of a wormhole extends infinitely far away, you will not be able to even define its total energy or momentum or angular momentum, so it will be impossible to say if they are conserved. But do note that all of the wormhole examples that we have discussed so far (except for the cylindrical wormhole of infinite extent) are asymptotically flat, and will obey the local conservation of all of these quantities.

===Wormhole dynamics===

A wormhole mouth has the gravitational field and conserved properties of mass, momentum, and angular momentum. By the equivalence principle of general relativity, this must mean that it behaves as if it had those properties. So a wormhole mouth will follow ballistic trajectories in vacuum through space-time as if it were an object in free fall. If it is around a concentrated mass, like a planet or sun, it will have a Keplerian orbit around that object (or, if the wormhole has more mass than the planet or sun, perhaps it would be more reasonable to say that the planet or sun would orbit it). The mass of the wormhole mouth will resist acceleration like any other mass, according to Newton's laws of motion. Force is the time rate of change of momentum, so as the wormhole mouth's momentum changes it will experience forces, and these will change its trajectory in the same way as if that momentum (or forces) acted on a material object with the same mass. And if the wormhole mouth has an electric charge, it will experience forces from electric and magnetic fields; again, these can change its trajectory.

In other words, a wormhole mouth acts like a physical object of the same mass, charge, energy, momentum, and angular momentum.

The motion of the wormhole mouths through our normal space does not affect their relative separation through the wormhole's throat. That can remain the same length regardless of the behavior of the mouths in our normal universe.

As we covered, if an object enters a wormhole mouth that mouth gains the object's momentum. This will give the wormhole a "kick" changing its motion in the direction that the entering object was originally moving. The dynamics work out the same as if the wormhole were a sticky blob and the object collided and stuck to it. In both cases, you get a final object with the combined masses of both initial things (the mouth and the entering object) that has been knocked off course a bit by the momentum of the entering object hitting it.

If that object comes out of the other mouth, then as that mouth gains the opposite of the objects momentum it gets a recoil kick in the opposite direction. Again, this results in dynamics that are similar to things we already know about – like a gun with a bullet loaded firing the bullet. As the bullet leaves, the gun + bullet system loses the mass of the bullet, and as the bullet shoots away the gun recoils in the opposite direction.

So now consider what happens if you direct a constant stream of material through a wormhole. As the jet of material exits its mouth, that mouth loses mass at the same rate as the mass flow rate of the jet. It also continually gains momentum by the recoil kick of the gas. These dynamics are identical to that of a rocket - as the propellant is ejected, the rocket loses the propellant's mass and the recoil momentum from the escaping gas jet pushes the rocket in the opposite direction. So by shooting stuff through a wormhole you can turn it into a rocket, with dynamics identical to a rocket, obeying the [https://en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation Tsiolkovsky rocket equation], and everything else that happens with rockets.

===Does gravity go through a wormhole?===

The above diagrams of the electric field of a charge as it approaches a wormhole mouth also hold for the gravitational field of a mass that is near a wormhole mouth. In the same way that the electric field cannot enter the wormhole's interior or leak through to the other side unless the charge enters the wormhole or goes through to the other side, so to will the gravitational field of a nearby mass curve around the wormhole. The gravity of a nearby planet will fall to zero inside the wormhole throat and will not affect those near the opposite mouth.

The caveat is that for a thin shell wormhole a small amount of field can bow out through the infinitesimally short throat to reach the immediate vicinity of the other side, corresponding to a field line that comes through and then loops back. These fringing fields leaking through may be noticeable very close to the mouth when the throat is much shorter than the width of the mouths. When the throat is long compared to the size of the mouth, this will not be a concern.

===Can waves go through a wormhole?===

So if gravity won't go through a wormhole, and static electric and magnetic fields can't go through a wormhole, then what about light? That's made up of electricity and magnetism, right?

So, turns out that light can go through a wormhole. So can other electromagnetic waves like radio and x-rays. So can gravitational waves. The rules for drawing field lines only really work for static charges; for radiating waves you also need to include the parts about how the changing field generates more field. In the end, waves that are smaller in wavelength than the size of the wormhole can go through just fine. You'll get some [[Diffraction|diffraction]] around the mouths, but for wavelengths much smaller than the mouth size even that might not be noticable.

===Can negative mass wormholes exist?===

So a wormhole mouth loses the mass of anything that comes out of it. What happens if you send something through a wormhole with more mass than the mouth which it exits from? This rule says that the mass must become negative, but as we saw earlier this leads to all kinds of problems (especially because the wormhole can also be charged) and is likely to be forbidden by the laws of nature.

If a wormhole's mouth can reach negative mass, you can also get other interesting instabilities. The positive mass mouth will attract matter into it, which will pass through, increasing the mass of the positive mass mouth and making the mass of the negative mass mouth even more negative. Meanwhile the gravity of the negative mass mouth will repel matter, preventing any return flow. Both the negative and positive masses will gradually increase over time.

So what's an aspiring science fiction writer to do? If you have a particular desire for negative mass things in the fiction you create, go ahead and allow it. Just be aware of the consequences. Otherwise, it is probably easier to say that negative mass wormhole mouths can't exist, and justify that with the quantum energy inequalities. Which, of course, brings up the question of what happens when you try to push an object through a wormhole that would make the mass of one of its mouths go negative? Two obvious possibilities are that either you get a back-reaction force that makes it so you can't ever make something go through the mouth to make its mass negative (perhaps by the formation of a horizon as the mass approaches zero), or the wormhole collapses before the mass can ever reach zero. Perhaps inventive science fiction authors and fans can come up with other plausible options as well.

==Wormholes and time travel==

Wormholes connect across both space and time. So it is natural to consider the possibility of using wormholes for time travel.

Now maybe you don't want to have to deal with time travel in your fiction. It introduces all kinds of opportunities for paradoxes, and you have to keep track of plot lines in a self-consistent way that go back and forth through time, influencing themselves in complicated ways. Okay, no problem. Lets just have our wormholes always connect to the same time. Easy right? In fact, it is natural to justify this. Physics is local, so if you can make wormholes you probably have to create both ends right next to each other at the same time. You can then move the wormhole ends away from each other – maybe putting one end on an interstellar spacecraft or something – but they stay at the same time. Right?

Right?

Except ...

In relativity, motion not only affects how fast you go through space, but also how fast you go through time. When things move, they experience time dilation, so if you move one wormhole rapidly far away and then bring it back, the time dilation it experienced will naturally allow it to form a time machine so you can send things and people back to interfere with their own pasts. Even worse, time can also be slowed down by gravity, so if your wormholes are in areas affected differently by your galaxy's own gravity, or around stars of different mass, or just at different elevations on the same planet, the rate they experience time will be different. Temporal complications ahoy!

Let's look at how this works in more detail. We'll take our standard science fiction empire, called The Empire. Like all empires, it has a central area of authority, the metropole (which we will place on a planet called Metropole), which extracts wealth from various colonies and subjugated client states for its own enrichment in the form of taxes, tribute, and forced favorable trading opportunities. Let's give our Empire two colony worlds which we'll call Colony A and Colony B (these Empire folks sure aren't very creative in their naming, are they?). Just for convenience, we'll put Metropole, Colony A, and Colony B all 100 light years away from each other, in an equilateral triangle.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:WormholesTimeTravel graphic1 wb.svg|600px|frameless]]
<td>[[File:WormholesTimeTravel graphic2 wb.svg|600px|frameless]]
<tr>
<td width=400>Metropole and its two colonies A and B
<td width=400>Metropole sending a wormhole to Colony A.
</table>

So Metropole launches a wormhole mouth out to Colony A. They want to get to Colony A quickly, so they'll make the wormhole go really fast. Let's say they fling it out at 99.9999% the speed of light. At this speed, it takes 100.0001 years for the wormhole mouth to reach its destination in the reference frame of Metropole and Colony A. But due to relativistic time dilation, in the reference frame of the projected wormhole mouth it only takes 0.1414 years to go from Metropole to Colony A. Because the techs at Metropole can look through the wormhole, they will see that 0.1414 years after launch it arrives at its destination. At that point, brave explorers from Metropole can go through and set foot on Colony A – they only need wait about a month and a half to feel alien soil under their feet, rather than a century. In Metropole's reference frame, going through the wormhole takes you 100 light years away, and 99.8596 years into the future. Going the other way, from Colony A to Metropole, takes you 100 light years away and 99.8596 years into the past.

[[File:WormholesTimeTravel graphic3 wb.svg|800px|frameless]]

One might think that these time warps would let you engage in all kinds of time travel. It is easy to see that the Metropole—Colony A situation described here doesn't allow these kinds of shenanigans. For practical purposes, you only have a time machine when you can go back to the place you left at a time before you left. And you can't do that here. Go from Colony A to Metropole and you go back in time 99.8596 years. Go back to Colony A through the wormhole, and you go forward in time the same amount, plus any time you spent on Metropole, so you get back after you left. If you go back through flat space-time, it will always take at least 100 years since you can't go faster than the speed of light so you also get back after you left. No paradoxes for you!

However, it is easy to imagine situations where a wormhole, or a configuration of wormholes, does make a time machine. For example, what happens if we immediately turn around and send the wormhole mouth from Colony A back to Metropole? In the reference frame of Metropole, the wormhole comes back with its "Return to sender" sticker 200.0002 years after it was launched. But going through from the left-behind mouth to the round trip mouth will take you 199.7192 years into the future. And if you step into the round trip mouth, you will go back in time by 199.7192 years into your own past. You now have a real honest to goodness time machine, complete with paradoxes.

[[File:WormholesTimeTravel graphic4 wb.svg|800px|frameless]]

Or maybe the wormhole mouth stays at Colony A. And Metropole sends another one to Colony B, that also goes 99.8596 years into the future. Now Colony A sends a wormhole to Colony B. This wormhole also goes 99.8564 years into the future as a consequence of its trip. This means if a traveler at Colony B went through the Colony A wormhole he would go back in time 99.8564 years. Then going from Colony A to Metropole he would go back in time another 99.8564 years. Then he could go from Metropole to Colony B and go forward in time 99.8564 years. The net result is that he ended up back where he started nearly a century before he left.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:WormholesTimeTravel graphic5 wb.svg|600px|frameless]]
<td>[[File:WormholesTimeTravel graphic6 wb.svg|600px|frameless]]
</table>

And it is not just motion that causes time dilation. Being different depths into a gravitational potential does this as well. If Colony A is deeper into the gravity well of the galaxy, or orbits a heavier star, it will experience more gravitational time dilation than Metropole. So eventually it will build up more than 100 years of time slip between the two end of the wormhole and you can have a time machine.

So is time travel inevitable if you have wormholes? What a nuisance!

Well, maybe not. Here's why. Think about what happens when the Colony A – Colony B wormhole has gone just far enough that a light signal going through the wormholes can get back to where it left just as it is leaving. Now, since the propagating signal and the newly transmitted signal are both leaving at the same time, you have double the intensity. So this doubled intensity signal goes around and meets itself again, quadrupling its intensity. And so on. At this point, just as the configuration is on the verge of becoming a time machine, it becomes a perfect resonator for light signals, which then build up to arbitrarily high intensities until something breaks and you don't have an incipient time machine any more.

Now, this won't always work if you just consider light signals. Light can get defocused or sent off in odd directions or something so that you don't have a good path back until after you end up with a time machine. But if you look at what happens in semiclassical quantum gravity, will get a buildup in the amplitude of quantum fluctuations as soon as you start to get a time machine, called vacuum polarization. These fluctuations are expected to build up to sufficient amplitude as to destroy the time machine to be, at least for the few simple cases studied. Or perhaps it just causes the wormholes to bounce away from each other so as to prevent time machine formation. This has not yet been shown to occur in general. But to does seem to happen for all possible configurations of one or two wormholes about to become time machines
<ref name="Visser_Lorentzian_wormholes"></ref>
.

But let us assume for the moment that wormholes cannot form time machines in any configuration due to this vacuum polarization mechanism. Now there are limits on just how you can place and move your wormholes. You don't want to accidentally almost form a time machine and break your transportation network. There are several ways to do this:
<ul>
<li> Move your wormholes slowly, so as not to build up much time dilation (or time lag, if you will).
<li> Don't form closed loops in your wormhole transportation network. The less distance you need to travel to go all the way around a loop to get back where you started, the less time lag is needed to make a time machine.
<li> "Discharge" the time lag by periodically rapidly moving the mouth of a wormhole that is "behind" in time in such a way that it comes back to where it was. Perhaps you can put a charge on it by sending a [[Particle_Beam_Weapons|particle beam]] through it, and then accelerate it up to relativistic speeds in a synchrotron for a while.
<li> Send new wormholes that you are going to place through the existing wormhole network. This way they acquire the same time lag across their mouths as all of the wormholes they traverse. Until they build up some additional time lag of their own, you provide a perfect return trip without making a time machine.
</ul>

In conclusion, if you want a story or setting that involves time travel, you can plausibly still use wormholes somehow to allow that (although it might take loops of several of them). If you want a story that avoids the complications of time travel, you can still plausibly have wormholes in that setting (although you now need to beware of the limitations on your wormhole transportation network imposed by the no time machine rule).

==Practical implications==

===Getting places===

Usually in science fiction, people want wormholes because they want to be able to get to cool places far away without taking forever. Very often, this involves the technology (or the protagonist, with some kind of psychic power) "opening" a wormhole from the current location to the desired location when there wasn't one before.

Sadly, the ability to just create a wormhole to anywhere you want to seems unlikely. Physics tends to be local, which means that if you can make a wormhole both ends will emerge right next to each other. You will need to move the ends to where you want them to be using other methods (mailing them by post, putting them on a spacecraft, etc.). As was mentioned earlier, if you shoot a stream of stuff through a wormhole, the end that the stuff comes out of becomes a rocket. So one way to move an end of a wormhole is to shoot stuff through the wormhole. Because you can leave all the big heavy equipment at home (generators, rocket motors, cooling systems, and everything else), if you can make your wormhole light enough this could be a low cost and efficient method of getting it places. Remember that you can exploit time dilation to make it seem like you can get someplace in much less time than you would expect light to take going to that place (at the expense of taking you some time forward in the future as well when you step through). If you can get your wormhole mass down to a few grams, or even several kilograms, using powerful lasers to shoot through the wormhole to make a photon rocket could be a highly efficient way to go places at relativistic speeds. Perhaps it could even fuel itself by collecting mass from the interstellar medium (ISM) as it flies along, and it might be able to brake against the ISM or solar wind of the destination star system to slow down, reducing mass loss even further.

===Keeping time===

We have already shown that wormholes connect across both space and time, so that a trip between star systems could take you hundreds of years into the future, and the return trip takes you hundreds of years back in time. And this is even before we throw in how time slips between planets when considering relativistic time dilation due to different speeds and gravitational potentials.

Fortunately, all the weirdness of different time rates and going backward and forward in time can be ignored by the average person. This is because you never need to go from one world to another, or back, across the vast gulfs of interstellar space. You just take the wormhole between them. All you ever need to worry about is the coordinate frame that goes across the wormhole. When considering this reference frame, you're not hopping all over the place in time. If it takes ten minutes to cross the wormhole between the two planets, when you get to your destination world the clocks will read ten minutes later than they did when you left your departure world. By coordinating their time-keeping across the wormhole network, all of the worlds of the network can agree on a common time to coordinate their activities. This is all travelers ever need to worry about, and they can then ignore all the relativistic weirdness. Your network engineers will still need to keep track of relative time drift and how close a given configuration is getting to a time loop. But unless your protagonist is a network engineer, they can just ignore all that stuff. And, as an author, so can you! Assume your engineers are competent, you have good regulatory bodies and standards institutions, and don't worry about any of this "time travel" that doesn't actually let you cause paradoxes.

===Keeping wormholes away from planets===

By now, people expect science fiction to have spaceships. Wormholes offer a convenient excuse to have spaceships, because it provides a way for those spaceships to get to other worlds that are not in our solar system. But if you have wormholes, sometimes you need to ask why you still have spaceships? Why not just put one end on Earth and the other on an alien planet and step through to get where you are going?

Sometimes there are good reasons for this. If your wormhole has as much mass as Jupiter, you are not just going to keep it in Topeka, Kansas. That would kind of end the Earth. So you would put it in orbit around the Sun, and send spaceships from Earth out to the wormhole and through it to wherever it goes.

But all too often we see authors using special pleading to pigeonhole in their spaceships without thinking about good physics. Things like "it can't work in atmosphere" (use an airlock), "it can't work within a certain number of planetary radii" (why? There is no physical plausibility to such a restriction), "It can't work in a gravitational field of more than X" (violates the equivalence principle), and so on. But if you insist on having spaceships, and you also insist on wormholes that mass less than a planet, there are a few plausible ways to justify your preconceived ideas that at least plausibly align with physics.

For example, the closest you can plausibly get to "X number of planetary radii" or "gravity field less than X" is a criterion that relies on local measurements of space-time curvature only. And it turns out that this quantity is the tidal forces (curvature is a direct measurement of tides). Tidal forces are going to be proportional to the mass of the nearby thing causing the tides, and inversely proportional to the cube of the distance. If R is a measure of your space-time curvature, M is your object's mass, and r is the distance from the object to the wormhole, then

<div align=center>R &Proportional; M / r³.</div>

So if you want your wormholes to be at least ten planetary radii from Earth (r = 64 × 106 m, M = 6 × 1024 kg), then the curvature limit is R = 23 kg/m³ in some convenient system of units where we don't have to include constants of proportionality. But now suppose you want to send a 26 × 106 kg ore freighter through that wormhole. You'll only be able to get your freighter within about 100 meters of the wormhole before the freighter's tides collapse the wormhole. This might work if the freighter were really long and skinny, much longer than 100 meters, to reduce its transverse tides; or if the wormhole was much larger than 100 meters in diameter. But in general, note that the curvature and tides have units of density, so if the average density in any volume around the wormhole exceeds the critical density – oops, there it goes.

Also, having a limit on the geometric curvature of space-time similar to what you get around mere planets doesn't end up being very plausible – the curvatures in the wormhole itself exceed that by so very many orders of magnitude that having so stringent of a curvature tolerance seems very implausible, and just dancing on the edge of disaster. For example, if your wormhole is about 100 meters across, its curvature in these same units will be approximately 500 billion trillion kg/m³. Keeping the tides to within 23 kg/m³ is the equivalent of an engineering tolerance of one part in 25 billion trillion.

So if you don't really have a good way to limit where you can place a wormhole based on the nearby "gravity", are there other ways you can exclude them from the vicinity of planets?

Safety might be one good reason. We have already discussed various ways a wormhole might collapse (forming a time machine, having its mass go negative). What happens when a wormhole collapses? If there are any parts of the wormhole or things in the wormhole that are charged or made of charged particles (like ordinary matter) you could justify a considerable fraction of the wormhole's mass-energy being converted to electromagnetic radiation. In addition, if the wormhole collapses into a black hole it will evaporate away its mass-energy primarily in the form of electromagnetic waves as it undergoes Hawking radiation. Large black holes will last a long time, but if the black hole left behind has a mass of 500 tons or less it will last less than 10 seconds. And 500 tons worth of energy delivered in 10 seconds would do quite a number on a planet. These sorts of considerations could lead to a regulatory environment that require wormholes to be located several light seconds away from inhabited places (like planets).

===Wormholes on planets===

But sometimes you might want to have the wormhole on a planet. Perhaps you are envisioning one of those circular or oval fantasy portals, looking like a window to an exotic foreign land, and when you step through you are actually there. We already know that thin-shell wormholes can be very low mass, and can have a circular shape. So that should work, right?

There are at least two issues to consider here. One is that even if both wormhole ends are on the same planet, natural variations in atmospheric pressure due to weather will lead to very significant differences in air pressure from one side of the wormhole to another (unless the wormholes are so close that they are within a small distance of the extent of the same weather system – say, within a few km of each other). This pressure difference will drive severe winds through the wormhole, usually comparable to the winds of a hurricane (although there will be periods of relative calm when the pressure at both wormholes just happens to be at the same pressure). This effect will only be exacerbated if the wormhole mouths are at different altitudes, due to the decrease in air pressure with altitude. And if the wormhole connects between different worlds, the effect will be much worse. In addition to making travel inconvenient and possibly damaging the equipment, the winds will produce large uncontrollable mass flows across the wormhole (air on earth has a density of about 1 kg/m³, so if you have 50 m/s winds blowing through a 2 meter diameter portal you have a mass flow of about 160 kg/s through your wormhole). This will make it difficult for mass balance, and keeping both ends having high enough masses that neither gets so close to zero that the wormhole breaks or forms a horizon or goes negative mass and causes a runaway planetary catastrophe or whatever wormholes do when one end gets driven to near negative mass by the stuff leaving it.

As a consequence, planetary wormhole mouths will probably be kept in airlocks.

The other issue is that different latitudes on a planet will be rotating at different speeds; at the poles they won't be moving at all while at the equator they will be going a full planetary circumference every day. This means wormhole mouths at different latitudes will be building up different time dilations with respect to each other. In addition, wormholes at different altitudes will experience different gravitational time dilation rates. If you let this go on for long enough, the differences in time across the wormhole mouths will form a time machine. Whether this collapses the wormholes, makes them "bounce" apart from each other (likely ruining a lot of property and infrastructure in the process), or just forms a time loop that leads to various inconvenient paradoxes depends on the assumptions you made for the physics of your world. But it will probably be something you will want to avoid. Re-balancing the time differences between wormhole ends will probably need to be done every few decades.

===Wormholes on spaceships===

The necessity for radiators on spacecraft is really annoying. Dealing with all that waste heat is not fun, and these issues are not awesome kewl stuff like gigawatts and megatons and hundreds of g's of thrust and other things that let you show everyone how much your spaceships totally rule and everyone else's spacecraft totally drool. Instead you need these big fragile stupid-looking things hanging off the side of your ship, ruining your aesthetics and efficiency and being vulnerable to your enemies blowing them up, and lighting you up so bright that you're easily visible from the other side of the solar system. Hey! I know! Let's put a tiny wormhole on our spaceships so we can just send all the heat someplace else. Great idea! That'll work, right?

Actually, yes. If you have small enough wormholes, it will work great. But there are consequences.

So you're running your coolant loops from your reactor through the wormhole to some cooling towers on your home planet. Much better than radiators. But – why do you still have that reactor on your spaceship? It's big and heavy and needs lots of big and heavy shielding and poses nasty radiation hazards. Surely you could leave it behind on your home planet as well, next to those cooling towers, and just run a power line through the wormhole? You'll be able to run a bigger, more powerful reactor that way anyway, and that extra weight won't be bogging your ship down.

And why does your spaceship have its lasers on board? You can have a much bigger laser on the planet that you don't need to lug around. Just shine the laser beam through the wormhole and re-direct it with mirrors.

And all of your missiles? Might as well just feed them through the wormhole when you need to fire them.

And then do you really need all that life support equipment? Just run an HVAC and plumbing that sends fresh air and water to the spacecraft and takes used air and water off. And remember to send the occasional snack, too.

Oh, but wait. Why is all your crew even on the spacecraft again? They have a wormhole to the spaceship. You can just have all your crew run the spacecraft remotely from a mission control station. Most of your equipment is on the planet anyway, so most of the engineering staff is already here. Now you don't actually need to put people in harm's way. If someone does need to fix something just put them in a space suit and send them through, let them do their job, then bring them back. Most of that can be done by a tele-operated robot, in any event. And being able to go back to their families when their shift is over is great for morale.

Hmmm, your sensors could just look through the wormhole instead of having all the vulnerable and heavy and expensive equipment actually on the spacecraft. And as long as you are doing that, you can just leave your laser beam pointer at home too, and aim the laser by pointing it through the wormhole.

So now what's actually left in your actual spacecraft? Oh, the rocket itself. But remember how we already talked about how we could just keep the rocket engine and propellant on the planet and shoot the rocket jet through the wormhole, and it will move just like a rocket? Yeah, so no rocket either.

Congratulations, you've successfully gotten rid of your spaceship. The wormhole itself is your spaceship. It's all you need, it's safer, it's more capable, it's stealthier, and just better all around than some contraption of steel and Mylar and fissioning uranium fuel rods and fragile people going through space.

===Adding mass to wormholes===

So you just made a pair of wormholes. Chances are, you pulled them apart as near Planck mass objects, with about 20 micrograms of mass. Well, that's not very useful! What if you want to send something through that's bigger than 20 micrograms? You're going to need to add mass to your wormhole mouths.

A similar issue comes up if you project a wormhole at some far away planet or star system using the wormhole-rocket trick already discussed, maybe by using powerful lasers to make a photon rocket. To be practical, your projected wormhole mouth will need to be very low mass. But once you get it to where you want to go, you'll need to give the mouth enough mass to let your explorers and colonists and equipment through.

A simple way to do this is to simply suck stuff into the wormhole. This adds mass to the wormhole mouth the stuff comes through. Of course, if you take it out from the other side, that other mouth will lose that much mass, and if it didn't have than much mass to begin with you'll likely have problems. So just leave that stuff in there! Shove it off to the side. Let it clutter up the unused areas out of the main traffic route in the way that basements and attics and the unused spaces of garages accumulate all that old junk that you never use any more but don't want to throw away. Build a wall inside of your wormhole, so travelers don't have to see all the asteroidal rock or super salty water or very surprised natives or whatever it was you scooped up on your alien world. Shove it into the undesirable high curvature places with nasty tides. This wall can also help keep the legitimate passengers away from the inner working of the wormhole, the strange energies that support it, the cables and pipes and crawlspaces that only your maintenance workers should access anyway, and the extreme space-time curvatures that would rip the passengers to shreds if they ever were directly exposed to them.

===Tiny wormholes===

Wormholes that are too small to fit people through could still be useful for sending information. You could use them to rapidly communicate over long distances. If the wormholes were light enough and cheap enough, they might even replace cell phones and laggy land lines with super high bit rate connections.

You could also send energy through tiny wormholes, in the form of light or electric current. Storms that knock down power lines no longer need to cut off your home's electricity. You could also use them to power machinery that need light weight and high power output – electric transportation like airplanes, shipping, or trucking; directed energy weapons; or high specific impulse rockets.

===Causality denial, causality attacks===

We have shown that moving wormholes around can lead to time travel, and that there are plausible physical mechanisms that can prevent time-travel allowing wormhole configurations from happening. People being people, you just know that if this is possible, that it will not be long before these effects are used in politics and warfare.

In a network of wormholes that is about to create a time machine, one likely resolution is that things break until a time machine is no longer possible. In any chain stressed to its point of failure, it is the weakest link that breaks. So in a path through wormholes and normal space-time that is about to form a time machine, it is the weakest wormhole that will collapse (or otherwise have something bad happen to it). So if your rival polity has a wormhole that allows then easy trade with a rich world, and you want to edge in on that trade and simultaneously deny that trade to your rival, you can just send a bigger, stronger wormhole in such a way that forces a time loop. Now your rival's wimpy wormhole breaks, and you get access to the trade opportunities with the world! This is the basics of a causality attack – use the causality-enforcing properties of wormholes to attack your enemy's wormhole infrastructure.

But what if your rival polity anticipated this? What if they have strong wormholes, with an arrangement in space-time that blocks off easy access of your wormholes to a profitable region of space? This is a causality denial action, or a causality fortress. By controlling the time displacement of the wormholes in their network, a polity can control what additional connections can be made. This can be used to enforce borders, prevent infiltration, or restrict civilian wormholes to legitimate uses.

===Moving one wormhole through another wormhole===

If you have a wormhole mouth, you can move it through another wormhole as long as the mouth (and any supporting structures) is small enough to fit through the throat of the wormhole it is going through. The geometry all works out. In fact, this has a rather nice side-effect. If you are looking through the wormhole being moved from back home, you only see it take as long to go through the other wormhole as a wormhole transit usually takes even if it got projected far into the future or past. So the moved wormhole acquires the same time differences as the wormhole it just went through (added on to any time differences it already had, it doesn't get reset or anything). This can let you easily build wormhole networks that connect to themselves in loops. It also makes the network more susceptible to breaking, because there is much less leeway for error with a closed loop as far as making a time machine and breaking the loop. So it can be done, but there are consequences.

===Dropping a wormhole into a black hole===

In principle, if you have a wormhole connection to the inside of a black hole, you can get information out from inside the event horizon. The theorem that prevents things from escaping a black hole was made under the assumption that the energy can't go negative. And as we've already seen, wormholes require regions of negative energy to work. You still wouldn't be able to see further toward the singularity at the center than the wormhole's location. And the wormhole would be inevitably dragged into the singularity in a finite amount of time. But, in principle, during that time you could see what is happening in the black hole.

But there's a big catch with this. You can't actually get the wormhole into the black hole. Not if you assume that time machines can't form, anyway. From the point of view of the wormhole mouth falling into the black hole, it will pass through the event horizon in a finite amount of time. From the point of view of someone watching from outside, the infalling mouth gets more and more time dilated to the point that it never crosses the horizon. This time dilation sets up the conditions for a time machine, and any time machine preventing physics happens on the wormhole before it can ever get in.

== Possible Networks ==
=== Acyclic wormhole networks ===

Recall that wormholes are space-time connections. They bridge space as well as time. Assuming we want to build this network in a stable manner, we want to avoid Closed Timelike Curves from appearing. Let us start with some simple anaylsis of consequences on this. Assume projected wormholes have one mouth sent at relativistic speeds to a distant target and the other mouth is kept at rest at home. From the perspective of the dispatching party, this will lead to the new connection opening very quickly. Done again and again, large volumes of space can be accessed in a short time, a powerful advantage. 

[[File:Wormholes PartialOrder.svg|400px|thumb|The partial order relationship of the wormhole connection]]

Every projected connection thus extends onto the future. Then we must avoid forming loops, or more generally "backtracking" back towards a dispatched node. This nets us a graph with no closed loops - an acyclical graph, also called a tree. Since connections may not return to previously established nodes if they are time-shifting from the past into the future, we also get a partial order: nodes u that dispatch wormholes have a higher order over the nodes v they establish: u < v. This lets us establish a direction on the whole graph. For a particular node n, its '''parent node''' established it, parent(n) < n, and it in turn establishes '''children nodes''', n < child(n). The node that only has children nodes is the tree's '''root node'''. Nodes that have no children are '''leafs''' of the tree. The trees '''height''' is the longest outward path one can take from the root to a leaf. 

In terms of traversal, such nodes have some interesting and useful properties. Since wormhole traversals may be assumed to be effectively instantaneously bridging large distances, travel time is mostly defined by the travel distance between wormhole mouths. And as such, we want to minimize the amount of transits we have to make to get from a node A to a node B the fastest. It turns out, in order to achieve this, we want to have as large a choice of connections at each node. In the ideal case, any node except root can be reached in two transits: one transit to root, and from there to the other node. In the less ideal case we have to make one further stop-over, choosing from a number of connections at a second node to reach our destinations. The less ideal case from that sees us make two transits through such "switch yards". Assuming each node except the leafs in such a wormhole network has exactly k connections and there are n nodes in the whole network, we can reach any destination from root in logk (n) choices of wormhole transit. This is a powerful way to get to connections quickly through centralization. (In computer science, such k-nary trees are of great interest for quickly search-able data structures for this reason.) Configurations that are less "balanced" net appropriately lesser advantages and longer transit times. 

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:Wormholes AcyclicGraphExample.svg |600px|frameless]]
<td>[[File:Wormholes Trees.svg|600px|frameless]]
<tr>
<td width=400>An example of an acyclic wormhole network in space-time.
<td width=400>The connected tree graph of the network exampke, and a k-nary tree example.
</table>

Also, traffic to most other nodes ''has'' to go through a root-wards node. This gives these nodes significant presence in the larger network. Routing through them is unavoidable if you want to reach most of the rest of the network. Root nodes can profit from taxation, services to travelers, control of routing infrastructure for information, and so on. But there is also a catch. Assuming leaf nodes each send the same amount of traffic inward. The amount of traffic load increases polynomial as you go towards root. Your wormhole infrastructure (and what transports cargo between wormholes, in space or on planets) has to cope with this scaling. Congestion might become an issue. 

=== Mesh wormhole networks ===

The acyclic network may have some bothersome problems. The central routing focuses power and can keep transit times down quite significantly depending on the specific network infrastructure, but it’s also congestion-prone, and some close locations may be widely separated on the network. Or maybe you want to avoid centralized routing dependencies for political reasons. Root might like being root; everyone else being subject to root’s whims, not so much. So, can we build networks where there are two routes between the same two nodes? We can, but it comes with some necessary adjustment. 

Recall why the acyclic network is acyclic: to avoid forming CTCs due to the time connection between future and past from the projected wormholes. But if we let both wormhole mouths experience the same time dilation, the problem that ultimately leads to CTCs - one connection routing displacing further into the future than the other connection, and thus allowing one to arrive at a node before ones departure - is avoided entirely, since there is no time displacement to be accumulated. Thus we get meshes, where there may be many connections and cycles between nodes. An observer at one node watching for his transition from another node 10 light years away would have to wait for ten years to watch his light arrive. Closed timelike loops are not possible so long as this network is kept balanced. 

[[File:Wormholes MeshNetwork.svg|thumb|center|700px|An example of such a mesh network]]

The disadvantage - this network must take great care to ensure that the time dilation that inenvitably occurs due to different star velocities and the galactic gravitational field don’t accidentally create a time shift and CTC in the network. And also we cannot exploit time dilation to open connections very quickly, from the home node’s perspective. One has to wait out the full years it takes the wormhole’s far mouth to get to a remote destination, before the connection can be used. 

=== Semi-privileged wormhole networks ===

The interesting thing is, it is very much possible to build networks that make use both of relativistic projection to seemingly accelerate the network construction time, ''and'' contain cycles, with alternate routes to get to another node. The acyclic construction is a strategy to avoid CTC formation, but we should remember the ultimate constraint reason: we avoid cycles so that there are no connections which, through a certain time shift on them, a CTC could be formed. Acyclic construction is one strategy. Only building cyclic connections with careful time shift balancing is another! 

The foundation for such networks is an acyclic "base graph" that may be isolated from the larger network graph. We also call this the '''red graph'''. The base graph maintains all the partially ordered properties of the simple acyclic graph. We can call this the "privileged" graph, since its nodes and their connections define the principle positions in space-time of the network, and we must avoid generating CTCs on this definition. How do we avoid forming CTCs but still create alternative paths and thus loops? 

Let’s assume we have two nodes A and B ten light years apart. They both had connections established from a parent node P. Those connections are time-shifted over 99 years and part of the privileged base graph.

We couldn’t build a time-shifted connection between A and B without causing a CTC to be formed. But we are entirely fine sending signals '''Fast-as-Light''' from A to B and back. This is not a violation. Now consider: how does a wormhole with no time shift across it behave? In the mesh network case, such connections don’t cause CTCs to appear. And it doesn’t cause one to appear here either. How can we build such connections? We can of course use projection with relativistic speeds at both ends, like how we would build Mesh networks. But there is an even faster solution. If we can move wormholes through wormholes, we can shift the new mouths through the existing network to create this connection! 

The main catch is that we have to tightly watch out for temporal drift on such new connections. Through adding such edges that overall form a "blue graph" on top of the red graph, we are creating loops on the red graph underpinning the network. If one end of a wormhole mouth drifts more in time than the other (which will inevitably happen because stars won’t have the same mass, experience the same galactic gravity, or have the same velocity relative to each other) the conditions for a CTC will start to appear. The only way to pre-empt this is with continuous maintenance, taking connections offline and adjusting their temporal position using synchrotons, and having "safety space-time" spacing between wormholes. In this way, there is some "room" where signals can arrive faster and faster, but not before they were sent, allowing the connection to be used most of the time with only short intervals of adjustment. 

This works even when both ends connected don’t have the same time distance from the root node - the important thing is that the new shortcut accomodates this difference with a time shift of its own. We can calculate this difference easily. For two non-root nodes A,B, for each node we sum the temporal difference over the path from root to the nodes along the red graph, then subtract the difference from each other. The resulting remaining temporal difference is that the new connection on the blue graphh must have to keep the network stable. This means we can even do things like connect systems far out directly to core systems, or bridge from one edge of the graph to the other - arbitrary routings are possible. These routings can allow faster signal transmission and mass transport than through root, mitigating bandwith issues. The main problem is to watch out for network stability. As nodes accumulate, CTC avoidance becomes more and more delicate, and safety spacing must be maintained diligently. This can add to the effective travel time through such alternate connections, and thus maintains some of the "privilege" of the acyclic base graph. 

[[File:Wormholes BuildingCyclics.svg|800px|center]]

Arguably this network configuration is the most powerful, marrying many advantages and removing some disadvantages. It is however also bound to the need for safety spacing. Depending on your infrastructure assumptions, this can pose issues. If you want to implement serious stretches of safety space, wormhole links not part of the acyclic base graph would likely have to be located at interplanetary distances away from the main links, with light hours to light days of safety spacing. If your main network runs on planetary trains, this is rather a bother. Adjusting the internal throat length of wormholes can help in this regard, but of course now poses the issues of how to safely navigate such a long wormhole. Maybe you could extend something like space elevator tethers through such stretchy throats. With no gravity inside the wormhole itself, such cables could probably stretch quite significant distances. If your culture is handling wormholes mostly in space, you still have to account for added travel times through the safety spacing. Communications links would probably be easier connected between arbitrary nodes than macroscopic, craft-traversable wormholes.

[[File:Wormholes ComplexExample.svg|800px|thumb|center|An example of a complex semi-privileged graph wormhole network]]

==Credit==
Authors: Luke Campbell and Sevoris
 Tshhmon

==References==

[[Category:Physics]][[Category:Infrastructure]][[Category:Physics & Engineering‏‎]][[Category:Physics & Math & Engineering]][[Category:Transportation & Infrastructure‏‎]][[Category:Metric Engineering]]

Warp Drives

2026-07-27T00:17:45Z

Lwcamp: /* Lentz warp drive */

{{PageConstructionNotice}}

Science fiction often features spacecraft that can seemingly move across space and get between the place of departure and the destination much faster than light could have done. This appears to contradict the theory of relativity, which predicts unequivocally that nothing can move through space faster than light. Because relativity has been incredibly successful at describing nature, with its many other predictions regularly being confirmed to extraordinary accuracy and within the bounds of uncertainty of all the experiments that tested them, it gives confidence that relativity is a correct description of reality. Which seems to rather throw a wet towel on our hopes for rapid travel between stars.

However, while relativity does not allow things to move through space faster than light, it places no such restrictions on how fast space-time itself can expand, contract, or move around. This leads to the idea of a warp drive – the spacecraft remains stationary within a region of highly curved space-time, and that region moves at super-luminal speeds rather than the spacecraft.

== The Alcubierre warp drive ==

The first warp drive geometry that satisfied the Einstein field equations of relativity was proposed by Miguel Alcubierre<ref>M. Alcubierre, "The warp drive: hyper-fast travel within general relativity." Classical and Quantum Gravity. 11 (5): L73–L77 (1994). [https://arxiv.org/abs/gr-qc/0009013 arXiv:gr-qc/0009013]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1994CQGra..11L..73A 1994CQGra..11L..73A]. doi:[https://doi.org/10.1088%2F0264-9381%2F11%2F5%2F001 10.1088/0264-9381/11/5/001]. S2CID [https://api.semanticscholar.org/CorpusID:4797900 4797900].</ref>. In this geometry, a sphere of space-time moves at an arbitrary speed (potentially but not necessarily a speed much faster than light). Objects within the sphere are moved along with the sphere; an object at rest within the sphere would be moved along with the sphere indefinitely. Space is expanding at the rear boundary and contracting at the front boundary in order to keep the sphere moving. In order to satisfy the Einstein field equations, the boundary of the sphere must have a negative energy density. The challenges of space-time geometries with negative energy densities are described in our page on [[Wormholes#Exotic_energy_conditions|wormholes]], for our purposes it is enough to note that negative energy densities can pose problems if not handled carefully, there may be limits on how much negative energy you can have without nearby positive energy density, and it may not be possible to get enough negative energy to support a warp drive; although none of this is ruled out by physics – yet!

<table align=center>
<tr><td width=650>[[File:Energy_magnitude_of_Alcubierre_drive.png]]
<td width=650>[[File:Expansion_of_Alcubierre_drive.png]]
<tr><td>The magnitude of the energy of the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. (The energy density is negative everywhere in the shell, here only the magnitude is shown to aid visual comprehension.)
<td>The expansion of space in the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. Negative expansion means that space is contracting in that direction, positive expansion that it is expanding.
</table>

NOTE: What the? The energy distribution is symmetric! How does the drive know which way to go? There must be significant contributions of other components of the stress-energy tensor, and those have got to be asymmetric along the forward/backward axis. Check this when I get time.

(And what's with the ring in all the artwork, anyway?)

=== So what is going on? ===

What is it that makes a warp drive a warp drive, and how does it work?

A warp drive is a concept that comes out of the science of general relativity. General relativity collects space and time together into a single "structure" called spacetime. This structure is curved, and its curvature is determined by the material that is in it. Absent forces acting between matter, particles of matter move through spacetime on the shortest possible paths in this curved environment.
<blockquote>
"Spacetime tells matter how to move; matter tells spacetime how to curve." - John Archibald Wheeler
</blockquote>
All of this is encompassed in the Einstein field equation:
<div align=center>
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 8πG
<td rowspan="2" nowrap="nowrap" align="right"> T = R -  
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right"> R g
<tr>
<td style="border-top:solid 1px black;"> c4
<td style="border-top:solid 1px black;"> 2
</table>
</div>
Unfortunately, by convention, both the symbols G and R get used for a lot of different things. Here, symbols in bold are second rank tensors – constructs of arrays of 4×4 numbers where 4 is necessary because that is the number of dimensions of spacetime. Conventionally, these numbers are indexed by a value going from 0 to 3, with 0 being the part corresponding to the coordinate of time and 1 to 3 being the Cartesian coordinates that you might know as x, y, and z.

T is the stress-energy tensor: the (0,0) component represents the energy density at a given point; the (0,i) and (i, 0) components (with i going from 1 to 3) represent the momentum density; and the (i,j) components represent the stresses present, such as pressure or tension or shear.

The other term, R - R g/2, involves a partial description of the curvature. A full description of the curvature requires a construct of 4×4×4×4 components; too much for the stress energy tensor to fully constrain. Which is okay, because this allows curvature to exist away from where matter is, allowing gravity to have long range effects across empty space as well as for phenomena like gravitational waves. But it is enough that a given distribution of matter T gives enough constraints (along with specifying how the geometry behaves far away) that you can generally figure out what the actual gravity is.

This is the usual story that we are told. But there is an alternate way of looking at the same geometry such that the math works out the same, but it is described conceptually in a very different way. This divides spacetime up into "layers", called foliations, that represent slices of space and going from one foliation to the next is increasing in time. This is useful for computation because you can assign coordinates to your system. There are two additional constructs – the lapse function and the shift vector field. Starting at an event on one foliation, the lapse function at that position tells you that after a certain "proper time" (time perceived by an observer at that event) the time coordinate of the observer will be the proper time times the lapse function at that event. And the new spatial coordinates of the observer will be the proper time times the shift vector at that event. (Strictly speaking, this is only exactly true in the limit of the proper time approaching zero or when the lapse function and shift vectors are constant. But hopefully this gives a good enough insight to what these entities are so that we can go on to describe warp drives.)

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Lapse_shift.png]]
<tr><td>A folliation of space-time into slices of constant time coordinates (grey). At an event (black circle), the lapse function (yellow) shows how far into the future the point at that event will be projected in a unit amount of proper time experienced by an observer initially at the event. The shift vector (red) shows how that point changes coordinates after the unit amount of proper time. The cyan vector shows the new coordinates the point is projected onto. The lower black surface indicates a foliation of spacetime at the same time coordinate; after unit proper time, the events on the lower foliation get projected to the curved upper black foliation according to the shift vectors and lapse functions at each of their respective points.
</table>
</table>

Warp drives are generally restricted to spacetime geometries with a constant lapse function - all observers go forward by the same time coordinate after experiencing the same amount of proper time. Thus, all warp drives have no time dilation.
Far from the warp bubble, the shift vector vanishes – each spatial coordinate is mapped to the same spatial coordinate after any amount of proper time so long as the warp drive is not near. But the inside of the warp bubble is an extended region where the shift vector field is non-zero and constant. In this bubble, all spatial coordinates are mapped to a coordinate that shifts linearly in time with the proper time of the observer. Astute readers will see that this linear change in position with time is very similar to a velocity. This region of constant shift vector then moves with the shift vector velocity to keep stuff originally in the bubble so that it stays in the bubble. In between the inside of the bubble and the far away regions with zero shift vector, the shift vector field falls away to match one region to the other.

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Warp_shift.png]]
<tr><td>The shift vector of a one-dimensional slice through warp geometry, in red. At the edges there is no shift. At the center (inside the warp bubble) there is a region of uniform shift so that at a later time the inside of the warp bubble is rigidly translated to different spatial coordinates (in blue). The two foliations remain the same distance apart, because warp drives maintain a unit lapse function.
</table>
</table>

Traditionally, scientists would start with an interesting distribution of matter given by T and then find out what kind of curvature it created. This works well for planets and stars and black holes and cosmic strings and other things which are observed in the universe and you want to see what sort of effect they had. But starting in the 1980's, some physicists started looking at Einstein's field equations the other way around. They start with a desired geometry, find the curvature part of the field equation R - R g/2, and use that to find out what T has to be in order to get that curvature. This was originally done for [[Wormholes|wormholes]], but in 1994 Miguel Alcubierre used this same trick to figure out what could create a region of constant shift vector that moved with the shift vector - the warp drive. With this demand for a warp geometry, the required distribution of matter and energy and momentum and stress can be determined.

The original Alcubierre shift vector N→ produces a spherical warp bubble with an axi-symmetric distribution of, well, everything along the direction of motion with warp velocity vector v→.
<div align="center">
N→ = v→ f(rs)
</div>
Here, rs is the distance from the center of the warp bubble, and f(rs) can be any function that is 1 near rs = 0 and is 0 for rs well beyond the radius of the warp bubble.

=== Challenges and possible resolutions ===

Almost as soon as Alcubierre proposed his warp drive, others began picking it apart.
<ol>
<li>Pfenning and Ford<ref name="PfenningFord_2001">M. J. Pfenning and L. H. Ford, "The unphysical nature of 'Warp Drive'", Classical and Quantum Gravity. 14 (7): 1743–1751 (1997). arXiv:[https://arxiv.org/abs/gr-qc/9702026 gr-qc/9702026]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1997CQGra..14.1743P 1997CQGra..14.1743P]. doi:[https://doi.org/10.1088%2F0264-9381%2F14%2F7%2F011 10.1088/0264-9381/14/7/011]. S2CID [https://api.semanticscholar.org/CorpusID:15279207 15279207].</ref> found that if the negative energy density wall around the bubble obeys [[Wormholes#Quantum_energy_inequalities|quantum energy inequalities]], then for warp speeds of around light speed the shell thickness would be on the order of a hundred Planck lengths; a distance far smaller than any other known physical phenomenon or object.
<li>In the same work, Pfenning and Ford also found that for a bubble a hundred meters in radius, a warp speed of around the speed of light, and the required thickness of about 100 Planck lengths the energy in the bubble shell would be E ≈ -1063 kg c2. The magnitude of this latter value is ten orders of magnitude larger than the energy of the entire visible universe. They do note, however, that if quantum energy inequalities can be ignored then a warp drive with a hundred meters radius but a shell thickness of one meter would "only" have an energy magnitude of about a quarter solar mass.
<li>As with any method of faster than light travel, the warp drive could be used to make a time machine.
<li>Perhaps most seriously, if the warp drive is going faster than light speed, the negative energy regions on the outside of the shell won't be in the warp parts of the bubble. They will have to be moving through space at faster than light speed if the bubble is to maintain its integrity, which is the very problem that the warp drive was designed to avoid. The outside of the warp bubble shell that maintains the warp bubble would fall away and the warp bubble would quickly erode to sub-luminal speeds.<ref name="Van Den Broeck 2000">Chris Van Den Broeck, "On the (im)possibility of warp bubbles", arXiv:gr-qc/9906050 https://doi.org/10.48550/arXiv.gr-qc/9906050</ref>
<li>A more general problem that includes the previous one is that the geometry of the warp drive determines the distribution of energy, momentum, and stress (pressure, shear, and tension) by Einstein's field equation, which states that the stress energy tensor is directly proportional to a reduced order curvature tensor of space-time. However, the stress-energy tensor must also obey the continuity equation – basically, for the energy to change in a given volume of space that energy must come in or go out of the boundary of that volume and similarly for the momentum. So a warp drive can have the warp condition for one moment in time. But after that, the stress-energy that maintains the warp will start to move according to the forces on it and according to the natural paths that it takes through the warped space-time. There is no guarantee that the configuration of the stress-energy at later times will be sufficient to maintain the warp.
<blockquote>
From the perspective of simulating the warp drive dynamically, the key challenge is stability. One can straightforwardly generate the initial metric that describes the Alcubierre metric and infer the instantaneous matter configuration that must support it using the Einstein Equation, but a time-domain evolution of the coupled matter-gravity system necessitates that we specify the equation of state of the matter, which in turn determines how it evolves. In a perfect fluid, this involves specifying the relation between the fluid pressure and density, but in a more general case, the full relation between the
stress-energy components must be given. There is (to our knowledge) no known equation of state that would maintain the warp drive metric in a stable configuration over time - therefore, whilst one can require that initially, the warp bubble is constant, it will quickly evolve away from that state and, in most cases, the warp fluid and spacetime deformations will disperse or collapse into a central point.
 
– Clough, Dietrich, and Khan<ref>Katy Clough, Tim Dietrich, and Sebastian Khan, "What no one has seen before: gravitational waveforms from warp drive collapse", arXiv:2406.02466 [gr-qc] https://doi.org/10.48550/arXiv.2406.02466</ref>
</blockquote>
<li>In order to get the matter (or more generally, stress-energy) that forms the warp shell up to speeds that could allow the shell to exist, you need to be able to, well, accelerate that matter to warp speeds. Which gives a chicken-and-egg problem - in order to achieve superluminal travel you need to be able to get matter or energy moving superluminally in the first place<ref name="Bobrick and Martire">Alexey Bobrick and Gianni Martire, "Introducing Physical Warp Drives", arXiv:2102.06824 [gr-qc] https://doi.org/10.48550/arXiv.2102.06824</ref><ref>D. H. Coule, "No warp drive", Class. Quantum Grav. 15 (1998) 2523–2527. https://doi.org/10.1088/0264-9381/15/8/026</ref>, which rather makes warp drives unnecessary from the get-go.
<li>A warp bubble moving faster than light will not be able to get any signals in front of it as it will overtake its own signal. In fact, anyone in the interior of the bubble would not be able to send signals to the front surface of the bubble. In science-speak, a horizon forms at the front of the bubble<ref name="Natario">José Natário, "Warp drive with zero expansion", Classical and Quantum Gravity. 19 (6): 1157–1166 (2002). arXiv:[https://arxiv.org/abs/gr-qc/0110086 gr-qc/0110086]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2002CQGra..19.1157N 2002CQGra..19.1157N]. doi:[https://doi.org/10.1088%2F0264-9381%2F19%2F6%2F308 10.1088/0264-9381/19/6/308]. S2CID [https://api.semanticscholar.org/CorpusID:15859984 15859984].</ref>. As a consequence, the bubble cannot be controlled, steered, nor stopped from within the bubble itself.
<li>A Horizon will naturally emit Hawking radiation. It is estimated that the radiation will raise the temperature of the interior of the bubble to approximately 1032 K<ref name="Finazzi 2009">Stefano Finazzi, Stefano Liberati, and Carlos Barceló, "Semiclassical instability of dynamical warp drives", Phys. Rev. D 79, 124017 (12 June, 2009) https://doi.org/10.1103/PhysRevD.79.124017 , arXiv:0904.0141 [gr-qc] https://doi.org/10.48550/arXiv.0904.0141</ref> (at this level of accuracy, the result is the same if you replace temperatures in K by the same number in °C or °F). This temperature, however, may be lower if quantum inequalities do not have to be fulfilled.
<li>Semi-classical instabilities at the horizon are expected to lead to a divergence of the stress-energy tensor at the boundary in a short period of time, thus leading to the destruction of the warp bubble<ref name="Finazzi 2009"></ref>.
</ol>
Helpfully, Van Den Broeck<ref name="VanDenBroeck_2000">C. Van Den Broeck, "Alcubierre’s Warp Drive: Problems and Prospects" AIP Conference Proceedings. 504: 1105–1110 (2000). Bibcode:[https://ui.adsabs.harvard.edu/abs/2000AIPC..504.1105V 2000AIPC..504.1105V]. doi:[https://doi.org/10.1063%2F1.1290913 10.1063/1.1290913].</ref> suggested several solutions – or at least mitigations – for some of these problems
<ol>
<li>Quantum inequalities had not been shown to be true in general for highly curved space-times.
<li>A proposed warp drive geometry<ref name="VanDenBroeck_1999">C. Van Den Broeck, "A 'warp drive' with more reasonable total energy requirements". Classical and Quantum Gravity. 16 (12): 3973–3979 (1999). arXiv:[https://arxiv.org/abs/gr-qc/9905084 gr-qc/9905084]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1999CQGra..16.3973V 1999CQGra..16.3973V]. doi:[https://doi.org/10.1088%2F0264-9381%2F16%2F12%2F314 10.1088/0264-9381/16/12/314]. S2CID [https://api.semanticscholar.org/CorpusID:15466313 15466313]. </ref> (see the van Den Broeck drive, below) would be able to minimize the magnitude of the energy required to about that of the mass of our sun. While still large, it is not unphysically large.
<li>Time travel is always going to be a worry with faster than light travel. There's no neat solution to this one.
<li>It may be possible for the negative energy outside of the superluminal region to compress into the superluminal region to form a shock in space-time. The front surface of the bubble would then be a singularity. It is not clear if such a sharp jump in physical properties is possible, but neither is sure that it is impossible, either. However, while this might help with the negative energy in front of the bubble getting swept up because it cannot move faster than light, it does little to help with the material at the back of the bubble on the outside getting left behind because it cannot keep up.
</ol>
Other mitigating research has also shown
<ol start=9>
<li>Semi-classical divergence only occurs in warp drives with only one spatial dimensions. Those with two or more spatial dimensions (as in our universe) only contain points where divergences occur, and these divergences can be managed. In fact, their effects can be reduced by making the warp shape more elongated<ref>Carlos Barceló, Valentin Boyanov, Luis J. Garay, Eduardo Martín-Martínez,
and Jose M. Sánchez Velázquez, "Warp drive aerodynamics", JHEP08 (2022) 288 https://doi.org/10.1007/JHEP08(2022)288 , arXiv:2207.06458 [gr-qc] https://doi.org/10.48550/arXiv.2207.06458</ref>.
</ol>

=== Alcubierre warp interactions with light and matter ===

Space is not entirely empty. It is filled with a diffuse plasma in the form of the interstellar medium, as well as cosmic radiation, light from stars, and cosmic microwave background radiation. A warp drive propagating through space will encounter this stuff. What happens when this matter and radiation have a warp bubble pass across them?

The first analysis of matter encountering a warp bubble was performed by Pfenning and Ford<ref name="PfenningFord_2001"></ref>. They looked at the warp bubble interaction with a massive object at rest with the frame of reference of the warp drive (keep in mind that the rest frame is the same inside and outside the warp bubble; in this rest frame objects inside the bubble have no momentum even though they are, in some sense, changing location rapidly with time). When the warp bubble passes, the object experiences an acceleration in the direction of the warp bubble motion. When the warp shell passes and the object is inside the bubble, it will be moving with approximately the speed of the bubble. Pfenning and Ford analyzed this problem with a continuous distribution of shell energy that, strictly speaking, never falls to zero except at the bubble center and at spatial infinity, so unless the object passes through the center in Pfenning and Ford's description it will never quite get up to the bubble's speed. In this case, the object will move almost as fast as the object but will pass through the bubble in a finite time, after which it will again be at rest with respect to the reference frame but displaced along the direction of the warp bubble motion by some distance. With this description of the warp bubble, a spacecraft of finite size will always be moving a little slower than the warp bubble and would have to use rockets to keep up with it. in addition, the spacecraft would experience tidal forces that would cause stress on the spacecraft's structure.

Pfenning and Ford also examined cases where the shell is of a finite (possibly infinitesimal) width and falls to zero both inside and outside the bubble. In this case any matter encountering the bubble would be collected at the bow of the bubble and thereafter move along with it.

McMonigal et al. analyzed the situation for both massive particles and light moving along the axis of travel of the bubble<ref name="McMonigalEtAl_2012">B. McMonigal, G. F. Lewis, and P. O'Byrne, "Alcubierre warp drive: On the matter of matter". Physical Review D. 85 (6) 064024 (20 March 2012). arXiv:[https://arxiv.org/abs/1202.5708 1202.5708]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2012PhRvD..85f4024M 2012PhRvD..85f4024M]. doi:[https://doi.org/10.1103%2FPhysRevD.85.064024 10.1103/PhysRevD.85.064024]. S2CID [https://api.semanticscholar.org/CorpusID:3993148 3993148].</ref>. They found that light moving opposite the warp direction passed through the bubble without incident, being only somewhat delayed by passing through the bubble. A warp bubble that is warping at sub-luminal speeds can have light catch up from behind it. This light is able to pass through the bubble, and is somewhat advanced in its path by the speed of the bubble. For super-luminal warp bubbles, however, the situation is different. The bubble will catch up to light moving it its own direction that is originally in front of it. This light cannot escape forward, the bubble being too fast. Nor can it escape backward, as the light is propagating forward and the interior of the bubble is at rest. Thus, the light gets caught at the bow of the warp bubble, unable to escape for so long as the warp bubble is active. This light is strongly blue shifted to extremely energetic x-rays and gamma rays. The space behind the bubble is swept clear of forward-moving light.

The situation for matter over-run by a super-luminal warp bubble is similar. Matter moving backward passes through the bubble, being only swept a ways along the bubble's path as sign of its passing but otherwise continuing on their way. Matter at rest behaves the same way as Pfenning and Ford discovered. Meanwhile, matter moving in the same direction of the warp is overtaken and collects at the bow of the bubble, unable to leave for so long as the bubble is warping. This matter "surfing" on the bubble bow is highly accelerated to relativistic speeds, experiencing extreme time dilation.

For a sub-luminal bubble, there is a speed faster than the warp speed where particles coming up from behind overtake the bubble and pass through, out the front. Particles below this speed but still faster than the warp speed still overtake the bubble, but then bounce backwards out of the bubble from behind. They are still moving forward, but are now moving more slowly than the warp speed. Particles moving forward but slower than the bubble will be overtaken and then bounced out the front with a speed higher than the warp speed. Those particles moving backward will pass through the bubble and exit with their initial speed.

The spacecraft inside the bubble will observe that light moving backward compared to the warp direction is blue-shifted, and matter moving backward passes through with increased energy. Meanwhile, a sub-luminal bubble will see light that was moving forward as red-shifted and matter catching up to the bubble from behind will pass by with reduced energy (superluminal bubbles, of course, do not have any matter or radiation catching up to them from behind).

When a superluminal warp bubble that has been collecting matter and radiation for the duration of its journey and blue-shifting it to much higher energies is turned off, all that matter and energy is released as a blast of radiation in the direction the bubble was warping.

McMonigal et al. conclude by noting that any spacecraft in the warp bubble would need shielding to protect against the blue-shifted radiation and matter of increased energy. In addition the destination would be "blasted into oblivion" by the release of matter and radiation that had been caught in the bubble during the trip.

Clark et al.<ref>Chad Clark, William A. Hiscock, and Shane L. Larson, "Null geodesics in the Alcubierre warp drive spacetime: the view from the bridge", arXiv:gr-qc/9907019 (2018) https://doi.org/10.48550/arXiv.gr-qc/9907019</ref> examine what it would look like from within an operating warp drive. They find that light moving at 90 degrees to the direction of motion will look like it is still coming from 90 degrees. However, light from the front hemisphere is squished progressively toward the forward point the faster you go. For a warp bubble moving faster than the speed of light, there is a cone behind the warp bubble where no light can reach it. However, you don't see a black spot behind you – light from the regions in the backward hemisphere from outside that cone that can reach you is distorted toward the direction behind you to fill in that spot. Light from the forward direction is blue-shifted and light from behind is red-shifted. At speeds of around 200 times light speed, the cosmic microwave background would be blue shifted to energies similar to that of the solar photosphere, heating anything in the warp bubble as if they were close to the sun.

== Van Den Broeck warp drive ==

If an Alcubierre warp bubble a hundred meters across requires a magnitude of energy greater than the entire energy in the observable universe, one option to reduce the magnitude of energy used is to make the warp bubble smaller. Much smaller. Pfenning and Ford<ref name="PfenningFord_2001"></ref> suggested making the warp bubble smaller than an atom. Of course, that brings up the problem of how to stuff a spacecraft in there. Van Den Broeck proposed a solution<ref name="VanDenBroeck_1999"></ref>: make the warp bubble only about the size of an atomic nucleus but expand the space inside the bubble enormously so that a spacecraft could fit in. This is somewhat like a nuclear sized wormhole that leads to a pocket universe that holds the spacecraft. Unfortunately, the metric doesn't fit neatly into any [[Wormholes#Three_dimensions|embedding diagram]] that I can figure out, but the math works out so that spatial coordinates inside the bubble are expanded by a factor of 1017 and a spacecraft would have a few hundred meters of bubble interior to putz around in. This trick manages to reduce the magnitude of energy to only about the mass energy of a few stars similar to our own. Other than that, it is otherwise a normal Alcubierre drive.

Further work<ref>S. Krasnikov, "The quantum inequalities do not forbid spacetime shortcuts", Physical Review D. 67 (10) 104013. https://doi.org/10.1103%2FPhysRevD.67.104013, arXiv:gr-qc/0207057 (2003) https://doi.org/10.48550/arXiv.gr-qc/0207057 </ref> has managed to use Van Den Broeck's trick to reduce the magnitude of the negative energy to only about a milligram.

== Natário warp drive ==

The Alcubierre drive is not the only way to construct a warp drive. José Natário<ref name="Natario"></ref> showed that it was just one example of an entire class of warp drives. These are space-time geometries with a smooth and localized shift vector field and a unit lapse function.

Natário then went on to find in that class a set of warp drives that do not have any expansion or contraction of space at all. Instead, space encountering the front boundary of the warp bubble instead "slides" around the outside of the bubble until it gets to the corresponding place on the back and is left behind there. Another way to think of it is that space entering the bubble shell is compressed in the radial direction of the shell but is simultaneously expanded in the tangential direction so that there is no net change in volume. When it gets to the back, the opposite occurs. In the language of mathematics, the shift vector field is divergenceless.

<table align=center>
<tr><td width=360>[[File:Natario_drive_flow_of_space.png]]
<td width=445>[[File:Natario_warp_drive_horizons.png]]
<tr><td style="vertical-align: top;">A representation of the motion of space in a zero-expansion Natário drive with a vanishingly thin warp shell around the bubble.
<td>The horizons of the generalized Natário class of warp drives (including the Alcubierre warp drive and all fell-Heisenberg drives) at superluminal speed in the direction of the green arrow. No event inside the bubble can affect events outside the bubble in front of the blue line. No events outside the bubble behind the red line can affect events inside the bubble. The lines have an angle α with respect to the direction of motion with sin(α) = 1/vs for vs the speed of the bubble relative to light speed. This is analogous to the Mach cone of supersonic objects.
</table>

NOTE: There are two different things that can be meant by the term "Natário warp drive". The generalized Natário warp drive is the general theory of all warp drives that includes within it the Alcubierre, Fell-Heisenberg, and Lentz warp drives, among infinitely many others. The zero-expansion Natário warp drive is the specific implementation of the generalized Natário warp drive that has a divergenceless shift vector field.

The generalized class of warp drives developed by Natário (including both the Alcubierre and this zero-expansion warp drive) are shown to always have regions in the warp shell where the energy density is negative to at least some observers. You can't get away from it – to warp, you need negative energy density.

The Natário class of warp drives blueshift light coming in from the front and redshift light catching up from the back. For the case of a super-luminal warp bubble, there will of course be a horizon at the back of the bubble and light will not be able to get through from behind. For a bubble speed vs relative to the speed of light, light coming from straight ahead will be blueshifted up in frequency and photon energy by a factor of 1 + vs. For sub-luminal travel, light coming from behind will be redshifted by a factor 1 - vs. In general for n as the unit direction along which the light is propagating, the blueshift factor will be 1 + vs ⋅ n. If the warp shell is infinitesimally thin, light from outside that reaches the center of the bubble will not be distorted in direction although it will be frequency shifted, but observers still inside the bubble but displaced from the center will see the field of view distorted as well as frequency shifted.

Most of the difficulties that apply to the Alcubierre warp drive also apply to the generalized Natário warp drive (and thus to the zero-expansion Natário drive and Fell-Heisenberg drive as well). In particular, quantum energy inequalities, high magnitudes of the energy required, formation of time machines, instabilities related to the stress-energy making up the warp shell being unable to keep up with the bubble, the chicken-and-egg problem of needing FTL to achieve FTL, the formation of horizons making warp bubbles uncontrollable, and Hawking radiation at the horizons, are all still concerns for generic warp drives.

== Fell-Heisenberg warp drives ==

S. D. B. Fell and L. Heisenberg investigated what sort of warp structure is necessary to produce a warp drive with positive energy density everywhere<ref name="Fell-Heisenberg">S. D. B. Fell and L> Heisenberg, "Positive energy warp drive from hidden geometric structures", Classical and Quantum Gravity 38 155020 (2021) https://doi.org/10.1088/1361-6382/ac0e47 https://arxiv.org/abs/2104.06488</ref>. They found that warp drives of the generic Natário form with irrotational shift vectors had a simple expression for the energy density and vanishing momentum (irrotational means that the shift vector can be described as the gradient of a scalar potential field or, equivalently, that the curl of the shift vector field vanishes everywhere). Fell and Heisenberg proposed several potential fields that either had positive energy density everywhere except at discontinuities, or that were continuous but had regions that violated the weak energy condition (i. e. negative energy densities in some frames of reference). One specific example they provided of a warp field had an energy that is about 10,000 times less than the mass-energy of our sun. Or only about half the mass-energy of Jupiter. This is a significant improvement over many other proposed drives.

Despite the introduction discussing warp drive configurations that satisfy the various energy conditions, the configurations described in the paper are shown to locally violate the weak and strong energy conditions. Nonetheless, the energy density is still mostly positive. Santiago, Schuster, and Visser's<ref name="Santiago Schuster Visser 2022">J. Santiago, S. Schuster, and M. Visser, "Generic warp drives violate the null energy condition", Physical Review D 105, 064038 (2022) https://doi.org/10.1103/PhysRevD.105.064038</ref> work shows that the various energy conditions must still be violated by this warp drive; to not violate these, energy density must be positive in all reference frames not just those of the co-moving observer.

Any vector field can be described by its divergence, its curl, and a constant. The constant part of a warp drive's shift vector can be discarded because then the warp drive would not be localized. It is interesting, therefore, that the Natário zero-expansion drive has zero divergence and can be described only by its curl, while the Fell-Heisenberg drive has zero curl and can be described only by its divergence. These two classes of drives thus rely on opposite choices of setting one of the two different parts describing a vector field to zero. As might be expected, the Fell-Heisenberg drive is not zero expansion – space-time will change in volume as it passes through the warp bubble (although after going all the way through it will return to its original volume). However, objects in space-time will retain their original orientation as they pass through the warp bubble.

Any vector field with zero curl can be represented as the gradient of a suitable scalar function. Thus, the Fell-Heisenberg drive can be completely described by just scalar field (one number at all points of space and time) rather than a vector field.

Previous examples of warp drives, such as the Alcubierre and zero-expansion Natário drives, tend toward a flat space-time as you go far enough away faster than would occur if there were any mass present in the warp drive. That is, the ADM mass of such drives is zero. However, one of the Fell-Heisenberg examples described a spacetime where the geometry approaches that of what you would get for a massive object at the warp location, giving one of the first examples of a warp drive with mass.

In the Fell-Heisenberg drive the momentum vanishes everywhere. Yet the energy occupies regions where the shift vector is varying rapidly. The lack of momentum means that the energy will not move to keep up with the differential expansion and movement of the spacetime. As a consequence, at later times the energy will have a different distribution than what is necessary to maintain the given warp configuration. The exact time evolution is not solved but this likely leads to collapse of warp bubble. Note that this is not a problem unique to the Fell-Heisenberg drive. Assuming a geometry that describes a warp produces a specific distribution of the stress-energy tensor that satisfies Einstein's equation. But there is no guarantee that this stress energy tensor satisfies the continuity equation, which describes how the distribution of matter and momentum change with time. The issue of stability of any warp drive is not well understood, but most warp drive geometries will be unstable in the sense that their matter that generates the warp fields will not be able to keep up with the motion of the warp. The Fell-Heisenberg drive is unique only in that, with vanishing momentum, there is no configuration that allows stable propagation.

(n. b. The Heisenberg here is Lavinia Heisenberg, not the Werner Heisenberg of quantum physics and uncertainty principle fame.)

== Lentz warp drive ==

E. Lentz<ref name="Lentz">E. W. Lentz, "Breaking the warp barrier: hyper-fast solitons in Einstein–Maxwell-plasma theory", Classical and Quantum Gravity. 38 075015 (2021). arXiv:[https://arxiv.org/abs/2006.07125 2006.07125]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2021CQGra..38g5015L 2021CQGra..38g5015L]. doi:[https://doi.org/10.1088%2F1361-6382%2Fabe692 10.1088/1361-6382/abe692]. ISSN [https://search.worldcat.org/issn/0264-9381 0264-9381]. S2CID [https://api.semanticscholar.org/CorpusID:219635854 219635854].</ref> investigated a particular configuration of the Fell-Heisenberg class of warp drives, where the scalar field forms a selection of rhomboid regions, resulting in an energy density that is everywhere positive and a shift vector with vanishing tidal force at the center for a comfortable ride.

However, Santiago, Schuster, and Visser<ref name="Santiago Schuster Visser 2022"></ref> dispute claims that the Lentz drive satisfies the energy conditions, noting that everywhere positive energy density in one frame of reference is insufficient to establish that the energy density is positive in all reference frames; knowledge of the Cauchy stress tensor is also needed. They show that any generic warp drive will violate the strong energy condition, null energy condition, and weak energy condition.

== Conservation laws ==

(Discussion of asymptotic flatness: what is it? Where does it apply?)

(The "big four" conservation laws and how they relate to asymptotic flatness.)

(ADM mass -> 0; conservation of energy/mass issues.)<ref>Sebastian Schuster, Jessica Santiago,and Matt Visser, "ADM mass in warp drive spacetimes" General Relativity and Gravitation (2023) 55:14 https://doi.org/10.1007/s10714-022-03061-9</ref>

(Discuss issues of conservation of angular momentum.)

(If angular momentum conservation is ignored, discuss how momentum & energy are affected by outside forces. In a gravitational field equivalent to inertial frame ... like being in an accelerated elevator; will acquire momentum buildup while staying "at rest".)

== Warp Railroads ==

Van Den Broeck<ref name="VanDenBroeck_2000"></ref> suggested a warp railroad,

You can set up devices ahead of time along the path of the bubble that produce the negative energy regions for the warp bubble at the appropriate time without needing the negative energy to ever move with the bubble at all. At least for bubbles on a predictable schedule and route.

helps with:

-stability (elaborate)

-conservation laws (elaborate)

Mention Krasnikov<ref name="Krasnikov 1998">S. V. Krasnikov, "Hyperfast travel in general relativity" Phys. Rev. D 57 4760</ref>

Hyperwave<ref name="Pieri 2024">Lorenzo Pieri, "Hyperwave: hyper-Fast Communication within General Relativity", arXiv:2311.12069 [gr-qc] https://doi.org/10.48550/arXiv.2311.12069</ref> makes use of this warp railroad idea.

== Quantum effects ==

(quantum stuff here<ref name="Hiscock1997">W. A. Hiscock, "Quantum effects in the Alcubierre warp drive spacetime", Classical and Quantum Gravity 14 L183 https://doi.org/10.1088/0264-9381/14/11/002 https://arxiv.org/abs/gr-qc/9707024</ref><ref name="Finazzi et al 2009">S. Finazzi, S. Liberati, C. Barceló, "Semiclassical instability of dynamical warp drives", Physical Review D 79, 124017 (2009)https://doi.org/10.1103/PhysRevD.79.124017 https://arxiv.org/abs/0904.0141</ref>)

From<ref name="Pieri 2024"></ref>
<blockquote>
It should be emphasised that Quantum Inequalities have not been proven for realistic interacting quantum fields, where the they are expected to be weaker if they exist [Cadamuro, 2019], and they often require Minkowskian conditions at infinity, a condition which is not appropriate for fields inside bounded regions, as in a Casimir devices.
</blockquote>

== Warp drives and black holes ==

What happens if you drive your warp drive into a black hole?

The pop-sci reason that nothing can escape a black hole is that at the event horizon the escape velocity is faster than light and nothing can go faster than light. A warp drive can go faster than light.

The more detailed reason from general relativity is that at the event horizon space and time are rotated sufficiently that "inward" becomes "forward in time." You can no more go farther away from the singularity in a black hole once you are inside the event horizon than you can go backward in time. Any super-luminal travel can go backward in time, and warp drives can engage in super-luminal travel.

The Hawking area theorem that says that black holes can only grow, never shrink relies on the condition that the energy is everywhere positive. Warp drives have negative energy density.

So in principle, there's nothing that prevents a warp drive from taking a dip into a black hole and coming back out again.

Garattini and Zatrimaylov looked into the problem of a warp drive near (and in) a black hole<ref name="Garattini and Zatrimaylov">R. Garattini and K. Zatrimaylov, "Black holes, warp drives, and energy conditions", Physics Letters B 856 138910 (2024) https://doi.org/10.1016/j.physletb.2024.138910</ref>. The found that the black hole reduces the negative energy requirement for the warp bubble to move inward toward the black hole singularity, but increases the negative energy needed to move away from the center of the black hole. In addition, a warp drive parked at the event horizon would allow light from inside the horizon to pass through the bubble and escape back outside.

== Credit ==
Author: Luke Campbell

== References ==

[[Category:Physics]][[Category:Infrastructure]][[Category:Physics & Engineering‏‎]][[Category:Physics & Math & Engineering]][[Category:Transportation & Infrastructure‏‎]][[Category:Metric Engineering]]

Warp Drives

2026-07-27T00:17:01Z

Lwcamp: /* Fell-Heisenberg warp drives */

{{PageConstructionNotice}}

Science fiction often features spacecraft that can seemingly move across space and get between the place of departure and the destination much faster than light could have done. This appears to contradict the theory of relativity, which predicts unequivocally that nothing can move through space faster than light. Because relativity has been incredibly successful at describing nature, with its many other predictions regularly being confirmed to extraordinary accuracy and within the bounds of uncertainty of all the experiments that tested them, it gives confidence that relativity is a correct description of reality. Which seems to rather throw a wet towel on our hopes for rapid travel between stars.

However, while relativity does not allow things to move through space faster than light, it places no such restrictions on how fast space-time itself can expand, contract, or move around. This leads to the idea of a warp drive – the spacecraft remains stationary within a region of highly curved space-time, and that region moves at super-luminal speeds rather than the spacecraft.

== The Alcubierre warp drive ==

The first warp drive geometry that satisfied the Einstein field equations of relativity was proposed by Miguel Alcubierre<ref>M. Alcubierre, "The warp drive: hyper-fast travel within general relativity." Classical and Quantum Gravity. 11 (5): L73–L77 (1994). [https://arxiv.org/abs/gr-qc/0009013 arXiv:gr-qc/0009013]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1994CQGra..11L..73A 1994CQGra..11L..73A]. doi:[https://doi.org/10.1088%2F0264-9381%2F11%2F5%2F001 10.1088/0264-9381/11/5/001]. S2CID [https://api.semanticscholar.org/CorpusID:4797900 4797900].</ref>. In this geometry, a sphere of space-time moves at an arbitrary speed (potentially but not necessarily a speed much faster than light). Objects within the sphere are moved along with the sphere; an object at rest within the sphere would be moved along with the sphere indefinitely. Space is expanding at the rear boundary and contracting at the front boundary in order to keep the sphere moving. In order to satisfy the Einstein field equations, the boundary of the sphere must have a negative energy density. The challenges of space-time geometries with negative energy densities are described in our page on [[Wormholes#Exotic_energy_conditions|wormholes]], for our purposes it is enough to note that negative energy densities can pose problems if not handled carefully, there may be limits on how much negative energy you can have without nearby positive energy density, and it may not be possible to get enough negative energy to support a warp drive; although none of this is ruled out by physics – yet!

<table align=center>
<tr><td width=650>[[File:Energy_magnitude_of_Alcubierre_drive.png]]
<td width=650>[[File:Expansion_of_Alcubierre_drive.png]]
<tr><td>The magnitude of the energy of the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. (The energy density is negative everywhere in the shell, here only the magnitude is shown to aid visual comprehension.)
<td>The expansion of space in the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. Negative expansion means that space is contracting in that direction, positive expansion that it is expanding.
</table>

NOTE: What the? The energy distribution is symmetric! How does the drive know which way to go? There must be significant contributions of other components of the stress-energy tensor, and those have got to be asymmetric along the forward/backward axis. Check this when I get time.

(And what's with the ring in all the artwork, anyway?)

=== So what is going on? ===

What is it that makes a warp drive a warp drive, and how does it work?

A warp drive is a concept that comes out of the science of general relativity. General relativity collects space and time together into a single "structure" called spacetime. This structure is curved, and its curvature is determined by the material that is in it. Absent forces acting between matter, particles of matter move through spacetime on the shortest possible paths in this curved environment.
<blockquote>
"Spacetime tells matter how to move; matter tells spacetime how to curve." - John Archibald Wheeler
</blockquote>
All of this is encompassed in the Einstein field equation:
<div align=center>
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 8πG
<td rowspan="2" nowrap="nowrap" align="right"> T = R -  
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right"> R g
<tr>
<td style="border-top:solid 1px black;"> c4
<td style="border-top:solid 1px black;"> 2
</table>
</div>
Unfortunately, by convention, both the symbols G and R get used for a lot of different things. Here, symbols in bold are second rank tensors – constructs of arrays of 4×4 numbers where 4 is necessary because that is the number of dimensions of spacetime. Conventionally, these numbers are indexed by a value going from 0 to 3, with 0 being the part corresponding to the coordinate of time and 1 to 3 being the Cartesian coordinates that you might know as x, y, and z.

T is the stress-energy tensor: the (0,0) component represents the energy density at a given point; the (0,i) and (i, 0) components (with i going from 1 to 3) represent the momentum density; and the (i,j) components represent the stresses present, such as pressure or tension or shear.

The other term, R - R g/2, involves a partial description of the curvature. A full description of the curvature requires a construct of 4×4×4×4 components; too much for the stress energy tensor to fully constrain. Which is okay, because this allows curvature to exist away from where matter is, allowing gravity to have long range effects across empty space as well as for phenomena like gravitational waves. But it is enough that a given distribution of matter T gives enough constraints (along with specifying how the geometry behaves far away) that you can generally figure out what the actual gravity is.

This is the usual story that we are told. But there is an alternate way of looking at the same geometry such that the math works out the same, but it is described conceptually in a very different way. This divides spacetime up into "layers", called foliations, that represent slices of space and going from one foliation to the next is increasing in time. This is useful for computation because you can assign coordinates to your system. There are two additional constructs – the lapse function and the shift vector field. Starting at an event on one foliation, the lapse function at that position tells you that after a certain "proper time" (time perceived by an observer at that event) the time coordinate of the observer will be the proper time times the lapse function at that event. And the new spatial coordinates of the observer will be the proper time times the shift vector at that event. (Strictly speaking, this is only exactly true in the limit of the proper time approaching zero or when the lapse function and shift vectors are constant. But hopefully this gives a good enough insight to what these entities are so that we can go on to describe warp drives.)

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Lapse_shift.png]]
<tr><td>A folliation of space-time into slices of constant time coordinates (grey). At an event (black circle), the lapse function (yellow) shows how far into the future the point at that event will be projected in a unit amount of proper time experienced by an observer initially at the event. The shift vector (red) shows how that point changes coordinates after the unit amount of proper time. The cyan vector shows the new coordinates the point is projected onto. The lower black surface indicates a foliation of spacetime at the same time coordinate; after unit proper time, the events on the lower foliation get projected to the curved upper black foliation according to the shift vectors and lapse functions at each of their respective points.
</table>
</table>

Warp drives are generally restricted to spacetime geometries with a constant lapse function - all observers go forward by the same time coordinate after experiencing the same amount of proper time. Thus, all warp drives have no time dilation.
Far from the warp bubble, the shift vector vanishes – each spatial coordinate is mapped to the same spatial coordinate after any amount of proper time so long as the warp drive is not near. But the inside of the warp bubble is an extended region where the shift vector field is non-zero and constant. In this bubble, all spatial coordinates are mapped to a coordinate that shifts linearly in time with the proper time of the observer. Astute readers will see that this linear change in position with time is very similar to a velocity. This region of constant shift vector then moves with the shift vector velocity to keep stuff originally in the bubble so that it stays in the bubble. In between the inside of the bubble and the far away regions with zero shift vector, the shift vector field falls away to match one region to the other.

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Warp_shift.png]]
<tr><td>The shift vector of a one-dimensional slice through warp geometry, in red. At the edges there is no shift. At the center (inside the warp bubble) there is a region of uniform shift so that at a later time the inside of the warp bubble is rigidly translated to different spatial coordinates (in blue). The two foliations remain the same distance apart, because warp drives maintain a unit lapse function.
</table>
</table>

Traditionally, scientists would start with an interesting distribution of matter given by T and then find out what kind of curvature it created. This works well for planets and stars and black holes and cosmic strings and other things which are observed in the universe and you want to see what sort of effect they had. But starting in the 1980's, some physicists started looking at Einstein's field equations the other way around. They start with a desired geometry, find the curvature part of the field equation R - R g/2, and use that to find out what T has to be in order to get that curvature. This was originally done for [[Wormholes|wormholes]], but in 1994 Miguel Alcubierre used this same trick to figure out what could create a region of constant shift vector that moved with the shift vector - the warp drive. With this demand for a warp geometry, the required distribution of matter and energy and momentum and stress can be determined.

The original Alcubierre shift vector N→ produces a spherical warp bubble with an axi-symmetric distribution of, well, everything along the direction of motion with warp velocity vector v→.
<div align="center">
N→ = v→ f(rs)
</div>
Here, rs is the distance from the center of the warp bubble, and f(rs) can be any function that is 1 near rs = 0 and is 0 for rs well beyond the radius of the warp bubble.

=== Challenges and possible resolutions ===

Almost as soon as Alcubierre proposed his warp drive, others began picking it apart.
<ol>
<li>Pfenning and Ford<ref name="PfenningFord_2001">M. J. Pfenning and L. H. Ford, "The unphysical nature of 'Warp Drive'", Classical and Quantum Gravity. 14 (7): 1743–1751 (1997). arXiv:[https://arxiv.org/abs/gr-qc/9702026 gr-qc/9702026]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1997CQGra..14.1743P 1997CQGra..14.1743P]. doi:[https://doi.org/10.1088%2F0264-9381%2F14%2F7%2F011 10.1088/0264-9381/14/7/011]. S2CID [https://api.semanticscholar.org/CorpusID:15279207 15279207].</ref> found that if the negative energy density wall around the bubble obeys [[Wormholes#Quantum_energy_inequalities|quantum energy inequalities]], then for warp speeds of around light speed the shell thickness would be on the order of a hundred Planck lengths; a distance far smaller than any other known physical phenomenon or object.
<li>In the same work, Pfenning and Ford also found that for a bubble a hundred meters in radius, a warp speed of around the speed of light, and the required thickness of about 100 Planck lengths the energy in the bubble shell would be E ≈ -1063 kg c2. The magnitude of this latter value is ten orders of magnitude larger than the energy of the entire visible universe. They do note, however, that if quantum energy inequalities can be ignored then a warp drive with a hundred meters radius but a shell thickness of one meter would "only" have an energy magnitude of about a quarter solar mass.
<li>As with any method of faster than light travel, the warp drive could be used to make a time machine.
<li>Perhaps most seriously, if the warp drive is going faster than light speed, the negative energy regions on the outside of the shell won't be in the warp parts of the bubble. They will have to be moving through space at faster than light speed if the bubble is to maintain its integrity, which is the very problem that the warp drive was designed to avoid. The outside of the warp bubble shell that maintains the warp bubble would fall away and the warp bubble would quickly erode to sub-luminal speeds.<ref name="Van Den Broeck 2000">Chris Van Den Broeck, "On the (im)possibility of warp bubbles", arXiv:gr-qc/9906050 https://doi.org/10.48550/arXiv.gr-qc/9906050</ref>
<li>A more general problem that includes the previous one is that the geometry of the warp drive determines the distribution of energy, momentum, and stress (pressure, shear, and tension) by Einstein's field equation, which states that the stress energy tensor is directly proportional to a reduced order curvature tensor of space-time. However, the stress-energy tensor must also obey the continuity equation – basically, for the energy to change in a given volume of space that energy must come in or go out of the boundary of that volume and similarly for the momentum. So a warp drive can have the warp condition for one moment in time. But after that, the stress-energy that maintains the warp will start to move according to the forces on it and according to the natural paths that it takes through the warped space-time. There is no guarantee that the configuration of the stress-energy at later times will be sufficient to maintain the warp.
<blockquote>
From the perspective of simulating the warp drive dynamically, the key challenge is stability. One can straightforwardly generate the initial metric that describes the Alcubierre metric and infer the instantaneous matter configuration that must support it using the Einstein Equation, but a time-domain evolution of the coupled matter-gravity system necessitates that we specify the equation of state of the matter, which in turn determines how it evolves. In a perfect fluid, this involves specifying the relation between the fluid pressure and density, but in a more general case, the full relation between the
stress-energy components must be given. There is (to our knowledge) no known equation of state that would maintain the warp drive metric in a stable configuration over time - therefore, whilst one can require that initially, the warp bubble is constant, it will quickly evolve away from that state and, in most cases, the warp fluid and spacetime deformations will disperse or collapse into a central point.
 
– Clough, Dietrich, and Khan<ref>Katy Clough, Tim Dietrich, and Sebastian Khan, "What no one has seen before: gravitational waveforms from warp drive collapse", arXiv:2406.02466 [gr-qc] https://doi.org/10.48550/arXiv.2406.02466</ref>
</blockquote>
<li>In order to get the matter (or more generally, stress-energy) that forms the warp shell up to speeds that could allow the shell to exist, you need to be able to, well, accelerate that matter to warp speeds. Which gives a chicken-and-egg problem - in order to achieve superluminal travel you need to be able to get matter or energy moving superluminally in the first place<ref name="Bobrick and Martire">Alexey Bobrick and Gianni Martire, "Introducing Physical Warp Drives", arXiv:2102.06824 [gr-qc] https://doi.org/10.48550/arXiv.2102.06824</ref><ref>D. H. Coule, "No warp drive", Class. Quantum Grav. 15 (1998) 2523–2527. https://doi.org/10.1088/0264-9381/15/8/026</ref>, which rather makes warp drives unnecessary from the get-go.
<li>A warp bubble moving faster than light will not be able to get any signals in front of it as it will overtake its own signal. In fact, anyone in the interior of the bubble would not be able to send signals to the front surface of the bubble. In science-speak, a horizon forms at the front of the bubble<ref name="Natario">José Natário, "Warp drive with zero expansion", Classical and Quantum Gravity. 19 (6): 1157–1166 (2002). arXiv:[https://arxiv.org/abs/gr-qc/0110086 gr-qc/0110086]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2002CQGra..19.1157N 2002CQGra..19.1157N]. doi:[https://doi.org/10.1088%2F0264-9381%2F19%2F6%2F308 10.1088/0264-9381/19/6/308]. S2CID [https://api.semanticscholar.org/CorpusID:15859984 15859984].</ref>. As a consequence, the bubble cannot be controlled, steered, nor stopped from within the bubble itself.
<li>A Horizon will naturally emit Hawking radiation. It is estimated that the radiation will raise the temperature of the interior of the bubble to approximately 1032 K<ref name="Finazzi 2009">Stefano Finazzi, Stefano Liberati, and Carlos Barceló, "Semiclassical instability of dynamical warp drives", Phys. Rev. D 79, 124017 (12 June, 2009) https://doi.org/10.1103/PhysRevD.79.124017 , arXiv:0904.0141 [gr-qc] https://doi.org/10.48550/arXiv.0904.0141</ref> (at this level of accuracy, the result is the same if you replace temperatures in K by the same number in °C or °F). This temperature, however, may be lower if quantum inequalities do not have to be fulfilled.
<li>Semi-classical instabilities at the horizon are expected to lead to a divergence of the stress-energy tensor at the boundary in a short period of time, thus leading to the destruction of the warp bubble<ref name="Finazzi 2009"></ref>.
</ol>
Helpfully, Van Den Broeck<ref name="VanDenBroeck_2000">C. Van Den Broeck, "Alcubierre’s Warp Drive: Problems and Prospects" AIP Conference Proceedings. 504: 1105–1110 (2000). Bibcode:[https://ui.adsabs.harvard.edu/abs/2000AIPC..504.1105V 2000AIPC..504.1105V]. doi:[https://doi.org/10.1063%2F1.1290913 10.1063/1.1290913].</ref> suggested several solutions – or at least mitigations – for some of these problems
<ol>
<li>Quantum inequalities had not been shown to be true in general for highly curved space-times.
<li>A proposed warp drive geometry<ref name="VanDenBroeck_1999">C. Van Den Broeck, "A 'warp drive' with more reasonable total energy requirements". Classical and Quantum Gravity. 16 (12): 3973–3979 (1999). arXiv:[https://arxiv.org/abs/gr-qc/9905084 gr-qc/9905084]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1999CQGra..16.3973V 1999CQGra..16.3973V]. doi:[https://doi.org/10.1088%2F0264-9381%2F16%2F12%2F314 10.1088/0264-9381/16/12/314]. S2CID [https://api.semanticscholar.org/CorpusID:15466313 15466313]. </ref> (see the van Den Broeck drive, below) would be able to minimize the magnitude of the energy required to about that of the mass of our sun. While still large, it is not unphysically large.
<li>Time travel is always going to be a worry with faster than light travel. There's no neat solution to this one.
<li>It may be possible for the negative energy outside of the superluminal region to compress into the superluminal region to form a shock in space-time. The front surface of the bubble would then be a singularity. It is not clear if such a sharp jump in physical properties is possible, but neither is sure that it is impossible, either. However, while this might help with the negative energy in front of the bubble getting swept up because it cannot move faster than light, it does little to help with the material at the back of the bubble on the outside getting left behind because it cannot keep up.
</ol>
Other mitigating research has also shown
<ol start=9>
<li>Semi-classical divergence only occurs in warp drives with only one spatial dimensions. Those with two or more spatial dimensions (as in our universe) only contain points where divergences occur, and these divergences can be managed. In fact, their effects can be reduced by making the warp shape more elongated<ref>Carlos Barceló, Valentin Boyanov, Luis J. Garay, Eduardo Martín-Martínez,
and Jose M. Sánchez Velázquez, "Warp drive aerodynamics", JHEP08 (2022) 288 https://doi.org/10.1007/JHEP08(2022)288 , arXiv:2207.06458 [gr-qc] https://doi.org/10.48550/arXiv.2207.06458</ref>.
</ol>

=== Alcubierre warp interactions with light and matter ===

Space is not entirely empty. It is filled with a diffuse plasma in the form of the interstellar medium, as well as cosmic radiation, light from stars, and cosmic microwave background radiation. A warp drive propagating through space will encounter this stuff. What happens when this matter and radiation have a warp bubble pass across them?

The first analysis of matter encountering a warp bubble was performed by Pfenning and Ford<ref name="PfenningFord_2001"></ref>. They looked at the warp bubble interaction with a massive object at rest with the frame of reference of the warp drive (keep in mind that the rest frame is the same inside and outside the warp bubble; in this rest frame objects inside the bubble have no momentum even though they are, in some sense, changing location rapidly with time). When the warp bubble passes, the object experiences an acceleration in the direction of the warp bubble motion. When the warp shell passes and the object is inside the bubble, it will be moving with approximately the speed of the bubble. Pfenning and Ford analyzed this problem with a continuous distribution of shell energy that, strictly speaking, never falls to zero except at the bubble center and at spatial infinity, so unless the object passes through the center in Pfenning and Ford's description it will never quite get up to the bubble's speed. In this case, the object will move almost as fast as the object but will pass through the bubble in a finite time, after which it will again be at rest with respect to the reference frame but displaced along the direction of the warp bubble motion by some distance. With this description of the warp bubble, a spacecraft of finite size will always be moving a little slower than the warp bubble and would have to use rockets to keep up with it. in addition, the spacecraft would experience tidal forces that would cause stress on the spacecraft's structure.

Pfenning and Ford also examined cases where the shell is of a finite (possibly infinitesimal) width and falls to zero both inside and outside the bubble. In this case any matter encountering the bubble would be collected at the bow of the bubble and thereafter move along with it.

McMonigal et al. analyzed the situation for both massive particles and light moving along the axis of travel of the bubble<ref name="McMonigalEtAl_2012">B. McMonigal, G. F. Lewis, and P. O'Byrne, "Alcubierre warp drive: On the matter of matter". Physical Review D. 85 (6) 064024 (20 March 2012). arXiv:[https://arxiv.org/abs/1202.5708 1202.5708]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2012PhRvD..85f4024M 2012PhRvD..85f4024M]. doi:[https://doi.org/10.1103%2FPhysRevD.85.064024 10.1103/PhysRevD.85.064024]. S2CID [https://api.semanticscholar.org/CorpusID:3993148 3993148].</ref>. They found that light moving opposite the warp direction passed through the bubble without incident, being only somewhat delayed by passing through the bubble. A warp bubble that is warping at sub-luminal speeds can have light catch up from behind it. This light is able to pass through the bubble, and is somewhat advanced in its path by the speed of the bubble. For super-luminal warp bubbles, however, the situation is different. The bubble will catch up to light moving it its own direction that is originally in front of it. This light cannot escape forward, the bubble being too fast. Nor can it escape backward, as the light is propagating forward and the interior of the bubble is at rest. Thus, the light gets caught at the bow of the warp bubble, unable to escape for so long as the warp bubble is active. This light is strongly blue shifted to extremely energetic x-rays and gamma rays. The space behind the bubble is swept clear of forward-moving light.

The situation for matter over-run by a super-luminal warp bubble is similar. Matter moving backward passes through the bubble, being only swept a ways along the bubble's path as sign of its passing but otherwise continuing on their way. Matter at rest behaves the same way as Pfenning and Ford discovered. Meanwhile, matter moving in the same direction of the warp is overtaken and collects at the bow of the bubble, unable to leave for so long as the bubble is warping. This matter "surfing" on the bubble bow is highly accelerated to relativistic speeds, experiencing extreme time dilation.

For a sub-luminal bubble, there is a speed faster than the warp speed where particles coming up from behind overtake the bubble and pass through, out the front. Particles below this speed but still faster than the warp speed still overtake the bubble, but then bounce backwards out of the bubble from behind. They are still moving forward, but are now moving more slowly than the warp speed. Particles moving forward but slower than the bubble will be overtaken and then bounced out the front with a speed higher than the warp speed. Those particles moving backward will pass through the bubble and exit with their initial speed.

The spacecraft inside the bubble will observe that light moving backward compared to the warp direction is blue-shifted, and matter moving backward passes through with increased energy. Meanwhile, a sub-luminal bubble will see light that was moving forward as red-shifted and matter catching up to the bubble from behind will pass by with reduced energy (superluminal bubbles, of course, do not have any matter or radiation catching up to them from behind).

When a superluminal warp bubble that has been collecting matter and radiation for the duration of its journey and blue-shifting it to much higher energies is turned off, all that matter and energy is released as a blast of radiation in the direction the bubble was warping.

McMonigal et al. conclude by noting that any spacecraft in the warp bubble would need shielding to protect against the blue-shifted radiation and matter of increased energy. In addition the destination would be "blasted into oblivion" by the release of matter and radiation that had been caught in the bubble during the trip.

Clark et al.<ref>Chad Clark, William A. Hiscock, and Shane L. Larson, "Null geodesics in the Alcubierre warp drive spacetime: the view from the bridge", arXiv:gr-qc/9907019 (2018) https://doi.org/10.48550/arXiv.gr-qc/9907019</ref> examine what it would look like from within an operating warp drive. They find that light moving at 90 degrees to the direction of motion will look like it is still coming from 90 degrees. However, light from the front hemisphere is squished progressively toward the forward point the faster you go. For a warp bubble moving faster than the speed of light, there is a cone behind the warp bubble where no light can reach it. However, you don't see a black spot behind you – light from the regions in the backward hemisphere from outside that cone that can reach you is distorted toward the direction behind you to fill in that spot. Light from the forward direction is blue-shifted and light from behind is red-shifted. At speeds of around 200 times light speed, the cosmic microwave background would be blue shifted to energies similar to that of the solar photosphere, heating anything in the warp bubble as if they were close to the sun.

== Van Den Broeck warp drive ==

If an Alcubierre warp bubble a hundred meters across requires a magnitude of energy greater than the entire energy in the observable universe, one option to reduce the magnitude of energy used is to make the warp bubble smaller. Much smaller. Pfenning and Ford<ref name="PfenningFord_2001"></ref> suggested making the warp bubble smaller than an atom. Of course, that brings up the problem of how to stuff a spacecraft in there. Van Den Broeck proposed a solution<ref name="VanDenBroeck_1999"></ref>: make the warp bubble only about the size of an atomic nucleus but expand the space inside the bubble enormously so that a spacecraft could fit in. This is somewhat like a nuclear sized wormhole that leads to a pocket universe that holds the spacecraft. Unfortunately, the metric doesn't fit neatly into any [[Wormholes#Three_dimensions|embedding diagram]] that I can figure out, but the math works out so that spatial coordinates inside the bubble are expanded by a factor of 1017 and a spacecraft would have a few hundred meters of bubble interior to putz around in. This trick manages to reduce the magnitude of energy to only about the mass energy of a few stars similar to our own. Other than that, it is otherwise a normal Alcubierre drive.

Further work<ref>S. Krasnikov, "The quantum inequalities do not forbid spacetime shortcuts", Physical Review D. 67 (10) 104013. https://doi.org/10.1103%2FPhysRevD.67.104013, arXiv:gr-qc/0207057 (2003) https://doi.org/10.48550/arXiv.gr-qc/0207057 </ref> has managed to use Van Den Broeck's trick to reduce the magnitude of the negative energy to only about a milligram.

== Natário warp drive ==

The Alcubierre drive is not the only way to construct a warp drive. José Natário<ref name="Natario"></ref> showed that it was just one example of an entire class of warp drives. These are space-time geometries with a smooth and localized shift vector field and a unit lapse function.

Natário then went on to find in that class a set of warp drives that do not have any expansion or contraction of space at all. Instead, space encountering the front boundary of the warp bubble instead "slides" around the outside of the bubble until it gets to the corresponding place on the back and is left behind there. Another way to think of it is that space entering the bubble shell is compressed in the radial direction of the shell but is simultaneously expanded in the tangential direction so that there is no net change in volume. When it gets to the back, the opposite occurs. In the language of mathematics, the shift vector field is divergenceless.

<table align=center>
<tr><td width=360>[[File:Natario_drive_flow_of_space.png]]
<td width=445>[[File:Natario_warp_drive_horizons.png]]
<tr><td style="vertical-align: top;">A representation of the motion of space in a zero-expansion Natário drive with a vanishingly thin warp shell around the bubble.
<td>The horizons of the generalized Natário class of warp drives (including the Alcubierre warp drive and all fell-Heisenberg drives) at superluminal speed in the direction of the green arrow. No event inside the bubble can affect events outside the bubble in front of the blue line. No events outside the bubble behind the red line can affect events inside the bubble. The lines have an angle α with respect to the direction of motion with sin(α) = 1/vs for vs the speed of the bubble relative to light speed. This is analogous to the Mach cone of supersonic objects.
</table>

NOTE: There are two different things that can be meant by the term "Natário warp drive". The generalized Natário warp drive is the general theory of all warp drives that includes within it the Alcubierre, Fell-Heisenberg, and Lentz warp drives, among infinitely many others. The zero-expansion Natário warp drive is the specific implementation of the generalized Natário warp drive that has a divergenceless shift vector field.

The generalized class of warp drives developed by Natário (including both the Alcubierre and this zero-expansion warp drive) are shown to always have regions in the warp shell where the energy density is negative to at least some observers. You can't get away from it – to warp, you need negative energy density.

The Natário class of warp drives blueshift light coming in from the front and redshift light catching up from the back. For the case of a super-luminal warp bubble, there will of course be a horizon at the back of the bubble and light will not be able to get through from behind. For a bubble speed vs relative to the speed of light, light coming from straight ahead will be blueshifted up in frequency and photon energy by a factor of 1 + vs. For sub-luminal travel, light coming from behind will be redshifted by a factor 1 - vs. In general for n as the unit direction along which the light is propagating, the blueshift factor will be 1 + vs ⋅ n. If the warp shell is infinitesimally thin, light from outside that reaches the center of the bubble will not be distorted in direction although it will be frequency shifted, but observers still inside the bubble but displaced from the center will see the field of view distorted as well as frequency shifted.

Most of the difficulties that apply to the Alcubierre warp drive also apply to the generalized Natário warp drive (and thus to the zero-expansion Natário drive and Fell-Heisenberg drive as well). In particular, quantum energy inequalities, high magnitudes of the energy required, formation of time machines, instabilities related to the stress-energy making up the warp shell being unable to keep up with the bubble, the chicken-and-egg problem of needing FTL to achieve FTL, the formation of horizons making warp bubbles uncontrollable, and Hawking radiation at the horizons, are all still concerns for generic warp drives.

== Fell-Heisenberg warp drives ==

S. D. B. Fell and L. Heisenberg investigated what sort of warp structure is necessary to produce a warp drive with positive energy density everywhere<ref name="Fell-Heisenberg">S. D. B. Fell and L> Heisenberg, "Positive energy warp drive from hidden geometric structures", Classical and Quantum Gravity 38 155020 (2021) https://doi.org/10.1088/1361-6382/ac0e47 https://arxiv.org/abs/2104.06488</ref>. They found that warp drives of the generic Natário form with irrotational shift vectors had a simple expression for the energy density and vanishing momentum (irrotational means that the shift vector can be described as the gradient of a scalar potential field or, equivalently, that the curl of the shift vector field vanishes everywhere). Fell and Heisenberg proposed several potential fields that either had positive energy density everywhere except at discontinuities, or that were continuous but had regions that violated the weak energy condition (i. e. negative energy densities in some frames of reference). One specific example they provided of a warp field had an energy that is about 10,000 times less than the mass-energy of our sun. Or only about half the mass-energy of Jupiter. This is a significant improvement over many other proposed drives.

Despite the introduction discussing warp drive configurations that satisfy the various energy conditions, the configurations described in the paper are shown to locally violate the weak and strong energy conditions. Nonetheless, the energy density is still mostly positive. Santiago, Schuster, and Visser's<ref name="Santiago Schuster Visser 2022">J. Santiago, S. Schuster, and M. Visser, "Generic warp drives violate the null energy condition", Physical Review D 105, 064038 (2022) https://doi.org/10.1103/PhysRevD.105.064038</ref> work shows that the various energy conditions must still be violated by this warp drive; to not violate these, energy density must be positive in all reference frames not just those of the co-moving observer.

Any vector field can be described by its divergence, its curl, and a constant. The constant part of a warp drive's shift vector can be discarded because then the warp drive would not be localized. It is interesting, therefore, that the Natário zero-expansion drive has zero divergence and can be described only by its curl, while the Fell-Heisenberg drive has zero curl and can be described only by its divergence. These two classes of drives thus rely on opposite choices of setting one of the two different parts describing a vector field to zero. As might be expected, the Fell-Heisenberg drive is not zero expansion – space-time will change in volume as it passes through the warp bubble (although after going all the way through it will return to its original volume). However, objects in space-time will retain their original orientation as they pass through the warp bubble.

Any vector field with zero curl can be represented as the gradient of a suitable scalar function. Thus, the Fell-Heisenberg drive can be completely described by just scalar field (one number at all points of space and time) rather than a vector field.

Previous examples of warp drives, such as the Alcubierre and zero-expansion Natário drives, tend toward a flat space-time as you go far enough away faster than would occur if there were any mass present in the warp drive. That is, the ADM mass of such drives is zero. However, one of the Fell-Heisenberg examples described a spacetime where the geometry approaches that of what you would get for a massive object at the warp location, giving one of the first examples of a warp drive with mass.

In the Fell-Heisenberg drive the momentum vanishes everywhere. Yet the energy occupies regions where the shift vector is varying rapidly. The lack of momentum means that the energy will not move to keep up with the differential expansion and movement of the spacetime. As a consequence, at later times the energy will have a different distribution than what is necessary to maintain the given warp configuration. The exact time evolution is not solved but this likely leads to collapse of warp bubble. Note that this is not a problem unique to the Fell-Heisenberg drive. Assuming a geometry that describes a warp produces a specific distribution of the stress-energy tensor that satisfies Einstein's equation. But there is no guarantee that this stress energy tensor satisfies the continuity equation, which describes how the distribution of matter and momentum change with time. The issue of stability of any warp drive is not well understood, but most warp drive geometries will be unstable in the sense that their matter that generates the warp fields will not be able to keep up with the motion of the warp. The Fell-Heisenberg drive is unique only in that, with vanishing momentum, there is no configuration that allows stable propagation.

(n. b. The Heisenberg here is Lavinia Heisenberg, not the Werner Heisenberg of quantum physics and uncertainty principle fame.)

== Lentz warp drive ==

E. Lentz<ref name="Lentz">E. W. Lentz, "Breaking the warp barrier: hyper-fast solitons in Einstein–Maxwell-plasma theory", Classical and Quantum Gravity. 38 075015 (2021). arXiv:[https://arxiv.org/abs/2006.07125 2006.07125]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2021CQGra..38g5015L 2021CQGra..38g5015L]. doi:[https://doi.org/10.1088%2F1361-6382%2Fabe692 10.1088/1361-6382/abe692]. ISSN [https://search.worldcat.org/issn/0264-9381 0264-9381]. S2CID [https://api.semanticscholar.org/CorpusID:219635854 219635854].</ref> investigated a particular configuration of the Fell-Heisenberg class of warp drives, where the scalar field has a hyperbolic shape. The necessary conditions were found to be satisfied by a selection of rhomboid regions of the scalar field, resulting in an energy density that is everywhere positive and a shift vector with vanishing tidal force at the center for a comfortable ride.

However, Santiago, Schuster, and Visser<ref name="Santiago Schuster Visser 2022"></ref> dispute claims that the Lentz drive satisfies the energy conditions, noting that everywhere positive energy density in one frame of reference is insufficient to establish that the energy density is positive in all reference frames; knowledge of the Cauchy stress tensor is also needed. They show that any generic warp drive will violate the strong energy condition, null energy condition, and weak energy condition.

== Conservation laws ==

(Discussion of asymptotic flatness: what is it? Where does it apply?)

(The "big four" conservation laws and how they relate to asymptotic flatness.)

(ADM mass -> 0; conservation of energy/mass issues.)<ref>Sebastian Schuster, Jessica Santiago,and Matt Visser, "ADM mass in warp drive spacetimes" General Relativity and Gravitation (2023) 55:14 https://doi.org/10.1007/s10714-022-03061-9</ref>

(Discuss issues of conservation of angular momentum.)

(If angular momentum conservation is ignored, discuss how momentum & energy are affected by outside forces. In a gravitational field equivalent to inertial frame ... like being in an accelerated elevator; will acquire momentum buildup while staying "at rest".)

== Warp Railroads ==

Van Den Broeck<ref name="VanDenBroeck_2000"></ref> suggested a warp railroad,

You can set up devices ahead of time along the path of the bubble that produce the negative energy regions for the warp bubble at the appropriate time without needing the negative energy to ever move with the bubble at all. At least for bubbles on a predictable schedule and route.

helps with:

-stability (elaborate)

-conservation laws (elaborate)

Mention Krasnikov<ref name="Krasnikov 1998">S. V. Krasnikov, "Hyperfast travel in general relativity" Phys. Rev. D 57 4760</ref>

Hyperwave<ref name="Pieri 2024">Lorenzo Pieri, "Hyperwave: hyper-Fast Communication within General Relativity", arXiv:2311.12069 [gr-qc] https://doi.org/10.48550/arXiv.2311.12069</ref> makes use of this warp railroad idea.

== Quantum effects ==

(quantum stuff here<ref name="Hiscock1997">W. A. Hiscock, "Quantum effects in the Alcubierre warp drive spacetime", Classical and Quantum Gravity 14 L183 https://doi.org/10.1088/0264-9381/14/11/002 https://arxiv.org/abs/gr-qc/9707024</ref><ref name="Finazzi et al 2009">S. Finazzi, S. Liberati, C. Barceló, "Semiclassical instability of dynamical warp drives", Physical Review D 79, 124017 (2009)https://doi.org/10.1103/PhysRevD.79.124017 https://arxiv.org/abs/0904.0141</ref>)

From<ref name="Pieri 2024"></ref>
<blockquote>
It should be emphasised that Quantum Inequalities have not been proven for realistic interacting quantum fields, where the they are expected to be weaker if they exist [Cadamuro, 2019], and they often require Minkowskian conditions at infinity, a condition which is not appropriate for fields inside bounded regions, as in a Casimir devices.
</blockquote>

== Warp drives and black holes ==

What happens if you drive your warp drive into a black hole?

The pop-sci reason that nothing can escape a black hole is that at the event horizon the escape velocity is faster than light and nothing can go faster than light. A warp drive can go faster than light.

The more detailed reason from general relativity is that at the event horizon space and time are rotated sufficiently that "inward" becomes "forward in time." You can no more go farther away from the singularity in a black hole once you are inside the event horizon than you can go backward in time. Any super-luminal travel can go backward in time, and warp drives can engage in super-luminal travel.

The Hawking area theorem that says that black holes can only grow, never shrink relies on the condition that the energy is everywhere positive. Warp drives have negative energy density.

So in principle, there's nothing that prevents a warp drive from taking a dip into a black hole and coming back out again.

Garattini and Zatrimaylov looked into the problem of a warp drive near (and in) a black hole<ref name="Garattini and Zatrimaylov">R. Garattini and K. Zatrimaylov, "Black holes, warp drives, and energy conditions", Physics Letters B 856 138910 (2024) https://doi.org/10.1016/j.physletb.2024.138910</ref>. The found that the black hole reduces the negative energy requirement for the warp bubble to move inward toward the black hole singularity, but increases the negative energy needed to move away from the center of the black hole. In addition, a warp drive parked at the event horizon would allow light from inside the horizon to pass through the bubble and escape back outside.

== Credit ==
Author: Luke Campbell

== References ==

[[Category:Physics]][[Category:Infrastructure]][[Category:Physics & Engineering‏‎]][[Category:Physics & Math & Engineering]][[Category:Transportation & Infrastructure‏‎]][[Category:Metric Engineering]]

Warp Drives

2026-07-27T00:09:25Z

Lwcamp: /* Natário warp drive */

{{PageConstructionNotice}}

Science fiction often features spacecraft that can seemingly move across space and get between the place of departure and the destination much faster than light could have done. This appears to contradict the theory of relativity, which predicts unequivocally that nothing can move through space faster than light. Because relativity has been incredibly successful at describing nature, with its many other predictions regularly being confirmed to extraordinary accuracy and within the bounds of uncertainty of all the experiments that tested them, it gives confidence that relativity is a correct description of reality. Which seems to rather throw a wet towel on our hopes for rapid travel between stars.

However, while relativity does not allow things to move through space faster than light, it places no such restrictions on how fast space-time itself can expand, contract, or move around. This leads to the idea of a warp drive – the spacecraft remains stationary within a region of highly curved space-time, and that region moves at super-luminal speeds rather than the spacecraft.

== The Alcubierre warp drive ==

The first warp drive geometry that satisfied the Einstein field equations of relativity was proposed by Miguel Alcubierre<ref>M. Alcubierre, "The warp drive: hyper-fast travel within general relativity." Classical and Quantum Gravity. 11 (5): L73–L77 (1994). [https://arxiv.org/abs/gr-qc/0009013 arXiv:gr-qc/0009013]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1994CQGra..11L..73A 1994CQGra..11L..73A]. doi:[https://doi.org/10.1088%2F0264-9381%2F11%2F5%2F001 10.1088/0264-9381/11/5/001]. S2CID [https://api.semanticscholar.org/CorpusID:4797900 4797900].</ref>. In this geometry, a sphere of space-time moves at an arbitrary speed (potentially but not necessarily a speed much faster than light). Objects within the sphere are moved along with the sphere; an object at rest within the sphere would be moved along with the sphere indefinitely. Space is expanding at the rear boundary and contracting at the front boundary in order to keep the sphere moving. In order to satisfy the Einstein field equations, the boundary of the sphere must have a negative energy density. The challenges of space-time geometries with negative energy densities are described in our page on [[Wormholes#Exotic_energy_conditions|wormholes]], for our purposes it is enough to note that negative energy densities can pose problems if not handled carefully, there may be limits on how much negative energy you can have without nearby positive energy density, and it may not be possible to get enough negative energy to support a warp drive; although none of this is ruled out by physics – yet!

<table align=center>
<tr><td width=650>[[File:Energy_magnitude_of_Alcubierre_drive.png]]
<td width=650>[[File:Expansion_of_Alcubierre_drive.png]]
<tr><td>The magnitude of the energy of the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. (The energy density is negative everywhere in the shell, here only the magnitude is shown to aid visual comprehension.)
<td>The expansion of space in the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. Negative expansion means that space is contracting in that direction, positive expansion that it is expanding.
</table>

NOTE: What the? The energy distribution is symmetric! How does the drive know which way to go? There must be significant contributions of other components of the stress-energy tensor, and those have got to be asymmetric along the forward/backward axis. Check this when I get time.

(And what's with the ring in all the artwork, anyway?)

=== So what is going on? ===

What is it that makes a warp drive a warp drive, and how does it work?

A warp drive is a concept that comes out of the science of general relativity. General relativity collects space and time together into a single "structure" called spacetime. This structure is curved, and its curvature is determined by the material that is in it. Absent forces acting between matter, particles of matter move through spacetime on the shortest possible paths in this curved environment.
<blockquote>
"Spacetime tells matter how to move; matter tells spacetime how to curve." - John Archibald Wheeler
</blockquote>
All of this is encompassed in the Einstein field equation:
<div align=center>
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 8πG
<td rowspan="2" nowrap="nowrap" align="right"> T = R -  
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right"> R g
<tr>
<td style="border-top:solid 1px black;"> c4
<td style="border-top:solid 1px black;"> 2
</table>
</div>
Unfortunately, by convention, both the symbols G and R get used for a lot of different things. Here, symbols in bold are second rank tensors – constructs of arrays of 4×4 numbers where 4 is necessary because that is the number of dimensions of spacetime. Conventionally, these numbers are indexed by a value going from 0 to 3, with 0 being the part corresponding to the coordinate of time and 1 to 3 being the Cartesian coordinates that you might know as x, y, and z.

T is the stress-energy tensor: the (0,0) component represents the energy density at a given point; the (0,i) and (i, 0) components (with i going from 1 to 3) represent the momentum density; and the (i,j) components represent the stresses present, such as pressure or tension or shear.

The other term, R - R g/2, involves a partial description of the curvature. A full description of the curvature requires a construct of 4×4×4×4 components; too much for the stress energy tensor to fully constrain. Which is okay, because this allows curvature to exist away from where matter is, allowing gravity to have long range effects across empty space as well as for phenomena like gravitational waves. But it is enough that a given distribution of matter T gives enough constraints (along with specifying how the geometry behaves far away) that you can generally figure out what the actual gravity is.

This is the usual story that we are told. But there is an alternate way of looking at the same geometry such that the math works out the same, but it is described conceptually in a very different way. This divides spacetime up into "layers", called foliations, that represent slices of space and going from one foliation to the next is increasing in time. This is useful for computation because you can assign coordinates to your system. There are two additional constructs – the lapse function and the shift vector field. Starting at an event on one foliation, the lapse function at that position tells you that after a certain "proper time" (time perceived by an observer at that event) the time coordinate of the observer will be the proper time times the lapse function at that event. And the new spatial coordinates of the observer will be the proper time times the shift vector at that event. (Strictly speaking, this is only exactly true in the limit of the proper time approaching zero or when the lapse function and shift vectors are constant. But hopefully this gives a good enough insight to what these entities are so that we can go on to describe warp drives.)

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Lapse_shift.png]]
<tr><td>A folliation of space-time into slices of constant time coordinates (grey). At an event (black circle), the lapse function (yellow) shows how far into the future the point at that event will be projected in a unit amount of proper time experienced by an observer initially at the event. The shift vector (red) shows how that point changes coordinates after the unit amount of proper time. The cyan vector shows the new coordinates the point is projected onto. The lower black surface indicates a foliation of spacetime at the same time coordinate; after unit proper time, the events on the lower foliation get projected to the curved upper black foliation according to the shift vectors and lapse functions at each of their respective points.
</table>
</table>

Warp drives are generally restricted to spacetime geometries with a constant lapse function - all observers go forward by the same time coordinate after experiencing the same amount of proper time. Thus, all warp drives have no time dilation.
Far from the warp bubble, the shift vector vanishes – each spatial coordinate is mapped to the same spatial coordinate after any amount of proper time so long as the warp drive is not near. But the inside of the warp bubble is an extended region where the shift vector field is non-zero and constant. In this bubble, all spatial coordinates are mapped to a coordinate that shifts linearly in time with the proper time of the observer. Astute readers will see that this linear change in position with time is very similar to a velocity. This region of constant shift vector then moves with the shift vector velocity to keep stuff originally in the bubble so that it stays in the bubble. In between the inside of the bubble and the far away regions with zero shift vector, the shift vector field falls away to match one region to the other.

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Warp_shift.png]]
<tr><td>The shift vector of a one-dimensional slice through warp geometry, in red. At the edges there is no shift. At the center (inside the warp bubble) there is a region of uniform shift so that at a later time the inside of the warp bubble is rigidly translated to different spatial coordinates (in blue). The two foliations remain the same distance apart, because warp drives maintain a unit lapse function.
</table>
</table>

Traditionally, scientists would start with an interesting distribution of matter given by T and then find out what kind of curvature it created. This works well for planets and stars and black holes and cosmic strings and other things which are observed in the universe and you want to see what sort of effect they had. But starting in the 1980's, some physicists started looking at Einstein's field equations the other way around. They start with a desired geometry, find the curvature part of the field equation R - R g/2, and use that to find out what T has to be in order to get that curvature. This was originally done for [[Wormholes|wormholes]], but in 1994 Miguel Alcubierre used this same trick to figure out what could create a region of constant shift vector that moved with the shift vector - the warp drive. With this demand for a warp geometry, the required distribution of matter and energy and momentum and stress can be determined.

The original Alcubierre shift vector N→ produces a spherical warp bubble with an axi-symmetric distribution of, well, everything along the direction of motion with warp velocity vector v→.
<div align="center">
N→ = v→ f(rs)
</div>
Here, rs is the distance from the center of the warp bubble, and f(rs) can be any function that is 1 near rs = 0 and is 0 for rs well beyond the radius of the warp bubble.

=== Challenges and possible resolutions ===

Almost as soon as Alcubierre proposed his warp drive, others began picking it apart.
<ol>
<li>Pfenning and Ford<ref name="PfenningFord_2001">M. J. Pfenning and L. H. Ford, "The unphysical nature of 'Warp Drive'", Classical and Quantum Gravity. 14 (7): 1743–1751 (1997). arXiv:[https://arxiv.org/abs/gr-qc/9702026 gr-qc/9702026]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1997CQGra..14.1743P 1997CQGra..14.1743P]. doi:[https://doi.org/10.1088%2F0264-9381%2F14%2F7%2F011 10.1088/0264-9381/14/7/011]. S2CID [https://api.semanticscholar.org/CorpusID:15279207 15279207].</ref> found that if the negative energy density wall around the bubble obeys [[Wormholes#Quantum_energy_inequalities|quantum energy inequalities]], then for warp speeds of around light speed the shell thickness would be on the order of a hundred Planck lengths; a distance far smaller than any other known physical phenomenon or object.
<li>In the same work, Pfenning and Ford also found that for a bubble a hundred meters in radius, a warp speed of around the speed of light, and the required thickness of about 100 Planck lengths the energy in the bubble shell would be E ≈ -1063 kg c2. The magnitude of this latter value is ten orders of magnitude larger than the energy of the entire visible universe. They do note, however, that if quantum energy inequalities can be ignored then a warp drive with a hundred meters radius but a shell thickness of one meter would "only" have an energy magnitude of about a quarter solar mass.
<li>As with any method of faster than light travel, the warp drive could be used to make a time machine.
<li>Perhaps most seriously, if the warp drive is going faster than light speed, the negative energy regions on the outside of the shell won't be in the warp parts of the bubble. They will have to be moving through space at faster than light speed if the bubble is to maintain its integrity, which is the very problem that the warp drive was designed to avoid. The outside of the warp bubble shell that maintains the warp bubble would fall away and the warp bubble would quickly erode to sub-luminal speeds.<ref name="Van Den Broeck 2000">Chris Van Den Broeck, "On the (im)possibility of warp bubbles", arXiv:gr-qc/9906050 https://doi.org/10.48550/arXiv.gr-qc/9906050</ref>
<li>A more general problem that includes the previous one is that the geometry of the warp drive determines the distribution of energy, momentum, and stress (pressure, shear, and tension) by Einstein's field equation, which states that the stress energy tensor is directly proportional to a reduced order curvature tensor of space-time. However, the stress-energy tensor must also obey the continuity equation – basically, for the energy to change in a given volume of space that energy must come in or go out of the boundary of that volume and similarly for the momentum. So a warp drive can have the warp condition for one moment in time. But after that, the stress-energy that maintains the warp will start to move according to the forces on it and according to the natural paths that it takes through the warped space-time. There is no guarantee that the configuration of the stress-energy at later times will be sufficient to maintain the warp.
<blockquote>
From the perspective of simulating the warp drive dynamically, the key challenge is stability. One can straightforwardly generate the initial metric that describes the Alcubierre metric and infer the instantaneous matter configuration that must support it using the Einstein Equation, but a time-domain evolution of the coupled matter-gravity system necessitates that we specify the equation of state of the matter, which in turn determines how it evolves. In a perfect fluid, this involves specifying the relation between the fluid pressure and density, but in a more general case, the full relation between the
stress-energy components must be given. There is (to our knowledge) no known equation of state that would maintain the warp drive metric in a stable configuration over time - therefore, whilst one can require that initially, the warp bubble is constant, it will quickly evolve away from that state and, in most cases, the warp fluid and spacetime deformations will disperse or collapse into a central point.
 
– Clough, Dietrich, and Khan<ref>Katy Clough, Tim Dietrich, and Sebastian Khan, "What no one has seen before: gravitational waveforms from warp drive collapse", arXiv:2406.02466 [gr-qc] https://doi.org/10.48550/arXiv.2406.02466</ref>
</blockquote>
<li>In order to get the matter (or more generally, stress-energy) that forms the warp shell up to speeds that could allow the shell to exist, you need to be able to, well, accelerate that matter to warp speeds. Which gives a chicken-and-egg problem - in order to achieve superluminal travel you need to be able to get matter or energy moving superluminally in the first place<ref name="Bobrick and Martire">Alexey Bobrick and Gianni Martire, "Introducing Physical Warp Drives", arXiv:2102.06824 [gr-qc] https://doi.org/10.48550/arXiv.2102.06824</ref><ref>D. H. Coule, "No warp drive", Class. Quantum Grav. 15 (1998) 2523–2527. https://doi.org/10.1088/0264-9381/15/8/026</ref>, which rather makes warp drives unnecessary from the get-go.
<li>A warp bubble moving faster than light will not be able to get any signals in front of it as it will overtake its own signal. In fact, anyone in the interior of the bubble would not be able to send signals to the front surface of the bubble. In science-speak, a horizon forms at the front of the bubble<ref name="Natario">José Natário, "Warp drive with zero expansion", Classical and Quantum Gravity. 19 (6): 1157–1166 (2002). arXiv:[https://arxiv.org/abs/gr-qc/0110086 gr-qc/0110086]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2002CQGra..19.1157N 2002CQGra..19.1157N]. doi:[https://doi.org/10.1088%2F0264-9381%2F19%2F6%2F308 10.1088/0264-9381/19/6/308]. S2CID [https://api.semanticscholar.org/CorpusID:15859984 15859984].</ref>. As a consequence, the bubble cannot be controlled, steered, nor stopped from within the bubble itself.
<li>A Horizon will naturally emit Hawking radiation. It is estimated that the radiation will raise the temperature of the interior of the bubble to approximately 1032 K<ref name="Finazzi 2009">Stefano Finazzi, Stefano Liberati, and Carlos Barceló, "Semiclassical instability of dynamical warp drives", Phys. Rev. D 79, 124017 (12 June, 2009) https://doi.org/10.1103/PhysRevD.79.124017 , arXiv:0904.0141 [gr-qc] https://doi.org/10.48550/arXiv.0904.0141</ref> (at this level of accuracy, the result is the same if you replace temperatures in K by the same number in °C or °F). This temperature, however, may be lower if quantum inequalities do not have to be fulfilled.
<li>Semi-classical instabilities at the horizon are expected to lead to a divergence of the stress-energy tensor at the boundary in a short period of time, thus leading to the destruction of the warp bubble<ref name="Finazzi 2009"></ref>.
</ol>
Helpfully, Van Den Broeck<ref name="VanDenBroeck_2000">C. Van Den Broeck, "Alcubierre’s Warp Drive: Problems and Prospects" AIP Conference Proceedings. 504: 1105–1110 (2000). Bibcode:[https://ui.adsabs.harvard.edu/abs/2000AIPC..504.1105V 2000AIPC..504.1105V]. doi:[https://doi.org/10.1063%2F1.1290913 10.1063/1.1290913].</ref> suggested several solutions – or at least mitigations – for some of these problems
<ol>
<li>Quantum inequalities had not been shown to be true in general for highly curved space-times.
<li>A proposed warp drive geometry<ref name="VanDenBroeck_1999">C. Van Den Broeck, "A 'warp drive' with more reasonable total energy requirements". Classical and Quantum Gravity. 16 (12): 3973–3979 (1999). arXiv:[https://arxiv.org/abs/gr-qc/9905084 gr-qc/9905084]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1999CQGra..16.3973V 1999CQGra..16.3973V]. doi:[https://doi.org/10.1088%2F0264-9381%2F16%2F12%2F314 10.1088/0264-9381/16/12/314]. S2CID [https://api.semanticscholar.org/CorpusID:15466313 15466313]. </ref> (see the van Den Broeck drive, below) would be able to minimize the magnitude of the energy required to about that of the mass of our sun. While still large, it is not unphysically large.
<li>Time travel is always going to be a worry with faster than light travel. There's no neat solution to this one.
<li>It may be possible for the negative energy outside of the superluminal region to compress into the superluminal region to form a shock in space-time. The front surface of the bubble would then be a singularity. It is not clear if such a sharp jump in physical properties is possible, but neither is sure that it is impossible, either. However, while this might help with the negative energy in front of the bubble getting swept up because it cannot move faster than light, it does little to help with the material at the back of the bubble on the outside getting left behind because it cannot keep up.
</ol>
Other mitigating research has also shown
<ol start=9>
<li>Semi-classical divergence only occurs in warp drives with only one spatial dimensions. Those with two or more spatial dimensions (as in our universe) only contain points where divergences occur, and these divergences can be managed. In fact, their effects can be reduced by making the warp shape more elongated<ref>Carlos Barceló, Valentin Boyanov, Luis J. Garay, Eduardo Martín-Martínez,
and Jose M. Sánchez Velázquez, "Warp drive aerodynamics", JHEP08 (2022) 288 https://doi.org/10.1007/JHEP08(2022)288 , arXiv:2207.06458 [gr-qc] https://doi.org/10.48550/arXiv.2207.06458</ref>.
</ol>

=== Alcubierre warp interactions with light and matter ===

Space is not entirely empty. It is filled with a diffuse plasma in the form of the interstellar medium, as well as cosmic radiation, light from stars, and cosmic microwave background radiation. A warp drive propagating through space will encounter this stuff. What happens when this matter and radiation have a warp bubble pass across them?

The first analysis of matter encountering a warp bubble was performed by Pfenning and Ford<ref name="PfenningFord_2001"></ref>. They looked at the warp bubble interaction with a massive object at rest with the frame of reference of the warp drive (keep in mind that the rest frame is the same inside and outside the warp bubble; in this rest frame objects inside the bubble have no momentum even though they are, in some sense, changing location rapidly with time). When the warp bubble passes, the object experiences an acceleration in the direction of the warp bubble motion. When the warp shell passes and the object is inside the bubble, it will be moving with approximately the speed of the bubble. Pfenning and Ford analyzed this problem with a continuous distribution of shell energy that, strictly speaking, never falls to zero except at the bubble center and at spatial infinity, so unless the object passes through the center in Pfenning and Ford's description it will never quite get up to the bubble's speed. In this case, the object will move almost as fast as the object but will pass through the bubble in a finite time, after which it will again be at rest with respect to the reference frame but displaced along the direction of the warp bubble motion by some distance. With this description of the warp bubble, a spacecraft of finite size will always be moving a little slower than the warp bubble and would have to use rockets to keep up with it. in addition, the spacecraft would experience tidal forces that would cause stress on the spacecraft's structure.

Pfenning and Ford also examined cases where the shell is of a finite (possibly infinitesimal) width and falls to zero both inside and outside the bubble. In this case any matter encountering the bubble would be collected at the bow of the bubble and thereafter move along with it.

McMonigal et al. analyzed the situation for both massive particles and light moving along the axis of travel of the bubble<ref name="McMonigalEtAl_2012">B. McMonigal, G. F. Lewis, and P. O'Byrne, "Alcubierre warp drive: On the matter of matter". Physical Review D. 85 (6) 064024 (20 March 2012). arXiv:[https://arxiv.org/abs/1202.5708 1202.5708]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2012PhRvD..85f4024M 2012PhRvD..85f4024M]. doi:[https://doi.org/10.1103%2FPhysRevD.85.064024 10.1103/PhysRevD.85.064024]. S2CID [https://api.semanticscholar.org/CorpusID:3993148 3993148].</ref>. They found that light moving opposite the warp direction passed through the bubble without incident, being only somewhat delayed by passing through the bubble. A warp bubble that is warping at sub-luminal speeds can have light catch up from behind it. This light is able to pass through the bubble, and is somewhat advanced in its path by the speed of the bubble. For super-luminal warp bubbles, however, the situation is different. The bubble will catch up to light moving it its own direction that is originally in front of it. This light cannot escape forward, the bubble being too fast. Nor can it escape backward, as the light is propagating forward and the interior of the bubble is at rest. Thus, the light gets caught at the bow of the warp bubble, unable to escape for so long as the warp bubble is active. This light is strongly blue shifted to extremely energetic x-rays and gamma rays. The space behind the bubble is swept clear of forward-moving light.

The situation for matter over-run by a super-luminal warp bubble is similar. Matter moving backward passes through the bubble, being only swept a ways along the bubble's path as sign of its passing but otherwise continuing on their way. Matter at rest behaves the same way as Pfenning and Ford discovered. Meanwhile, matter moving in the same direction of the warp is overtaken and collects at the bow of the bubble, unable to leave for so long as the bubble is warping. This matter "surfing" on the bubble bow is highly accelerated to relativistic speeds, experiencing extreme time dilation.

For a sub-luminal bubble, there is a speed faster than the warp speed where particles coming up from behind overtake the bubble and pass through, out the front. Particles below this speed but still faster than the warp speed still overtake the bubble, but then bounce backwards out of the bubble from behind. They are still moving forward, but are now moving more slowly than the warp speed. Particles moving forward but slower than the bubble will be overtaken and then bounced out the front with a speed higher than the warp speed. Those particles moving backward will pass through the bubble and exit with their initial speed.

The spacecraft inside the bubble will observe that light moving backward compared to the warp direction is blue-shifted, and matter moving backward passes through with increased energy. Meanwhile, a sub-luminal bubble will see light that was moving forward as red-shifted and matter catching up to the bubble from behind will pass by with reduced energy (superluminal bubbles, of course, do not have any matter or radiation catching up to them from behind).

When a superluminal warp bubble that has been collecting matter and radiation for the duration of its journey and blue-shifting it to much higher energies is turned off, all that matter and energy is released as a blast of radiation in the direction the bubble was warping.

McMonigal et al. conclude by noting that any spacecraft in the warp bubble would need shielding to protect against the blue-shifted radiation and matter of increased energy. In addition the destination would be "blasted into oblivion" by the release of matter and radiation that had been caught in the bubble during the trip.

Clark et al.<ref>Chad Clark, William A. Hiscock, and Shane L. Larson, "Null geodesics in the Alcubierre warp drive spacetime: the view from the bridge", arXiv:gr-qc/9907019 (2018) https://doi.org/10.48550/arXiv.gr-qc/9907019</ref> examine what it would look like from within an operating warp drive. They find that light moving at 90 degrees to the direction of motion will look like it is still coming from 90 degrees. However, light from the front hemisphere is squished progressively toward the forward point the faster you go. For a warp bubble moving faster than the speed of light, there is a cone behind the warp bubble where no light can reach it. However, you don't see a black spot behind you – light from the regions in the backward hemisphere from outside that cone that can reach you is distorted toward the direction behind you to fill in that spot. Light from the forward direction is blue-shifted and light from behind is red-shifted. At speeds of around 200 times light speed, the cosmic microwave background would be blue shifted to energies similar to that of the solar photosphere, heating anything in the warp bubble as if they were close to the sun.

== Van Den Broeck warp drive ==

If an Alcubierre warp bubble a hundred meters across requires a magnitude of energy greater than the entire energy in the observable universe, one option to reduce the magnitude of energy used is to make the warp bubble smaller. Much smaller. Pfenning and Ford<ref name="PfenningFord_2001"></ref> suggested making the warp bubble smaller than an atom. Of course, that brings up the problem of how to stuff a spacecraft in there. Van Den Broeck proposed a solution<ref name="VanDenBroeck_1999"></ref>: make the warp bubble only about the size of an atomic nucleus but expand the space inside the bubble enormously so that a spacecraft could fit in. This is somewhat like a nuclear sized wormhole that leads to a pocket universe that holds the spacecraft. Unfortunately, the metric doesn't fit neatly into any [[Wormholes#Three_dimensions|embedding diagram]] that I can figure out, but the math works out so that spatial coordinates inside the bubble are expanded by a factor of 1017 and a spacecraft would have a few hundred meters of bubble interior to putz around in. This trick manages to reduce the magnitude of energy to only about the mass energy of a few stars similar to our own. Other than that, it is otherwise a normal Alcubierre drive.

Further work<ref>S. Krasnikov, "The quantum inequalities do not forbid spacetime shortcuts", Physical Review D. 67 (10) 104013. https://doi.org/10.1103%2FPhysRevD.67.104013, arXiv:gr-qc/0207057 (2003) https://doi.org/10.48550/arXiv.gr-qc/0207057 </ref> has managed to use Van Den Broeck's trick to reduce the magnitude of the negative energy to only about a milligram.

== Natário warp drive ==

The Alcubierre drive is not the only way to construct a warp drive. José Natário<ref name="Natario"></ref> showed that it was just one example of an entire class of warp drives. These are space-time geometries with a smooth and localized shift vector field and a unit lapse function.

Natário then went on to find in that class a set of warp drives that do not have any expansion or contraction of space at all. Instead, space encountering the front boundary of the warp bubble instead "slides" around the outside of the bubble until it gets to the corresponding place on the back and is left behind there. Another way to think of it is that space entering the bubble shell is compressed in the radial direction of the shell but is simultaneously expanded in the tangential direction so that there is no net change in volume. When it gets to the back, the opposite occurs. In the language of mathematics, the shift vector field is divergenceless.

<table align=center>
<tr><td width=360>[[File:Natario_drive_flow_of_space.png]]
<td width=445>[[File:Natario_warp_drive_horizons.png]]
<tr><td style="vertical-align: top;">A representation of the motion of space in a zero-expansion Natário drive with a vanishingly thin warp shell around the bubble.
<td>The horizons of the generalized Natário class of warp drives (including the Alcubierre warp drive and all fell-Heisenberg drives) at superluminal speed in the direction of the green arrow. No event inside the bubble can affect events outside the bubble in front of the blue line. No events outside the bubble behind the red line can affect events inside the bubble. The lines have an angle α with respect to the direction of motion with sin(α) = 1/vs for vs the speed of the bubble relative to light speed. This is analogous to the Mach cone of supersonic objects.
</table>

NOTE: There are two different things that can be meant by the term "Natário warp drive". The generalized Natário warp drive is the general theory of all warp drives that includes within it the Alcubierre, Fell-Heisenberg, and Lentz warp drives, among infinitely many others. The zero-expansion Natário warp drive is the specific implementation of the generalized Natário warp drive that has a divergenceless shift vector field.

The generalized class of warp drives developed by Natário (including both the Alcubierre and this zero-expansion warp drive) are shown to always have regions in the warp shell where the energy density is negative to at least some observers. You can't get away from it – to warp, you need negative energy density.

The Natário class of warp drives blueshift light coming in from the front and redshift light catching up from the back. For the case of a super-luminal warp bubble, there will of course be a horizon at the back of the bubble and light will not be able to get through from behind. For a bubble speed vs relative to the speed of light, light coming from straight ahead will be blueshifted up in frequency and photon energy by a factor of 1 + vs. For sub-luminal travel, light coming from behind will be redshifted by a factor 1 - vs. In general for n as the unit direction along which the light is propagating, the blueshift factor will be 1 + vs ⋅ n. If the warp shell is infinitesimally thin, light from outside that reaches the center of the bubble will not be distorted in direction although it will be frequency shifted, but observers still inside the bubble but displaced from the center will see the field of view distorted as well as frequency shifted.

Most of the difficulties that apply to the Alcubierre warp drive also apply to the generalized Natário warp drive (and thus to the zero-expansion Natário drive and Fell-Heisenberg drive as well). In particular, quantum energy inequalities, high magnitudes of the energy required, formation of time machines, instabilities related to the stress-energy making up the warp shell being unable to keep up with the bubble, the chicken-and-egg problem of needing FTL to achieve FTL, the formation of horizons making warp bubbles uncontrollable, and Hawking radiation at the horizons, are all still concerns for generic warp drives.

== Fell-Heisenberg warp drives ==

S. D. B. Fell and L. Heisenberg investigated what sort of warp structure is necessary to produce a warp drive with positive energy density everywhere<ref name="Fell-Heisenberg">S. D. B. Fell and L> Heisenberg, "Positive energy warp drive from hidden geometric structures", Classical and Quantum Gravity 38 155020 (2021) https://doi.org/10.1088/1361-6382/ac0e47 https://arxiv.org/abs/2104.06488</ref>. They found that warp drives of the generic Natário form with irrotational shift vectors had a simple expression for the energy density and vanishing momentum (irrotational means that the shift vector can be described as the gradient of a scalar potential field or, equivalently, that the curl of the shift vector field vanishes everywhere). Fell and Heisenberg proposed several potential fields that either had positive energy density everywhere except at discontinuities, or that were continuous but had regions that violated the weak energy condition (i. e. negative energy densities in some frames of reference). One specific example they provided of a warp field had an energy that is about 10,000 times less than the mass-energy of our sun. Or only about half the mass-energy of Jupiter. This is a significant improvement over many other proposed drives.

Despite the introduction discussing warp drive configurations that satisfy the various energy conditions, the configurations described in the paper are shown to locally violate the weak and strong energy conditions. Nonetheless, the energy density is still mostly positive. Santiago, Schuster, and Visser's<ref name="Santiago Schuster Visser 2022">J. Santiago, S. Schuster, and M. Visser, "Generic warp drives violate the null energy condition", Physical Review D 105, 064038 (2022) https://doi.org/10.1103/PhysRevD.105.064038</ref> work shows that the various energy conditions must still be violated by this warp drive; to not violate these, energy density must be positive in all reference frames not just those of the co-moving observer.

Any vector field can be described by its divergence, its curl, and a constant. The constant part of a warp drive's shift vector can be discarded because then the warp drive would not be localized. It is interesting, therefore, that the Natário zero-expansion drive has zero divergence and can be described only by its curl, while the Fell-Heisenberg drive has zero curl and can be described only by its divergence. These two classes of drives thus rely on opposite choices of setting one of the two different parts describing a vector field to zero. As might be expected, the Fell-Heisenberg drive is not zero expansion – space-time will change in volume as it passes through the warp bubble (although after going all the way through it will return to its original volume). However, objects in space-time will retain their original orientation as they pass through the warp bubble.

Previous examples of warp drives, such as the Alcubierre and zero-expansion Natário drives, tend toward a flat space-time as you go far enough away faster than would occur if there were any mass present in the warp drive. That is, the ADM mass of such drives is zero. However, one of the Fell-Heisenberg examples described a spacetime where the geometry approaches that of what you would get for a massive object at the warp location, giving one of the first examples of a warp drive with mass.

In the Fell-Heisenberg drive the momentum vanishes everywhere. Yet the energy occupies regions where the shift vector is varying rapidly. The lack of momentum means that the energy will not move to keep up with the differential expansion and movement of the spacetime. As a consequence, at later times the energy will have a different distribution than what is necessary to maintain the given warp configuration. The exact time evolution is not solved but this likely leads to collapse of warp bubble. Note that this is not a problem unique to the Fell-Heisenberg drive. Assuming a geometry that describes a warp produces a specific distribution of the stress-energy tensor that satisfies Einstein's equation. But there is no guarantee that this stress energy tensor satisfies the continuity equation, which describes how the distribution of matter and momentum change with time. The issue of stability of any warp drive is not well understood, but most warp drive geometries will be unstable in the sense that their matter that generates the warp fields will not be able to keep up with the motion of the warp. The Fell-Heisenberg drive is unique only in that, with vanishing momentum, there is no configuration that allows stable propagation.

(n. b. The Heisenberg here is Lavinia Heisenberg, not the Werner Heisenberg of quantum physics and uncertainty principle fame.)

== Lentz warp drive ==

E. Lentz<ref name="Lentz">E. W. Lentz, "Breaking the warp barrier: hyper-fast solitons in Einstein–Maxwell-plasma theory", Classical and Quantum Gravity. 38 075015 (2021). arXiv:[https://arxiv.org/abs/2006.07125 2006.07125]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2021CQGra..38g5015L 2021CQGra..38g5015L]. doi:[https://doi.org/10.1088%2F1361-6382%2Fabe692 10.1088/1361-6382/abe692]. ISSN [https://search.worldcat.org/issn/0264-9381 0264-9381]. S2CID [https://api.semanticscholar.org/CorpusID:219635854 219635854].</ref> investigated a particular configuration of the Fell-Heisenberg class of warp drives, where the scalar field has a hyperbolic shape. The necessary conditions were found to be satisfied by a selection of rhomboid regions of the scalar field, resulting in an energy density that is everywhere positive and a shift vector with vanishing tidal force at the center for a comfortable ride.

However, Santiago, Schuster, and Visser<ref name="Santiago Schuster Visser 2022"></ref> dispute claims that the Lentz drive satisfies the energy conditions, noting that everywhere positive energy density in one frame of reference is insufficient to establish that the energy density is positive in all reference frames; knowledge of the Cauchy stress tensor is also needed. They show that any generic warp drive will violate the strong energy condition, null energy condition, and weak energy condition.

== Conservation laws ==

(Discussion of asymptotic flatness: what is it? Where does it apply?)

(The "big four" conservation laws and how they relate to asymptotic flatness.)

(ADM mass -> 0; conservation of energy/mass issues.)<ref>Sebastian Schuster, Jessica Santiago,and Matt Visser, "ADM mass in warp drive spacetimes" General Relativity and Gravitation (2023) 55:14 https://doi.org/10.1007/s10714-022-03061-9</ref>

(Discuss issues of conservation of angular momentum.)

(If angular momentum conservation is ignored, discuss how momentum & energy are affected by outside forces. In a gravitational field equivalent to inertial frame ... like being in an accelerated elevator; will acquire momentum buildup while staying "at rest".)

== Warp Railroads ==

Van Den Broeck<ref name="VanDenBroeck_2000"></ref> suggested a warp railroad,

You can set up devices ahead of time along the path of the bubble that produce the negative energy regions for the warp bubble at the appropriate time without needing the negative energy to ever move with the bubble at all. At least for bubbles on a predictable schedule and route.

helps with:

-stability (elaborate)

-conservation laws (elaborate)

Mention Krasnikov<ref name="Krasnikov 1998">S. V. Krasnikov, "Hyperfast travel in general relativity" Phys. Rev. D 57 4760</ref>

Hyperwave<ref name="Pieri 2024">Lorenzo Pieri, "Hyperwave: hyper-Fast Communication within General Relativity", arXiv:2311.12069 [gr-qc] https://doi.org/10.48550/arXiv.2311.12069</ref> makes use of this warp railroad idea.

== Quantum effects ==

(quantum stuff here<ref name="Hiscock1997">W. A. Hiscock, "Quantum effects in the Alcubierre warp drive spacetime", Classical and Quantum Gravity 14 L183 https://doi.org/10.1088/0264-9381/14/11/002 https://arxiv.org/abs/gr-qc/9707024</ref><ref name="Finazzi et al 2009">S. Finazzi, S. Liberati, C. Barceló, "Semiclassical instability of dynamical warp drives", Physical Review D 79, 124017 (2009)https://doi.org/10.1103/PhysRevD.79.124017 https://arxiv.org/abs/0904.0141</ref>)

From<ref name="Pieri 2024"></ref>
<blockquote>
It should be emphasised that Quantum Inequalities have not been proven for realistic interacting quantum fields, where the they are expected to be weaker if they exist [Cadamuro, 2019], and they often require Minkowskian conditions at infinity, a condition which is not appropriate for fields inside bounded regions, as in a Casimir devices.
</blockquote>

== Warp drives and black holes ==

What happens if you drive your warp drive into a black hole?

The pop-sci reason that nothing can escape a black hole is that at the event horizon the escape velocity is faster than light and nothing can go faster than light. A warp drive can go faster than light.

The more detailed reason from general relativity is that at the event horizon space and time are rotated sufficiently that "inward" becomes "forward in time." You can no more go farther away from the singularity in a black hole once you are inside the event horizon than you can go backward in time. Any super-luminal travel can go backward in time, and warp drives can engage in super-luminal travel.

The Hawking area theorem that says that black holes can only grow, never shrink relies on the condition that the energy is everywhere positive. Warp drives have negative energy density.

So in principle, there's nothing that prevents a warp drive from taking a dip into a black hole and coming back out again.

Garattini and Zatrimaylov looked into the problem of a warp drive near (and in) a black hole<ref name="Garattini and Zatrimaylov">R. Garattini and K. Zatrimaylov, "Black holes, warp drives, and energy conditions", Physics Letters B 856 138910 (2024) https://doi.org/10.1016/j.physletb.2024.138910</ref>. The found that the black hole reduces the negative energy requirement for the warp bubble to move inward toward the black hole singularity, but increases the negative energy needed to move away from the center of the black hole. In addition, a warp drive parked at the event horizon would allow light from inside the horizon to pass through the bubble and escape back outside.

== Credit ==
Author: Luke Campbell

== References ==

[[Category:Physics]][[Category:Infrastructure]][[Category:Physics & Engineering‏‎]][[Category:Physics & Math & Engineering]][[Category:Transportation & Infrastructure‏‎]][[Category:Metric Engineering]]

Warp Drives

2026-07-27T00:08:17Z

Lwcamp: /* The Alcubierre warp drive */

{{PageConstructionNotice}}

Science fiction often features spacecraft that can seemingly move across space and get between the place of departure and the destination much faster than light could have done. This appears to contradict the theory of relativity, which predicts unequivocally that nothing can move through space faster than light. Because relativity has been incredibly successful at describing nature, with its many other predictions regularly being confirmed to extraordinary accuracy and within the bounds of uncertainty of all the experiments that tested them, it gives confidence that relativity is a correct description of reality. Which seems to rather throw a wet towel on our hopes for rapid travel between stars.

However, while relativity does not allow things to move through space faster than light, it places no such restrictions on how fast space-time itself can expand, contract, or move around. This leads to the idea of a warp drive – the spacecraft remains stationary within a region of highly curved space-time, and that region moves at super-luminal speeds rather than the spacecraft.

== The Alcubierre warp drive ==

The first warp drive geometry that satisfied the Einstein field equations of relativity was proposed by Miguel Alcubierre<ref>M. Alcubierre, "The warp drive: hyper-fast travel within general relativity." Classical and Quantum Gravity. 11 (5): L73–L77 (1994). [https://arxiv.org/abs/gr-qc/0009013 arXiv:gr-qc/0009013]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1994CQGra..11L..73A 1994CQGra..11L..73A]. doi:[https://doi.org/10.1088%2F0264-9381%2F11%2F5%2F001 10.1088/0264-9381/11/5/001]. S2CID [https://api.semanticscholar.org/CorpusID:4797900 4797900].</ref>. In this geometry, a sphere of space-time moves at an arbitrary speed (potentially but not necessarily a speed much faster than light). Objects within the sphere are moved along with the sphere; an object at rest within the sphere would be moved along with the sphere indefinitely. Space is expanding at the rear boundary and contracting at the front boundary in order to keep the sphere moving. In order to satisfy the Einstein field equations, the boundary of the sphere must have a negative energy density. The challenges of space-time geometries with negative energy densities are described in our page on [[Wormholes#Exotic_energy_conditions|wormholes]], for our purposes it is enough to note that negative energy densities can pose problems if not handled carefully, there may be limits on how much negative energy you can have without nearby positive energy density, and it may not be possible to get enough negative energy to support a warp drive; although none of this is ruled out by physics – yet!

<table align=center>
<tr><td width=650>[[File:Energy_magnitude_of_Alcubierre_drive.png]]
<td width=650>[[File:Expansion_of_Alcubierre_drive.png]]
<tr><td>The magnitude of the energy of the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. (The energy density is negative everywhere in the shell, here only the magnitude is shown to aid visual comprehension.)
<td>The expansion of space in the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. Negative expansion means that space is contracting in that direction, positive expansion that it is expanding.
</table>

NOTE: What the? The energy distribution is symmetric! How does the drive know which way to go? There must be significant contributions of other components of the stress-energy tensor, and those have got to be asymmetric along the forward/backward axis. Check this when I get time.

(And what's with the ring in all the artwork, anyway?)

=== So what is going on? ===

What is it that makes a warp drive a warp drive, and how does it work?

A warp drive is a concept that comes out of the science of general relativity. General relativity collects space and time together into a single "structure" called spacetime. This structure is curved, and its curvature is determined by the material that is in it. Absent forces acting between matter, particles of matter move through spacetime on the shortest possible paths in this curved environment.
<blockquote>
"Spacetime tells matter how to move; matter tells spacetime how to curve." - John Archibald Wheeler
</blockquote>
All of this is encompassed in the Einstein field equation:
<div align=center>
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 8πG
<td rowspan="2" nowrap="nowrap" align="right"> T = R -  
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right"> R g
<tr>
<td style="border-top:solid 1px black;"> c4
<td style="border-top:solid 1px black;"> 2
</table>
</div>
Unfortunately, by convention, both the symbols G and R get used for a lot of different things. Here, symbols in bold are second rank tensors – constructs of arrays of 4×4 numbers where 4 is necessary because that is the number of dimensions of spacetime. Conventionally, these numbers are indexed by a value going from 0 to 3, with 0 being the part corresponding to the coordinate of time and 1 to 3 being the Cartesian coordinates that you might know as x, y, and z.

T is the stress-energy tensor: the (0,0) component represents the energy density at a given point; the (0,i) and (i, 0) components (with i going from 1 to 3) represent the momentum density; and the (i,j) components represent the stresses present, such as pressure or tension or shear.

The other term, R - R g/2, involves a partial description of the curvature. A full description of the curvature requires a construct of 4×4×4×4 components; too much for the stress energy tensor to fully constrain. Which is okay, because this allows curvature to exist away from where matter is, allowing gravity to have long range effects across empty space as well as for phenomena like gravitational waves. But it is enough that a given distribution of matter T gives enough constraints (along with specifying how the geometry behaves far away) that you can generally figure out what the actual gravity is.

This is the usual story that we are told. But there is an alternate way of looking at the same geometry such that the math works out the same, but it is described conceptually in a very different way. This divides spacetime up into "layers", called foliations, that represent slices of space and going from one foliation to the next is increasing in time. This is useful for computation because you can assign coordinates to your system. There are two additional constructs – the lapse function and the shift vector field. Starting at an event on one foliation, the lapse function at that position tells you that after a certain "proper time" (time perceived by an observer at that event) the time coordinate of the observer will be the proper time times the lapse function at that event. And the new spatial coordinates of the observer will be the proper time times the shift vector at that event. (Strictly speaking, this is only exactly true in the limit of the proper time approaching zero or when the lapse function and shift vectors are constant. But hopefully this gives a good enough insight to what these entities are so that we can go on to describe warp drives.)

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Lapse_shift.png]]
<tr><td>A folliation of space-time into slices of constant time coordinates (grey). At an event (black circle), the lapse function (yellow) shows how far into the future the point at that event will be projected in a unit amount of proper time experienced by an observer initially at the event. The shift vector (red) shows how that point changes coordinates after the unit amount of proper time. The cyan vector shows the new coordinates the point is projected onto. The lower black surface indicates a foliation of spacetime at the same time coordinate; after unit proper time, the events on the lower foliation get projected to the curved upper black foliation according to the shift vectors and lapse functions at each of their respective points.
</table>
</table>

Warp drives are generally restricted to spacetime geometries with a constant lapse function - all observers go forward by the same time coordinate after experiencing the same amount of proper time. Thus, all warp drives have no time dilation.
Far from the warp bubble, the shift vector vanishes – each spatial coordinate is mapped to the same spatial coordinate after any amount of proper time so long as the warp drive is not near. But the inside of the warp bubble is an extended region where the shift vector field is non-zero and constant. In this bubble, all spatial coordinates are mapped to a coordinate that shifts linearly in time with the proper time of the observer. Astute readers will see that this linear change in position with time is very similar to a velocity. This region of constant shift vector then moves with the shift vector velocity to keep stuff originally in the bubble so that it stays in the bubble. In between the inside of the bubble and the far away regions with zero shift vector, the shift vector field falls away to match one region to the other.

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Warp_shift.png]]
<tr><td>The shift vector of a one-dimensional slice through warp geometry, in red. At the edges there is no shift. At the center (inside the warp bubble) there is a region of uniform shift so that at a later time the inside of the warp bubble is rigidly translated to different spatial coordinates (in blue). The two foliations remain the same distance apart, because warp drives maintain a unit lapse function.
</table>
</table>

Traditionally, scientists would start with an interesting distribution of matter given by T and then find out what kind of curvature it created. This works well for planets and stars and black holes and cosmic strings and other things which are observed in the universe and you want to see what sort of effect they had. But starting in the 1980's, some physicists started looking at Einstein's field equations the other way around. They start with a desired geometry, find the curvature part of the field equation R - R g/2, and use that to find out what T has to be in order to get that curvature. This was originally done for [[Wormholes|wormholes]], but in 1994 Miguel Alcubierre used this same trick to figure out what could create a region of constant shift vector that moved with the shift vector - the warp drive. With this demand for a warp geometry, the required distribution of matter and energy and momentum and stress can be determined.

The original Alcubierre shift vector N→ produces a spherical warp bubble with an axi-symmetric distribution of, well, everything along the direction of motion with warp velocity vector v→.
<div align="center">
N→ = v→ f(rs)
</div>
Here, rs is the distance from the center of the warp bubble, and f(rs) can be any function that is 1 near rs = 0 and is 0 for rs well beyond the radius of the warp bubble.

=== Challenges and possible resolutions ===

Almost as soon as Alcubierre proposed his warp drive, others began picking it apart.
<ol>
<li>Pfenning and Ford<ref name="PfenningFord_2001">M. J. Pfenning and L. H. Ford, "The unphysical nature of 'Warp Drive'", Classical and Quantum Gravity. 14 (7): 1743–1751 (1997). arXiv:[https://arxiv.org/abs/gr-qc/9702026 gr-qc/9702026]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1997CQGra..14.1743P 1997CQGra..14.1743P]. doi:[https://doi.org/10.1088%2F0264-9381%2F14%2F7%2F011 10.1088/0264-9381/14/7/011]. S2CID [https://api.semanticscholar.org/CorpusID:15279207 15279207].</ref> found that if the negative energy density wall around the bubble obeys [[Wormholes#Quantum_energy_inequalities|quantum energy inequalities]], then for warp speeds of around light speed the shell thickness would be on the order of a hundred Planck lengths; a distance far smaller than any other known physical phenomenon or object.
<li>In the same work, Pfenning and Ford also found that for a bubble a hundred meters in radius, a warp speed of around the speed of light, and the required thickness of about 100 Planck lengths the energy in the bubble shell would be E ≈ -1063 kg c2. The magnitude of this latter value is ten orders of magnitude larger than the energy of the entire visible universe. They do note, however, that if quantum energy inequalities can be ignored then a warp drive with a hundred meters radius but a shell thickness of one meter would "only" have an energy magnitude of about a quarter solar mass.
<li>As with any method of faster than light travel, the warp drive could be used to make a time machine.
<li>Perhaps most seriously, if the warp drive is going faster than light speed, the negative energy regions on the outside of the shell won't be in the warp parts of the bubble. They will have to be moving through space at faster than light speed if the bubble is to maintain its integrity, which is the very problem that the warp drive was designed to avoid. The outside of the warp bubble shell that maintains the warp bubble would fall away and the warp bubble would quickly erode to sub-luminal speeds.<ref name="Van Den Broeck 2000">Chris Van Den Broeck, "On the (im)possibility of warp bubbles", arXiv:gr-qc/9906050 https://doi.org/10.48550/arXiv.gr-qc/9906050</ref>
<li>A more general problem that includes the previous one is that the geometry of the warp drive determines the distribution of energy, momentum, and stress (pressure, shear, and tension) by Einstein's field equation, which states that the stress energy tensor is directly proportional to a reduced order curvature tensor of space-time. However, the stress-energy tensor must also obey the continuity equation – basically, for the energy to change in a given volume of space that energy must come in or go out of the boundary of that volume and similarly for the momentum. So a warp drive can have the warp condition for one moment in time. But after that, the stress-energy that maintains the warp will start to move according to the forces on it and according to the natural paths that it takes through the warped space-time. There is no guarantee that the configuration of the stress-energy at later times will be sufficient to maintain the warp.
<blockquote>
From the perspective of simulating the warp drive dynamically, the key challenge is stability. One can straightforwardly generate the initial metric that describes the Alcubierre metric and infer the instantaneous matter configuration that must support it using the Einstein Equation, but a time-domain evolution of the coupled matter-gravity system necessitates that we specify the equation of state of the matter, which in turn determines how it evolves. In a perfect fluid, this involves specifying the relation between the fluid pressure and density, but in a more general case, the full relation between the
stress-energy components must be given. There is (to our knowledge) no known equation of state that would maintain the warp drive metric in a stable configuration over time - therefore, whilst one can require that initially, the warp bubble is constant, it will quickly evolve away from that state and, in most cases, the warp fluid and spacetime deformations will disperse or collapse into a central point.
 
– Clough, Dietrich, and Khan<ref>Katy Clough, Tim Dietrich, and Sebastian Khan, "What no one has seen before: gravitational waveforms from warp drive collapse", arXiv:2406.02466 [gr-qc] https://doi.org/10.48550/arXiv.2406.02466</ref>
</blockquote>
<li>In order to get the matter (or more generally, stress-energy) that forms the warp shell up to speeds that could allow the shell to exist, you need to be able to, well, accelerate that matter to warp speeds. Which gives a chicken-and-egg problem - in order to achieve superluminal travel you need to be able to get matter or energy moving superluminally in the first place<ref name="Bobrick and Martire">Alexey Bobrick and Gianni Martire, "Introducing Physical Warp Drives", arXiv:2102.06824 [gr-qc] https://doi.org/10.48550/arXiv.2102.06824</ref><ref>D. H. Coule, "No warp drive", Class. Quantum Grav. 15 (1998) 2523–2527. https://doi.org/10.1088/0264-9381/15/8/026</ref>, which rather makes warp drives unnecessary from the get-go.
<li>A warp bubble moving faster than light will not be able to get any signals in front of it as it will overtake its own signal. In fact, anyone in the interior of the bubble would not be able to send signals to the front surface of the bubble. In science-speak, a horizon forms at the front of the bubble<ref name="Natario">José Natário, "Warp drive with zero expansion", Classical and Quantum Gravity. 19 (6): 1157–1166 (2002). arXiv:[https://arxiv.org/abs/gr-qc/0110086 gr-qc/0110086]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2002CQGra..19.1157N 2002CQGra..19.1157N]. doi:[https://doi.org/10.1088%2F0264-9381%2F19%2F6%2F308 10.1088/0264-9381/19/6/308]. S2CID [https://api.semanticscholar.org/CorpusID:15859984 15859984].</ref>. As a consequence, the bubble cannot be controlled, steered, nor stopped from within the bubble itself.
<li>A Horizon will naturally emit Hawking radiation. It is estimated that the radiation will raise the temperature of the interior of the bubble to approximately 1032 K<ref name="Finazzi 2009">Stefano Finazzi, Stefano Liberati, and Carlos Barceló, "Semiclassical instability of dynamical warp drives", Phys. Rev. D 79, 124017 (12 June, 2009) https://doi.org/10.1103/PhysRevD.79.124017 , arXiv:0904.0141 [gr-qc] https://doi.org/10.48550/arXiv.0904.0141</ref> (at this level of accuracy, the result is the same if you replace temperatures in K by the same number in °C or °F). This temperature, however, may be lower if quantum inequalities do not have to be fulfilled.
<li>Semi-classical instabilities at the horizon are expected to lead to a divergence of the stress-energy tensor at the boundary in a short period of time, thus leading to the destruction of the warp bubble<ref name="Finazzi 2009"></ref>.
</ol>
Helpfully, Van Den Broeck<ref name="VanDenBroeck_2000">C. Van Den Broeck, "Alcubierre’s Warp Drive: Problems and Prospects" AIP Conference Proceedings. 504: 1105–1110 (2000). Bibcode:[https://ui.adsabs.harvard.edu/abs/2000AIPC..504.1105V 2000AIPC..504.1105V]. doi:[https://doi.org/10.1063%2F1.1290913 10.1063/1.1290913].</ref> suggested several solutions – or at least mitigations – for some of these problems
<ol>
<li>Quantum inequalities had not been shown to be true in general for highly curved space-times.
<li>A proposed warp drive geometry<ref name="VanDenBroeck_1999">C. Van Den Broeck, "A 'warp drive' with more reasonable total energy requirements". Classical and Quantum Gravity. 16 (12): 3973–3979 (1999). arXiv:[https://arxiv.org/abs/gr-qc/9905084 gr-qc/9905084]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1999CQGra..16.3973V 1999CQGra..16.3973V]. doi:[https://doi.org/10.1088%2F0264-9381%2F16%2F12%2F314 10.1088/0264-9381/16/12/314]. S2CID [https://api.semanticscholar.org/CorpusID:15466313 15466313]. </ref> (see the van Den Broeck drive, below) would be able to minimize the magnitude of the energy required to about that of the mass of our sun. While still large, it is not unphysically large.
<li>Time travel is always going to be a worry with faster than light travel. There's no neat solution to this one.
<li>It may be possible for the negative energy outside of the superluminal region to compress into the superluminal region to form a shock in space-time. The front surface of the bubble would then be a singularity. It is not clear if such a sharp jump in physical properties is possible, but neither is sure that it is impossible, either. However, while this might help with the negative energy in front of the bubble getting swept up because it cannot move faster than light, it does little to help with the material at the back of the bubble on the outside getting left behind because it cannot keep up.
</ol>
Other mitigating research has also shown
<ol start=9>
<li>Semi-classical divergence only occurs in warp drives with only one spatial dimensions. Those with two or more spatial dimensions (as in our universe) only contain points where divergences occur, and these divergences can be managed. In fact, their effects can be reduced by making the warp shape more elongated<ref>Carlos Barceló, Valentin Boyanov, Luis J. Garay, Eduardo Martín-Martínez,
and Jose M. Sánchez Velázquez, "Warp drive aerodynamics", JHEP08 (2022) 288 https://doi.org/10.1007/JHEP08(2022)288 , arXiv:2207.06458 [gr-qc] https://doi.org/10.48550/arXiv.2207.06458</ref>.
</ol>

=== Alcubierre warp interactions with light and matter ===

Space is not entirely empty. It is filled with a diffuse plasma in the form of the interstellar medium, as well as cosmic radiation, light from stars, and cosmic microwave background radiation. A warp drive propagating through space will encounter this stuff. What happens when this matter and radiation have a warp bubble pass across them?

The first analysis of matter encountering a warp bubble was performed by Pfenning and Ford<ref name="PfenningFord_2001"></ref>. They looked at the warp bubble interaction with a massive object at rest with the frame of reference of the warp drive (keep in mind that the rest frame is the same inside and outside the warp bubble; in this rest frame objects inside the bubble have no momentum even though they are, in some sense, changing location rapidly with time). When the warp bubble passes, the object experiences an acceleration in the direction of the warp bubble motion. When the warp shell passes and the object is inside the bubble, it will be moving with approximately the speed of the bubble. Pfenning and Ford analyzed this problem with a continuous distribution of shell energy that, strictly speaking, never falls to zero except at the bubble center and at spatial infinity, so unless the object passes through the center in Pfenning and Ford's description it will never quite get up to the bubble's speed. In this case, the object will move almost as fast as the object but will pass through the bubble in a finite time, after which it will again be at rest with respect to the reference frame but displaced along the direction of the warp bubble motion by some distance. With this description of the warp bubble, a spacecraft of finite size will always be moving a little slower than the warp bubble and would have to use rockets to keep up with it. in addition, the spacecraft would experience tidal forces that would cause stress on the spacecraft's structure.

Pfenning and Ford also examined cases where the shell is of a finite (possibly infinitesimal) width and falls to zero both inside and outside the bubble. In this case any matter encountering the bubble would be collected at the bow of the bubble and thereafter move along with it.

McMonigal et al. analyzed the situation for both massive particles and light moving along the axis of travel of the bubble<ref name="McMonigalEtAl_2012">B. McMonigal, G. F. Lewis, and P. O'Byrne, "Alcubierre warp drive: On the matter of matter". Physical Review D. 85 (6) 064024 (20 March 2012). arXiv:[https://arxiv.org/abs/1202.5708 1202.5708]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2012PhRvD..85f4024M 2012PhRvD..85f4024M]. doi:[https://doi.org/10.1103%2FPhysRevD.85.064024 10.1103/PhysRevD.85.064024]. S2CID [https://api.semanticscholar.org/CorpusID:3993148 3993148].</ref>. They found that light moving opposite the warp direction passed through the bubble without incident, being only somewhat delayed by passing through the bubble. A warp bubble that is warping at sub-luminal speeds can have light catch up from behind it. This light is able to pass through the bubble, and is somewhat advanced in its path by the speed of the bubble. For super-luminal warp bubbles, however, the situation is different. The bubble will catch up to light moving it its own direction that is originally in front of it. This light cannot escape forward, the bubble being too fast. Nor can it escape backward, as the light is propagating forward and the interior of the bubble is at rest. Thus, the light gets caught at the bow of the warp bubble, unable to escape for so long as the warp bubble is active. This light is strongly blue shifted to extremely energetic x-rays and gamma rays. The space behind the bubble is swept clear of forward-moving light.

The situation for matter over-run by a super-luminal warp bubble is similar. Matter moving backward passes through the bubble, being only swept a ways along the bubble's path as sign of its passing but otherwise continuing on their way. Matter at rest behaves the same way as Pfenning and Ford discovered. Meanwhile, matter moving in the same direction of the warp is overtaken and collects at the bow of the bubble, unable to leave for so long as the bubble is warping. This matter "surfing" on the bubble bow is highly accelerated to relativistic speeds, experiencing extreme time dilation.

For a sub-luminal bubble, there is a speed faster than the warp speed where particles coming up from behind overtake the bubble and pass through, out the front. Particles below this speed but still faster than the warp speed still overtake the bubble, but then bounce backwards out of the bubble from behind. They are still moving forward, but are now moving more slowly than the warp speed. Particles moving forward but slower than the bubble will be overtaken and then bounced out the front with a speed higher than the warp speed. Those particles moving backward will pass through the bubble and exit with their initial speed.

The spacecraft inside the bubble will observe that light moving backward compared to the warp direction is blue-shifted, and matter moving backward passes through with increased energy. Meanwhile, a sub-luminal bubble will see light that was moving forward as red-shifted and matter catching up to the bubble from behind will pass by with reduced energy (superluminal bubbles, of course, do not have any matter or radiation catching up to them from behind).

When a superluminal warp bubble that has been collecting matter and radiation for the duration of its journey and blue-shifting it to much higher energies is turned off, all that matter and energy is released as a blast of radiation in the direction the bubble was warping.

McMonigal et al. conclude by noting that any spacecraft in the warp bubble would need shielding to protect against the blue-shifted radiation and matter of increased energy. In addition the destination would be "blasted into oblivion" by the release of matter and radiation that had been caught in the bubble during the trip.

Clark et al.<ref>Chad Clark, William A. Hiscock, and Shane L. Larson, "Null geodesics in the Alcubierre warp drive spacetime: the view from the bridge", arXiv:gr-qc/9907019 (2018) https://doi.org/10.48550/arXiv.gr-qc/9907019</ref> examine what it would look like from within an operating warp drive. They find that light moving at 90 degrees to the direction of motion will look like it is still coming from 90 degrees. However, light from the front hemisphere is squished progressively toward the forward point the faster you go. For a warp bubble moving faster than the speed of light, there is a cone behind the warp bubble where no light can reach it. However, you don't see a black spot behind you – light from the regions in the backward hemisphere from outside that cone that can reach you is distorted toward the direction behind you to fill in that spot. Light from the forward direction is blue-shifted and light from behind is red-shifted. At speeds of around 200 times light speed, the cosmic microwave background would be blue shifted to energies similar to that of the solar photosphere, heating anything in the warp bubble as if they were close to the sun.

== Van Den Broeck warp drive ==

If an Alcubierre warp bubble a hundred meters across requires a magnitude of energy greater than the entire energy in the observable universe, one option to reduce the magnitude of energy used is to make the warp bubble smaller. Much smaller. Pfenning and Ford<ref name="PfenningFord_2001"></ref> suggested making the warp bubble smaller than an atom. Of course, that brings up the problem of how to stuff a spacecraft in there. Van Den Broeck proposed a solution<ref name="VanDenBroeck_1999"></ref>: make the warp bubble only about the size of an atomic nucleus but expand the space inside the bubble enormously so that a spacecraft could fit in. This is somewhat like a nuclear sized wormhole that leads to a pocket universe that holds the spacecraft. Unfortunately, the metric doesn't fit neatly into any [[Wormholes#Three_dimensions|embedding diagram]] that I can figure out, but the math works out so that spatial coordinates inside the bubble are expanded by a factor of 1017 and a spacecraft would have a few hundred meters of bubble interior to putz around in. This trick manages to reduce the magnitude of energy to only about the mass energy of a few stars similar to our own. Other than that, it is otherwise a normal Alcubierre drive.

Further work<ref>S. Krasnikov, "The quantum inequalities do not forbid spacetime shortcuts", Physical Review D. 67 (10) 104013. https://doi.org/10.1103%2FPhysRevD.67.104013, arXiv:gr-qc/0207057 (2003) https://doi.org/10.48550/arXiv.gr-qc/0207057 </ref> has managed to use Van Den Broeck's trick to reduce the magnitude of the negative energy to only about a milligram.

== Natário warp drive ==

The Alcubierre drive is not the only way to construct a warp drive. José Natário<ref name="Natario"></ref> showed that it was just one example of an entire class of warp drives. These are space-time geometries with a smooth and localized shift vector field a shift vector is a vector field that describes how one "slice" of space out of space-time connects to the next) and a unit lapse function (the lapse function describes how the time you experience changes when going from a "slice" of space-time to the next).

Natário then went on to find in that class a set of warp drives that do not have any expansion or contraction of space at all. Instead, space encountering the front boundary of the warp bubble instead "slides" around the outside of the bubble until it gets to the corresponding place on the back and is left behind there. Another way to think of it is that space entering the bubble shell is compressed in the radial direction of the shell but is simultaneously expanded in the tangential direction so that there is no net change in volume. When it gets to the back, the opposite occurs. In the language of mathematics, the shift vector field is divergenceless.

<table align=center>
<tr><td width=360>[[File:Natario_drive_flow_of_space.png]]
<td width=445>[[File:Natario_warp_drive_horizons.png]]
<tr><td style="vertical-align: top;">A representation of the motion of space in a zero-expansion Natário drive with a vanishingly thin warp shell around the bubble.
<td>The horizons of the generalized Natário class of warp drives (including the Alcubierre warp drive and all fell-Heisenberg drives) at superluminal speed in the direction of the green arrow. No event inside the bubble can affect events outside the bubble in front of the blue line. No events outside the bubble behind the red line can affect events inside the bubble. The lines have an angle α with respect to the direction of motion with sin(α) = 1/vs for vs the speed of the bubble relative to light speed. This is analogous to the Mach cone of supersonic objects.
</table>

NOTE: There are two different things that can be meant by the term "Natário warp drive". The generalized Natário warp drive is the general theory of all warp drives that includes within it the Alcubierre, Fell-Heisenberg, and Lentz warp drives, among infinitely many others. The zero-expansion Natário warp drive is the specific implementation of the generalized Natário warp drive that has a divergenceless shift vector field.

The generalized class of warp drives developed by Natário (including both the Alcubierre and this zero-expansion warp drive) are shown to always have regions in the warp shell where the energy density is negative to at least some observers. You can't get away from it – to warp, you need negative energy density.

The Natário class of warp drives blueshift light coming in from the front and redshift light catching up from the back. For the case of a super-luminal warp bubble, there will of course be a horizon at the back of the bubble and light will not be able to get through from behind. For a bubble speed vs relative to the speed of light, light coming from straight ahead will be blueshifted up in frequency and photon energy by a factor of 1 + vs. For sub-luminal travel, light coming from behind will be redshifted by a factor 1 - vs. In general for n as the unit direction along which the light is propagating, the blueshift factor will be 1 + vs ⋅ n. If the warp shell is infinitesimally thin, light from outside that reaches the center of the bubble will not be distorted in direction although it will be frequency shifted, but observers still inside the bubble but displaced from the center will see the field of view distorted as well as frequency shifted.

Most of the difficulties that apply to the Alcubierre warp drive also apply to the generalized Natário warp drive (and thus to the zero-expansion Natário drive and Fell-Heisenberg drive as well). In particular, quantum energy inequalities, high magnitudes of the energy required, formation of time machines, instabilities related to the stress-energy making up the warp shell being unable to keep up with the bubble, the chicken-and-egg problem of needing FTL to achieve FTL, the formation of horizons making warp bubbles uncontrollable, and Hawking radiation at the horizons, are all still concerns for generic warp drives.

== Fell-Heisenberg warp drives ==

S. D. B. Fell and L. Heisenberg investigated what sort of warp structure is necessary to produce a warp drive with positive energy density everywhere<ref name="Fell-Heisenberg">S. D. B. Fell and L> Heisenberg, "Positive energy warp drive from hidden geometric structures", Classical and Quantum Gravity 38 155020 (2021) https://doi.org/10.1088/1361-6382/ac0e47 https://arxiv.org/abs/2104.06488</ref>. They found that warp drives of the generic Natário form with irrotational shift vectors had a simple expression for the energy density and vanishing momentum (irrotational means that the shift vector can be described as the gradient of a scalar potential field or, equivalently, that the curl of the shift vector field vanishes everywhere). Fell and Heisenberg proposed several potential fields that either had positive energy density everywhere except at discontinuities, or that were continuous but had regions that violated the weak energy condition (i. e. negative energy densities in some frames of reference). One specific example they provided of a warp field had an energy that is about 10,000 times less than the mass-energy of our sun. Or only about half the mass-energy of Jupiter. This is a significant improvement over many other proposed drives.

Despite the introduction discussing warp drive configurations that satisfy the various energy conditions, the configurations described in the paper are shown to locally violate the weak and strong energy conditions. Nonetheless, the energy density is still mostly positive. Santiago, Schuster, and Visser's<ref name="Santiago Schuster Visser 2022">J. Santiago, S. Schuster, and M. Visser, "Generic warp drives violate the null energy condition", Physical Review D 105, 064038 (2022) https://doi.org/10.1103/PhysRevD.105.064038</ref> work shows that the various energy conditions must still be violated by this warp drive; to not violate these, energy density must be positive in all reference frames not just those of the co-moving observer.

Any vector field can be described by its divergence, its curl, and a constant. The constant part of a warp drive's shift vector can be discarded because then the warp drive would not be localized. It is interesting, therefore, that the Natário zero-expansion drive has zero divergence and can be described only by its curl, while the Fell-Heisenberg drive has zero curl and can be described only by its divergence. These two classes of drives thus rely on opposite choices of setting one of the two different parts describing a vector field to zero. As might be expected, the Fell-Heisenberg drive is not zero expansion – space-time will change in volume as it passes through the warp bubble (although after going all the way through it will return to its original volume). However, objects in space-time will retain their original orientation as they pass through the warp bubble.

Previous examples of warp drives, such as the Alcubierre and zero-expansion Natário drives, tend toward a flat space-time as you go far enough away faster than would occur if there were any mass present in the warp drive. That is, the ADM mass of such drives is zero. However, one of the Fell-Heisenberg examples described a spacetime where the geometry approaches that of what you would get for a massive object at the warp location, giving one of the first examples of a warp drive with mass.

In the Fell-Heisenberg drive the momentum vanishes everywhere. Yet the energy occupies regions where the shift vector is varying rapidly. The lack of momentum means that the energy will not move to keep up with the differential expansion and movement of the spacetime. As a consequence, at later times the energy will have a different distribution than what is necessary to maintain the given warp configuration. The exact time evolution is not solved but this likely leads to collapse of warp bubble. Note that this is not a problem unique to the Fell-Heisenberg drive. Assuming a geometry that describes a warp produces a specific distribution of the stress-energy tensor that satisfies Einstein's equation. But there is no guarantee that this stress energy tensor satisfies the continuity equation, which describes how the distribution of matter and momentum change with time. The issue of stability of any warp drive is not well understood, but most warp drive geometries will be unstable in the sense that their matter that generates the warp fields will not be able to keep up with the motion of the warp. The Fell-Heisenberg drive is unique only in that, with vanishing momentum, there is no configuration that allows stable propagation.

(n. b. The Heisenberg here is Lavinia Heisenberg, not the Werner Heisenberg of quantum physics and uncertainty principle fame.)

== Lentz warp drive ==

E. Lentz<ref name="Lentz">E. W. Lentz, "Breaking the warp barrier: hyper-fast solitons in Einstein–Maxwell-plasma theory", Classical and Quantum Gravity. 38 075015 (2021). arXiv:[https://arxiv.org/abs/2006.07125 2006.07125]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2021CQGra..38g5015L 2021CQGra..38g5015L]. doi:[https://doi.org/10.1088%2F1361-6382%2Fabe692 10.1088/1361-6382/abe692]. ISSN [https://search.worldcat.org/issn/0264-9381 0264-9381]. S2CID [https://api.semanticscholar.org/CorpusID:219635854 219635854].</ref> investigated a particular configuration of the Fell-Heisenberg class of warp drives, where the scalar field has a hyperbolic shape. The necessary conditions were found to be satisfied by a selection of rhomboid regions of the scalar field, resulting in an energy density that is everywhere positive and a shift vector with vanishing tidal force at the center for a comfortable ride.

However, Santiago, Schuster, and Visser<ref name="Santiago Schuster Visser 2022"></ref> dispute claims that the Lentz drive satisfies the energy conditions, noting that everywhere positive energy density in one frame of reference is insufficient to establish that the energy density is positive in all reference frames; knowledge of the Cauchy stress tensor is also needed. They show that any generic warp drive will violate the strong energy condition, null energy condition, and weak energy condition.

== Conservation laws ==

(Discussion of asymptotic flatness: what is it? Where does it apply?)

(The "big four" conservation laws and how they relate to asymptotic flatness.)

(ADM mass -> 0; conservation of energy/mass issues.)<ref>Sebastian Schuster, Jessica Santiago,and Matt Visser, "ADM mass in warp drive spacetimes" General Relativity and Gravitation (2023) 55:14 https://doi.org/10.1007/s10714-022-03061-9</ref>

(Discuss issues of conservation of angular momentum.)

(If angular momentum conservation is ignored, discuss how momentum & energy are affected by outside forces. In a gravitational field equivalent to inertial frame ... like being in an accelerated elevator; will acquire momentum buildup while staying "at rest".)

== Warp Railroads ==

Van Den Broeck<ref name="VanDenBroeck_2000"></ref> suggested a warp railroad,

You can set up devices ahead of time along the path of the bubble that produce the negative energy regions for the warp bubble at the appropriate time without needing the negative energy to ever move with the bubble at all. At least for bubbles on a predictable schedule and route.

helps with:

-stability (elaborate)

-conservation laws (elaborate)

Mention Krasnikov<ref name="Krasnikov 1998">S. V. Krasnikov, "Hyperfast travel in general relativity" Phys. Rev. D 57 4760</ref>

Hyperwave<ref name="Pieri 2024">Lorenzo Pieri, "Hyperwave: hyper-Fast Communication within General Relativity", arXiv:2311.12069 [gr-qc] https://doi.org/10.48550/arXiv.2311.12069</ref> makes use of this warp railroad idea.

== Quantum effects ==

(quantum stuff here<ref name="Hiscock1997">W. A. Hiscock, "Quantum effects in the Alcubierre warp drive spacetime", Classical and Quantum Gravity 14 L183 https://doi.org/10.1088/0264-9381/14/11/002 https://arxiv.org/abs/gr-qc/9707024</ref><ref name="Finazzi et al 2009">S. Finazzi, S. Liberati, C. Barceló, "Semiclassical instability of dynamical warp drives", Physical Review D 79, 124017 (2009)https://doi.org/10.1103/PhysRevD.79.124017 https://arxiv.org/abs/0904.0141</ref>)

From<ref name="Pieri 2024"></ref>
<blockquote>
It should be emphasised that Quantum Inequalities have not been proven for realistic interacting quantum fields, where the they are expected to be weaker if they exist [Cadamuro, 2019], and they often require Minkowskian conditions at infinity, a condition which is not appropriate for fields inside bounded regions, as in a Casimir devices.
</blockquote>

== Warp drives and black holes ==

What happens if you drive your warp drive into a black hole?

The pop-sci reason that nothing can escape a black hole is that at the event horizon the escape velocity is faster than light and nothing can go faster than light. A warp drive can go faster than light.

The more detailed reason from general relativity is that at the event horizon space and time are rotated sufficiently that "inward" becomes "forward in time." You can no more go farther away from the singularity in a black hole once you are inside the event horizon than you can go backward in time. Any super-luminal travel can go backward in time, and warp drives can engage in super-luminal travel.

The Hawking area theorem that says that black holes can only grow, never shrink relies on the condition that the energy is everywhere positive. Warp drives have negative energy density.

So in principle, there's nothing that prevents a warp drive from taking a dip into a black hole and coming back out again.

Garattini and Zatrimaylov looked into the problem of a warp drive near (and in) a black hole<ref name="Garattini and Zatrimaylov">R. Garattini and K. Zatrimaylov, "Black holes, warp drives, and energy conditions", Physics Letters B 856 138910 (2024) https://doi.org/10.1016/j.physletb.2024.138910</ref>. The found that the black hole reduces the negative energy requirement for the warp bubble to move inward toward the black hole singularity, but increases the negative energy needed to move away from the center of the black hole. In addition, a warp drive parked at the event horizon would allow light from inside the horizon to pass through the bubble and escape back outside.

== Credit ==
Author: Luke Campbell

== References ==

[[Category:Physics]][[Category:Infrastructure]][[Category:Physics & Engineering‏‎]][[Category:Physics & Math & Engineering]][[Category:Transportation & Infrastructure‏‎]][[Category:Metric Engineering]]

Warp Drives

2026-07-27T00:06:20Z

Lwcamp: /* So what is going on? */

{{PageConstructionNotice}}

Science fiction often features spacecraft that can seemingly move across space and get between the place of departure and the destination much faster than light could have done. This appears to contradict the theory of relativity, which predicts unequivocally that nothing can move through space faster than light. Because relativity has been incredibly successful at describing nature, with its many other predictions regularly being confirmed to extraordinary accuracy and within the bounds of uncertainty of all the experiments that tested them, it gives confidence that relativity is a correct description of reality. Which seems to rather throw a wet towel on our hopes for rapid travel between stars.

However, while relativity does not allow things to move through space faster than light, it places no such restrictions on how fast space-time itself can expand, contract, or move around. This leads to the idea of a warp drive – the spacecraft remains stationary within a region of highly curved space-time, and that region moves at super-luminal speeds rather than the spacecraft.

== The Alcubierre warp drive ==

The first warp drive geometry that satisfied the Einstein field equations of relativity was proposed by Miguel Alcubierre<ref>M. Alcubierre, "The warp drive: hyper-fast travel within general relativity." Classical and Quantum Gravity. 11 (5): L73–L77 (1994). [https://arxiv.org/abs/gr-qc/0009013 arXiv:gr-qc/0009013]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1994CQGra..11L..73A 1994CQGra..11L..73A]. doi:[https://doi.org/10.1088%2F0264-9381%2F11%2F5%2F001 10.1088/0264-9381/11/5/001]. S2CID [https://api.semanticscholar.org/CorpusID:4797900 4797900].</ref>. In this geometry, a sphere of space-time moves at an arbitrary speed (potentially but not necessarily a speed much faster than light). Objects within the sphere are moved along with the sphere; an object at rest within the sphere would be moved along with the sphere indefinitely. Space is expanding at the rear boundary and contracting at the front boundary in order to keep the sphere moving. In order to satisfy the Einstein field equations, the boundary of the sphere must have a negative energy density. The challenges of space-time geometries with negative energy densities are described in our page on [[Wormholes#Exotic_energy_conditions|wormholes]], for our purposes it is enough to note that negative energy densities can pose problems if not handled carefully, there may be limits on how much negative energy you can have without nearby positive energy density, and it may not be possible to get enough negative energy to support a warp drive; although none of this is ruled out by physics – yet!

<table align=center>
<tr><td width=650>[[File:Energy_magnitude_of_Alcubierre_drive.png]]
<td width=650>[[File:Expansion_of_Alcubierre_drive.png]]
<tr><td>The magnitude of the energy of the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. (The energy density is negative everywhere in the shell, here only the magnitude is shown to aid visual comprehension.)
<td>The expansion of space in the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. Negative expansion means that space is contracting in that direction, positive expansion that it is expanding.
</table>

NOTE: What the? The energy distribution is symmetric! How does the drive know which way to go? There must be significant contributions of other components of the stress-energy tensor, and those have got to be asymmetric along the forward/backward axis. Check this when I get time.

Almost as soon as Alcubierre proposed his warp drive, others began picking it apart.
<ol>
<li>Pfenning and Ford<ref name="PfenningFord_2001">M. J. Pfenning and L. H. Ford, "The unphysical nature of 'Warp Drive'", Classical and Quantum Gravity. 14 (7): 1743–1751 (1997). arXiv:[https://arxiv.org/abs/gr-qc/9702026 gr-qc/9702026]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1997CQGra..14.1743P 1997CQGra..14.1743P]. doi:[https://doi.org/10.1088%2F0264-9381%2F14%2F7%2F011 10.1088/0264-9381/14/7/011]. S2CID [https://api.semanticscholar.org/CorpusID:15279207 15279207].</ref> found that if the negative energy density wall around the bubble obeys [[Wormholes#Quantum_energy_inequalities|quantum energy inequalities]], then for warp speeds of around light speed the shell thickness would be on the order of a hundred Planck lengths; a distance far smaller than any other known physical phenomenon or object.
<li>In the same work, Pfenning and Ford also found that for a bubble a hundred meters in radius, a warp speed of around the speed of light, and the required thickness of about 100 Planck lengths the energy in the bubble shell would be E ≈ -1063 kg c2. The magnitude of this latter value is ten orders of magnitude larger than the energy of the entire visible universe. They do note, however, that if quantum energy inequalities can be ignored then a warp drive with a hundred meters radius but a shell thickness of one meter would "only" have an energy magnitude of about a quarter solar mass.
<li>As with any method of faster than light travel, the warp drive could be used to make a time machine.
<li>Perhaps most seriously, if the warp drive is going faster than light speed, the negative energy regions on the outside of the shell won't be in the warp parts of the bubble. They will have to be moving through space at faster than light speed if the bubble is to maintain its integrity, which is the very problem that the warp drive was designed to avoid. The outside of the warp bubble shell that maintains the warp bubble would fall away and the warp bubble would quickly erode to sub-luminal speeds.<ref name="Van Den Broeck 2000">Chris Van Den Broeck, "On the (im)possibility of warp bubbles", arXiv:gr-qc/9906050 https://doi.org/10.48550/arXiv.gr-qc/9906050</ref>
<li>A more general problem that includes the previous one is that the geometry of the warp drive determines the distribution of energy, momentum, and stress (pressure, shear, and tension) by Einstein's field equation, which states that the stress energy tensor is directly proportional to a reduced order curvature tensor of space-time. However, the stress-energy tensor must also obey the continuity equation – basically, for the energy to change in a given volume of space that energy must come in or go out of the boundary of that volume and similarly for the momentum. So a warp drive can have the warp condition for one moment in time. But after that, the stress-energy that maintains the warp will start to move according to the forces on it and according to the natural paths that it takes through the warped space-time. There is no guarantee that the configuration of the stress-energy at later times will be sufficient to maintain the warp.
<blockquote>
From the perspective of simulating the warp drive dynamically, the key challenge is stability. One can straightforwardly generate the initial metric that describes the Alcubierre metric and infer the instantaneous matter configuration that must support it using the Einstein Equation, but a time-domain evolution of the coupled matter-gravity system necessitates that we specify the equation of state of the matter, which in turn determines how it evolves. In a perfect fluid, this involves specifying the relation between the fluid pressure and density, but in a more general case, the full relation between the
stress-energy components must be given. There is (to our knowledge) no known equation of state that would maintain the warp drive metric in a stable configuration over time - therefore, whilst one can require that initially, the warp bubble is constant, it will quickly evolve away from that state and, in most cases, the warp fluid and spacetime deformations will disperse or collapse into a central point.
 
– Clough, Dietrich, and Khan<ref>Katy Clough, Tim Dietrich, and Sebastian Khan, "What no one has seen before: gravitational waveforms from warp drive collapse", arXiv:2406.02466 [gr-qc] https://doi.org/10.48550/arXiv.2406.02466</ref>
</blockquote>
<li>In order to get the matter (or more generally, stress-energy) that forms the warp shell up to speeds that could allow the shell to exist, you need to be able to, well, accelerate that matter to warp speeds. Which gives a chicken-and-egg problem - in order to achieve superluminal travel you need to be able to get matter or energy moving superluminally in the first place<ref name="Bobrick and Martire">Alexey Bobrick and Gianni Martire, "Introducing Physical Warp Drives", arXiv:2102.06824 [gr-qc] https://doi.org/10.48550/arXiv.2102.06824</ref><ref>D. H. Coule, "No warp drive", Class. Quantum Grav. 15 (1998) 2523–2527. https://doi.org/10.1088/0264-9381/15/8/026</ref>, which rather makes warp drives unnecessary from the get-go.
<li>A warp bubble moving faster than light will not be able to get any signals in front of it as it will overtake its own signal. In fact, anyone in the interior of the bubble would not be able to send signals to the front surface of the bubble. In science-speak, a horizon forms at the front of the bubble<ref name="Natario">José Natário, "Warp drive with zero expansion", Classical and Quantum Gravity. 19 (6): 1157–1166 (2002). arXiv:[https://arxiv.org/abs/gr-qc/0110086 gr-qc/0110086]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2002CQGra..19.1157N 2002CQGra..19.1157N]. doi:[https://doi.org/10.1088%2F0264-9381%2F19%2F6%2F308 10.1088/0264-9381/19/6/308]. S2CID [https://api.semanticscholar.org/CorpusID:15859984 15859984].</ref>. As a consequence, the bubble cannot be controlled, steered, nor stopped from within the bubble itself.
<li>A Horizon will naturally emit Hawking radiation. It is estimated that the radiation will raise the temperature of the interior of the bubble to approximately 1032 K<ref name="Finazzi 2009">Stefano Finazzi, Stefano Liberati, and Carlos Barceló, "Semiclassical instability of dynamical warp drives", Phys. Rev. D 79, 124017 (12 June, 2009) https://doi.org/10.1103/PhysRevD.79.124017 , arXiv:0904.0141 [gr-qc] https://doi.org/10.48550/arXiv.0904.0141</ref> (at this level of accuracy, the result is the same if you replace temperatures in K by the same number in °C or °F). This temperature, however, may be lower if quantum inequalities do not have to be fulfilled.
<li>Semi-classical instabilities at the horizon are expected to lead to a divergence of the stress-energy tensor at the boundary in a short period of time, thus leading to the destruction of the warp bubble<ref name="Finazzi 2009"></ref>.
</ol>
Helpfully, Van Den Broeck<ref name="VanDenBroeck_2000">C. Van Den Broeck, "Alcubierre’s Warp Drive: Problems and Prospects" AIP Conference Proceedings. 504: 1105–1110 (2000). Bibcode:[https://ui.adsabs.harvard.edu/abs/2000AIPC..504.1105V 2000AIPC..504.1105V]. doi:[https://doi.org/10.1063%2F1.1290913 10.1063/1.1290913].</ref> suggested several solutions – or at least mitigations – for some of these problems
<ol>
<li>Quantum inequalities had not been shown to be true in general for highly curved space-times.
<li>A proposed warp drive geometry<ref name="VanDenBroeck_1999">C. Van Den Broeck, "A 'warp drive' with more reasonable total energy requirements". Classical and Quantum Gravity. 16 (12): 3973–3979 (1999). arXiv:[https://arxiv.org/abs/gr-qc/9905084 gr-qc/9905084]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1999CQGra..16.3973V 1999CQGra..16.3973V]. doi:[https://doi.org/10.1088%2F0264-9381%2F16%2F12%2F314 10.1088/0264-9381/16/12/314]. S2CID [https://api.semanticscholar.org/CorpusID:15466313 15466313]. </ref> (see the van Den Broeck drive, below) would be able to minimize the magnitude of the energy required to about that of the mass of our sun. While still large, it is not unphysically large.
<li>Time travel is always going to be a worry with faster than light travel. There's no neat solution to this one.
<li>It may be possible for the negative energy outside of the superluminal region to compress into the superluminal region to form a shock in space-time. The front surface of the bubble would then be a singularity. It is not clear if such a sharp jump in physical properties is possible, but neither is sure that it is impossible, either. However, while this might help with the negative energy in front of the bubble getting swept up because it cannot move faster than light, it does little to help with the material at the back of the bubble on the outside getting left behind because it cannot keep up.
</ol>
Other mitigating research has also shown
<ol start=9>
<li>Semi-classical divergence only occurs in warp drives with only one spatial dimensions. Those with two or more spatial dimensions (as in our universe) only contain points where divergences occur, and these divergences can be managed. In fact, their effects can be reduced by making the warp shape more elongated<ref>Carlos Barceló, Valentin Boyanov, Luis J. Garay, Eduardo Martín-Martínez,
and Jose M. Sánchez Velázquez, "Warp drive aerodynamics", JHEP08 (2022) 288 https://doi.org/10.1007/JHEP08(2022)288 , arXiv:2207.06458 [gr-qc] https://doi.org/10.48550/arXiv.2207.06458</ref>.
</ol>

(And what's with the ring in all the artwork, anyway?)

=== So what is going on? ===

What is it that makes a warp drive a warp drive, and how does it work?

A warp drive is a concept that comes out of the science of general relativity. General relativity collects space and time together into a single "structure" called spacetime. This structure is curved, and its curvature is determined by the material that is in it. Absent forces acting between matter, particles of matter move through spacetime on the shortest possible paths in this curved environment.
<blockquote>
"Spacetime tells matter how to move; matter tells spacetime how to curve." - John Archibald Wheeler
</blockquote>
All of this is encompassed in the Einstein field equation:
<div align=center>
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 8πG
<td rowspan="2" nowrap="nowrap" align="right"> T = R -  
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right"> R g
<tr>
<td style="border-top:solid 1px black;"> c4
<td style="border-top:solid 1px black;"> 2
</table>
</div>
Unfortunately, by convention, both the symbols G and R get used for a lot of different things. Here, symbols in bold are second rank tensors – constructs of arrays of 4×4 numbers where 4 is necessary because that is the number of dimensions of spacetime. Conventionally, these numbers are indexed by a value going from 0 to 3, with 0 being the part corresponding to the coordinate of time and 1 to 3 being the Cartesian coordinates that you might know as x, y, and z.

T is the stress-energy tensor: the (0,0) component represents the energy density at a given point; the (0,i) and (i, 0) components (with i going from 1 to 3) represent the momentum density; and the (i,j) components represent the stresses present, such as pressure or tension or shear.

The other term, R - R g/2, involves a partial description of the curvature. A full description of the curvature requires a construct of 4×4×4×4 components; too much for the stress energy tensor to fully constrain. Which is okay, because this allows curvature to exist away from where matter is, allowing gravity to have long range effects across empty space as well as for phenomena like gravitational waves. But it is enough that a given distribution of matter T gives enough constraints (along with specifying how the geometry behaves far away) that you can generally figure out what the actual gravity is.

This is the usual story that we are told. But there is an alternate way of looking at the same geometry such that the math works out the same, but it is described conceptually in a very different way. This divides spacetime up into "layers", called foliations, that represent slices of space and going from one foliation to the next is increasing in time. This is useful for computation because you can assign coordinates to your system. There are two additional constructs – the lapse function and the shift vector field. Starting at an event on one foliation, the lapse function at that position tells you that after a certain "proper time" (time perceived by an observer at that event) the time coordinate of the observer will be the proper time times the lapse function at that event. And the new spatial coordinates of the observer will be the proper time times the shift vector at that event. (Strictly speaking, this is only exactly true in the limit of the proper time approaching zero or when the lapse function and shift vectors are constant. But hopefully this gives a good enough insight to what these entities are so that we can go on to describe warp drives.)

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Lapse_shift.png]]
<tr><td>A folliation of space-time into slices of constant time coordinates (grey). At an event (black circle), the lapse function (yellow) shows how far into the future the point at that event will be projected in a unit amount of proper time experienced by an observer initially at the event. The shift vector (red) shows how that point changes coordinates after the unit amount of proper time. The cyan vector shows the new coordinates the point is projected onto. The lower black surface indicates a foliation of spacetime at the same time coordinate; after unit proper time, the events on the lower foliation get projected to the curved upper black foliation according to the shift vectors and lapse functions at each of their respective points.
</table>
</table>

Warp drives are generally restricted to spacetime geometries with a constant lapse function - all observers go forward by the same time coordinate after experiencing the same amount of proper time. Thus, all warp drives have no time dilation.
Far from the warp bubble, the shift vector vanishes – each spatial coordinate is mapped to the same spatial coordinate after any amount of proper time so long as the warp drive is not near. But the inside of the warp bubble is an extended region where the shift vector field is non-zero and constant. In this bubble, all spatial coordinates are mapped to a coordinate that shifts linearly in time with the proper time of the observer. Astute readers will see that this linear change in position with time is very similar to a velocity. This region of constant shift vector then moves with the shift vector velocity to keep stuff originally in the bubble so that it stays in the bubble. In between the inside of the bubble and the far away regions with zero shift vector, the shift vector field falls away to match one region to the other.

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Warp_shift.png]]
<tr><td>The shift vector of a one-dimensional slice through warp geometry, in red. At the edges there is no shift. At the center (inside the warp bubble) there is a region of uniform shift so that at a later time the inside of the warp bubble is rigidly translated to different spatial coordinates (in blue). The two foliations remain the same distance apart, because warp drives maintain a unit lapse function.
</table>
</table>

Traditionally, scientists would start with an interesting distribution of matter given by T and then find out what kind of curvature it created. This works well for planets and stars and black holes and cosmic strings and other things which are observed in the universe and you want to see what sort of effect they had. But starting in the 1980's, some physicists started looking at Einstein's field equations the other way around. They start with a desired geometry, find the curvature part of the field equation R - R g/2, and use that to find out what T has to be in order to get that curvature. This was originally done for [[Wormholes|wormholes]], but in 1994 Miguel Alcubierre used this same trick to figure out what could create a region of constant shift vector that moved with the shift vector - the warp drive. With this demand for a warp geometry, the required distribution of matter and energy and momentum and stress can be determined.

The original Alcubierre shift vector N→ produces a spherical warp bubble with an axi-symmetric distribution of, well, everything along the direction of motion with warp velocity vector v→.
<div align="center">
N→ = v→ f(rs)
</div>
Here, rs is the distance from the center of the warp bubble, and f(rs) can be any function that is 1 near rs = 0 and is 0 for rs well beyond the radius of the warp bubble.

=== Alcubierre warp interactions with light and matter ===

Space is not entirely empty. It is filled with a diffuse plasma in the form of the interstellar medium, as well as cosmic radiation, light from stars, and cosmic microwave background radiation. A warp drive propagating through space will encounter this stuff. What happens when this matter and radiation have a warp bubble pass across them?

The first analysis of matter encountering a warp bubble was performed by Pfenning and Ford<ref name="PfenningFord_2001"></ref>. They looked at the warp bubble interaction with a massive object at rest with the frame of reference of the warp drive (keep in mind that the rest frame is the same inside and outside the warp bubble; in this rest frame objects inside the bubble have no momentum even though they are, in some sense, changing location rapidly with time). When the warp bubble passes, the object experiences an acceleration in the direction of the warp bubble motion. When the warp shell passes and the object is inside the bubble, it will be moving with approximately the speed of the bubble. Pfenning and Ford analyzed this problem with a continuous distribution of shell energy that, strictly speaking, never falls to zero except at the bubble center and at spatial infinity, so unless the object passes through the center in Pfenning and Ford's description it will never quite get up to the bubble's speed. In this case, the object will move almost as fast as the object but will pass through the bubble in a finite time, after which it will again be at rest with respect to the reference frame but displaced along the direction of the warp bubble motion by some distance. With this description of the warp bubble, a spacecraft of finite size will always be moving a little slower than the warp bubble and would have to use rockets to keep up with it. in addition, the spacecraft would experience tidal forces that would cause stress on the spacecraft's structure.

Pfenning and Ford also examined cases where the shell is of a finite (possibly infinitesimal) width and falls to zero both inside and outside the bubble. In this case any matter encountering the bubble would be collected at the bow of the bubble and thereafter move along with it.

McMonigal et al. analyzed the situation for both massive particles and light moving along the axis of travel of the bubble<ref name="McMonigalEtAl_2012">B. McMonigal, G. F. Lewis, and P. O'Byrne, "Alcubierre warp drive: On the matter of matter". Physical Review D. 85 (6) 064024 (20 March 2012). arXiv:[https://arxiv.org/abs/1202.5708 1202.5708]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2012PhRvD..85f4024M 2012PhRvD..85f4024M]. doi:[https://doi.org/10.1103%2FPhysRevD.85.064024 10.1103/PhysRevD.85.064024]. S2CID [https://api.semanticscholar.org/CorpusID:3993148 3993148].</ref>. They found that light moving opposite the warp direction passed through the bubble without incident, being only somewhat delayed by passing through the bubble. A warp bubble that is warping at sub-luminal speeds can have light catch up from behind it. This light is able to pass through the bubble, and is somewhat advanced in its path by the speed of the bubble. For super-luminal warp bubbles, however, the situation is different. The bubble will catch up to light moving it its own direction that is originally in front of it. This light cannot escape forward, the bubble being too fast. Nor can it escape backward, as the light is propagating forward and the interior of the bubble is at rest. Thus, the light gets caught at the bow of the warp bubble, unable to escape for so long as the warp bubble is active. This light is strongly blue shifted to extremely energetic x-rays and gamma rays. The space behind the bubble is swept clear of forward-moving light.

The situation for matter over-run by a super-luminal warp bubble is similar. Matter moving backward passes through the bubble, being only swept a ways along the bubble's path as sign of its passing but otherwise continuing on their way. Matter at rest behaves the same way as Pfenning and Ford discovered. Meanwhile, matter moving in the same direction of the warp is overtaken and collects at the bow of the bubble, unable to leave for so long as the bubble is warping. This matter "surfing" on the bubble bow is highly accelerated to relativistic speeds, experiencing extreme time dilation.

For a sub-luminal bubble, there is a speed faster than the warp speed where particles coming up from behind overtake the bubble and pass through, out the front. Particles below this speed but still faster than the warp speed still overtake the bubble, but then bounce backwards out of the bubble from behind. They are still moving forward, but are now moving more slowly than the warp speed. Particles moving forward but slower than the bubble will be overtaken and then bounced out the front with a speed higher than the warp speed. Those particles moving backward will pass through the bubble and exit with their initial speed.

The spacecraft inside the bubble will observe that light moving backward compared to the warp direction is blue-shifted, and matter moving backward passes through with increased energy. Meanwhile, a sub-luminal bubble will see light that was moving forward as red-shifted and matter catching up to the bubble from behind will pass by with reduced energy (superluminal bubbles, of course, do not have any matter or radiation catching up to them from behind).

When a superluminal warp bubble that has been collecting matter and radiation for the duration of its journey and blue-shifting it to much higher energies is turned off, all that matter and energy is released as a blast of radiation in the direction the bubble was warping.

McMonigal et al. conclude by noting that any spacecraft in the warp bubble would need shielding to protect against the blue-shifted radiation and matter of increased energy. In addition the destination would be "blasted into oblivion" by the release of matter and radiation that had been caught in the bubble during the trip.

Clark et al.<ref>Chad Clark, William A. Hiscock, and Shane L. Larson, "Null geodesics in the Alcubierre warp drive spacetime: the view from the bridge", arXiv:gr-qc/9907019 (2018) https://doi.org/10.48550/arXiv.gr-qc/9907019</ref> examine what it would look like from within an operating warp drive. They find that light moving at 90 degrees to the direction of motion will look like it is still coming from 90 degrees. However, light from the front hemisphere is squished progressively toward the forward point the faster you go. For a warp bubble moving faster than the speed of light, there is a cone behind the warp bubble where no light can reach it. However, you don't see a black spot behind you – light from the regions in the backward hemisphere from outside that cone that can reach you is distorted toward the direction behind you to fill in that spot. Light from the forward direction is blue-shifted and light from behind is red-shifted. At speeds of around 200 times light speed, the cosmic microwave background would be blue shifted to energies similar to that of the solar photosphere, heating anything in the warp bubble as if they were close to the sun.

== Van Den Broeck warp drive ==

If an Alcubierre warp bubble a hundred meters across requires a magnitude of energy greater than the entire energy in the observable universe, one option to reduce the magnitude of energy used is to make the warp bubble smaller. Much smaller. Pfenning and Ford<ref name="PfenningFord_2001"></ref> suggested making the warp bubble smaller than an atom. Of course, that brings up the problem of how to stuff a spacecraft in there. Van Den Broeck proposed a solution<ref name="VanDenBroeck_1999"></ref>: make the warp bubble only about the size of an atomic nucleus but expand the space inside the bubble enormously so that a spacecraft could fit in. This is somewhat like a nuclear sized wormhole that leads to a pocket universe that holds the spacecraft. Unfortunately, the metric doesn't fit neatly into any [[Wormholes#Three_dimensions|embedding diagram]] that I can figure out, but the math works out so that spatial coordinates inside the bubble are expanded by a factor of 1017 and a spacecraft would have a few hundred meters of bubble interior to putz around in. This trick manages to reduce the magnitude of energy to only about the mass energy of a few stars similar to our own. Other than that, it is otherwise a normal Alcubierre drive.

Further work<ref>S. Krasnikov, "The quantum inequalities do not forbid spacetime shortcuts", Physical Review D. 67 (10) 104013. https://doi.org/10.1103%2FPhysRevD.67.104013, arXiv:gr-qc/0207057 (2003) https://doi.org/10.48550/arXiv.gr-qc/0207057 </ref> has managed to use Van Den Broeck's trick to reduce the magnitude of the negative energy to only about a milligram.

== Natário warp drive ==

The Alcubierre drive is not the only way to construct a warp drive. José Natário<ref name="Natario"></ref> showed that it was just one example of an entire class of warp drives. These are space-time geometries with a smooth and localized shift vector field a shift vector is a vector field that describes how one "slice" of space out of space-time connects to the next) and a unit lapse function (the lapse function describes how the time you experience changes when going from a "slice" of space-time to the next).

Natário then went on to find in that class a set of warp drives that do not have any expansion or contraction of space at all. Instead, space encountering the front boundary of the warp bubble instead "slides" around the outside of the bubble until it gets to the corresponding place on the back and is left behind there. Another way to think of it is that space entering the bubble shell is compressed in the radial direction of the shell but is simultaneously expanded in the tangential direction so that there is no net change in volume. When it gets to the back, the opposite occurs. In the language of mathematics, the shift vector field is divergenceless.

<table align=center>
<tr><td width=360>[[File:Natario_drive_flow_of_space.png]]
<td width=445>[[File:Natario_warp_drive_horizons.png]]
<tr><td style="vertical-align: top;">A representation of the motion of space in a zero-expansion Natário drive with a vanishingly thin warp shell around the bubble.
<td>The horizons of the generalized Natário class of warp drives (including the Alcubierre warp drive and all fell-Heisenberg drives) at superluminal speed in the direction of the green arrow. No event inside the bubble can affect events outside the bubble in front of the blue line. No events outside the bubble behind the red line can affect events inside the bubble. The lines have an angle α with respect to the direction of motion with sin(α) = 1/vs for vs the speed of the bubble relative to light speed. This is analogous to the Mach cone of supersonic objects.
</table>

NOTE: There are two different things that can be meant by the term "Natário warp drive". The generalized Natário warp drive is the general theory of all warp drives that includes within it the Alcubierre, Fell-Heisenberg, and Lentz warp drives, among infinitely many others. The zero-expansion Natário warp drive is the specific implementation of the generalized Natário warp drive that has a divergenceless shift vector field.

The generalized class of warp drives developed by Natário (including both the Alcubierre and this zero-expansion warp drive) are shown to always have regions in the warp shell where the energy density is negative to at least some observers. You can't get away from it – to warp, you need negative energy density.

The Natário class of warp drives blueshift light coming in from the front and redshift light catching up from the back. For the case of a super-luminal warp bubble, there will of course be a horizon at the back of the bubble and light will not be able to get through from behind. For a bubble speed vs relative to the speed of light, light coming from straight ahead will be blueshifted up in frequency and photon energy by a factor of 1 + vs. For sub-luminal travel, light coming from behind will be redshifted by a factor 1 - vs. In general for n as the unit direction along which the light is propagating, the blueshift factor will be 1 + vs ⋅ n. If the warp shell is infinitesimally thin, light from outside that reaches the center of the bubble will not be distorted in direction although it will be frequency shifted, but observers still inside the bubble but displaced from the center will see the field of view distorted as well as frequency shifted.

Most of the difficulties that apply to the Alcubierre warp drive also apply to the generalized Natário warp drive (and thus to the zero-expansion Natário drive and Fell-Heisenberg drive as well). In particular, quantum energy inequalities, high magnitudes of the energy required, formation of time machines, instabilities related to the stress-energy making up the warp shell being unable to keep up with the bubble, the chicken-and-egg problem of needing FTL to achieve FTL, the formation of horizons making warp bubbles uncontrollable, and Hawking radiation at the horizons, are all still concerns for generic warp drives.

== Fell-Heisenberg warp drives ==

S. D. B. Fell and L. Heisenberg investigated what sort of warp structure is necessary to produce a warp drive with positive energy density everywhere<ref name="Fell-Heisenberg">S. D. B. Fell and L> Heisenberg, "Positive energy warp drive from hidden geometric structures", Classical and Quantum Gravity 38 155020 (2021) https://doi.org/10.1088/1361-6382/ac0e47 https://arxiv.org/abs/2104.06488</ref>. They found that warp drives of the generic Natário form with irrotational shift vectors had a simple expression for the energy density and vanishing momentum (irrotational means that the shift vector can be described as the gradient of a scalar potential field or, equivalently, that the curl of the shift vector field vanishes everywhere). Fell and Heisenberg proposed several potential fields that either had positive energy density everywhere except at discontinuities, or that were continuous but had regions that violated the weak energy condition (i. e. negative energy densities in some frames of reference). One specific example they provided of a warp field had an energy that is about 10,000 times less than the mass-energy of our sun. Or only about half the mass-energy of Jupiter. This is a significant improvement over many other proposed drives.

Despite the introduction discussing warp drive configurations that satisfy the various energy conditions, the configurations described in the paper are shown to locally violate the weak and strong energy conditions. Nonetheless, the energy density is still mostly positive. Santiago, Schuster, and Visser's<ref name="Santiago Schuster Visser 2022">J. Santiago, S. Schuster, and M. Visser, "Generic warp drives violate the null energy condition", Physical Review D 105, 064038 (2022) https://doi.org/10.1103/PhysRevD.105.064038</ref> work shows that the various energy conditions must still be violated by this warp drive; to not violate these, energy density must be positive in all reference frames not just those of the co-moving observer.

Any vector field can be described by its divergence, its curl, and a constant. The constant part of a warp drive's shift vector can be discarded because then the warp drive would not be localized. It is interesting, therefore, that the Natário zero-expansion drive has zero divergence and can be described only by its curl, while the Fell-Heisenberg drive has zero curl and can be described only by its divergence. These two classes of drives thus rely on opposite choices of setting one of the two different parts describing a vector field to zero. As might be expected, the Fell-Heisenberg drive is not zero expansion – space-time will change in volume as it passes through the warp bubble (although after going all the way through it will return to its original volume). However, objects in space-time will retain their original orientation as they pass through the warp bubble.

Previous examples of warp drives, such as the Alcubierre and zero-expansion Natário drives, tend toward a flat space-time as you go far enough away faster than would occur if there were any mass present in the warp drive. That is, the ADM mass of such drives is zero. However, one of the Fell-Heisenberg examples described a spacetime where the geometry approaches that of what you would get for a massive object at the warp location, giving one of the first examples of a warp drive with mass.

In the Fell-Heisenberg drive the momentum vanishes everywhere. Yet the energy occupies regions where the shift vector is varying rapidly. The lack of momentum means that the energy will not move to keep up with the differential expansion and movement of the spacetime. As a consequence, at later times the energy will have a different distribution than what is necessary to maintain the given warp configuration. The exact time evolution is not solved but this likely leads to collapse of warp bubble. Note that this is not a problem unique to the Fell-Heisenberg drive. Assuming a geometry that describes a warp produces a specific distribution of the stress-energy tensor that satisfies Einstein's equation. But there is no guarantee that this stress energy tensor satisfies the continuity equation, which describes how the distribution of matter and momentum change with time. The issue of stability of any warp drive is not well understood, but most warp drive geometries will be unstable in the sense that their matter that generates the warp fields will not be able to keep up with the motion of the warp. The Fell-Heisenberg drive is unique only in that, with vanishing momentum, there is no configuration that allows stable propagation.

(n. b. The Heisenberg here is Lavinia Heisenberg, not the Werner Heisenberg of quantum physics and uncertainty principle fame.)

== Lentz warp drive ==

E. Lentz<ref name="Lentz">E. W. Lentz, "Breaking the warp barrier: hyper-fast solitons in Einstein–Maxwell-plasma theory", Classical and Quantum Gravity. 38 075015 (2021). arXiv:[https://arxiv.org/abs/2006.07125 2006.07125]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2021CQGra..38g5015L 2021CQGra..38g5015L]. doi:[https://doi.org/10.1088%2F1361-6382%2Fabe692 10.1088/1361-6382/abe692]. ISSN [https://search.worldcat.org/issn/0264-9381 0264-9381]. S2CID [https://api.semanticscholar.org/CorpusID:219635854 219635854].</ref> investigated a particular configuration of the Fell-Heisenberg class of warp drives, where the scalar field has a hyperbolic shape. The necessary conditions were found to be satisfied by a selection of rhomboid regions of the scalar field, resulting in an energy density that is everywhere positive and a shift vector with vanishing tidal force at the center for a comfortable ride.

However, Santiago, Schuster, and Visser<ref name="Santiago Schuster Visser 2022"></ref> dispute claims that the Lentz drive satisfies the energy conditions, noting that everywhere positive energy density in one frame of reference is insufficient to establish that the energy density is positive in all reference frames; knowledge of the Cauchy stress tensor is also needed. They show that any generic warp drive will violate the strong energy condition, null energy condition, and weak energy condition.

== Conservation laws ==

(Discussion of asymptotic flatness: what is it? Where does it apply?)

(The "big four" conservation laws and how they relate to asymptotic flatness.)

(ADM mass -> 0; conservation of energy/mass issues.)<ref>Sebastian Schuster, Jessica Santiago,and Matt Visser, "ADM mass in warp drive spacetimes" General Relativity and Gravitation (2023) 55:14 https://doi.org/10.1007/s10714-022-03061-9</ref>

(Discuss issues of conservation of angular momentum.)

(If angular momentum conservation is ignored, discuss how momentum & energy are affected by outside forces. In a gravitational field equivalent to inertial frame ... like being in an accelerated elevator; will acquire momentum buildup while staying "at rest".)

== Warp Railroads ==

Van Den Broeck<ref name="VanDenBroeck_2000"></ref> suggested a warp railroad,

You can set up devices ahead of time along the path of the bubble that produce the negative energy regions for the warp bubble at the appropriate time without needing the negative energy to ever move with the bubble at all. At least for bubbles on a predictable schedule and route.

helps with:

-stability (elaborate)

-conservation laws (elaborate)

Mention Krasnikov<ref name="Krasnikov 1998">S. V. Krasnikov, "Hyperfast travel in general relativity" Phys. Rev. D 57 4760</ref>

Hyperwave<ref name="Pieri 2024">Lorenzo Pieri, "Hyperwave: hyper-Fast Communication within General Relativity", arXiv:2311.12069 [gr-qc] https://doi.org/10.48550/arXiv.2311.12069</ref> makes use of this warp railroad idea.

== Quantum effects ==

(quantum stuff here<ref name="Hiscock1997">W. A. Hiscock, "Quantum effects in the Alcubierre warp drive spacetime", Classical and Quantum Gravity 14 L183 https://doi.org/10.1088/0264-9381/14/11/002 https://arxiv.org/abs/gr-qc/9707024</ref><ref name="Finazzi et al 2009">S. Finazzi, S. Liberati, C. Barceló, "Semiclassical instability of dynamical warp drives", Physical Review D 79, 124017 (2009)https://doi.org/10.1103/PhysRevD.79.124017 https://arxiv.org/abs/0904.0141</ref>)

From<ref name="Pieri 2024"></ref>
<blockquote>
It should be emphasised that Quantum Inequalities have not been proven for realistic interacting quantum fields, where the they are expected to be weaker if they exist [Cadamuro, 2019], and they often require Minkowskian conditions at infinity, a condition which is not appropriate for fields inside bounded regions, as in a Casimir devices.
</blockquote>

== Warp drives and black holes ==

What happens if you drive your warp drive into a black hole?

The pop-sci reason that nothing can escape a black hole is that at the event horizon the escape velocity is faster than light and nothing can go faster than light. A warp drive can go faster than light.

The more detailed reason from general relativity is that at the event horizon space and time are rotated sufficiently that "inward" becomes "forward in time." You can no more go farther away from the singularity in a black hole once you are inside the event horizon than you can go backward in time. Any super-luminal travel can go backward in time, and warp drives can engage in super-luminal travel.

The Hawking area theorem that says that black holes can only grow, never shrink relies on the condition that the energy is everywhere positive. Warp drives have negative energy density.

So in principle, there's nothing that prevents a warp drive from taking a dip into a black hole and coming back out again.

Garattini and Zatrimaylov looked into the problem of a warp drive near (and in) a black hole<ref name="Garattini and Zatrimaylov">R. Garattini and K. Zatrimaylov, "Black holes, warp drives, and energy conditions", Physics Letters B 856 138910 (2024) https://doi.org/10.1016/j.physletb.2024.138910</ref>. The found that the black hole reduces the negative energy requirement for the warp bubble to move inward toward the black hole singularity, but increases the negative energy needed to move away from the center of the black hole. In addition, a warp drive parked at the event horizon would allow light from inside the horizon to pass through the bubble and escape back outside.

== Credit ==
Author: Luke Campbell

== References ==

[[Category:Physics]][[Category:Infrastructure]][[Category:Physics & Engineering‏‎]][[Category:Physics & Math & Engineering]][[Category:Transportation & Infrastructure‏‎]][[Category:Metric Engineering]]

Warp Drives

2026-07-27T00:05:25Z

Lwcamp: /* So what is going on? */

{{PageConstructionNotice}}

Science fiction often features spacecraft that can seemingly move across space and get between the place of departure and the destination much faster than light could have done. This appears to contradict the theory of relativity, which predicts unequivocally that nothing can move through space faster than light. Because relativity has been incredibly successful at describing nature, with its many other predictions regularly being confirmed to extraordinary accuracy and within the bounds of uncertainty of all the experiments that tested them, it gives confidence that relativity is a correct description of reality. Which seems to rather throw a wet towel on our hopes for rapid travel between stars.

However, while relativity does not allow things to move through space faster than light, it places no such restrictions on how fast space-time itself can expand, contract, or move around. This leads to the idea of a warp drive – the spacecraft remains stationary within a region of highly curved space-time, and that region moves at super-luminal speeds rather than the spacecraft.

== The Alcubierre warp drive ==

The first warp drive geometry that satisfied the Einstein field equations of relativity was proposed by Miguel Alcubierre<ref>M. Alcubierre, "The warp drive: hyper-fast travel within general relativity." Classical and Quantum Gravity. 11 (5): L73–L77 (1994). [https://arxiv.org/abs/gr-qc/0009013 arXiv:gr-qc/0009013]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1994CQGra..11L..73A 1994CQGra..11L..73A]. doi:[https://doi.org/10.1088%2F0264-9381%2F11%2F5%2F001 10.1088/0264-9381/11/5/001]. S2CID [https://api.semanticscholar.org/CorpusID:4797900 4797900].</ref>. In this geometry, a sphere of space-time moves at an arbitrary speed (potentially but not necessarily a speed much faster than light). Objects within the sphere are moved along with the sphere; an object at rest within the sphere would be moved along with the sphere indefinitely. Space is expanding at the rear boundary and contracting at the front boundary in order to keep the sphere moving. In order to satisfy the Einstein field equations, the boundary of the sphere must have a negative energy density. The challenges of space-time geometries with negative energy densities are described in our page on [[Wormholes#Exotic_energy_conditions|wormholes]], for our purposes it is enough to note that negative energy densities can pose problems if not handled carefully, there may be limits on how much negative energy you can have without nearby positive energy density, and it may not be possible to get enough negative energy to support a warp drive; although none of this is ruled out by physics – yet!

<table align=center>
<tr><td width=650>[[File:Energy_magnitude_of_Alcubierre_drive.png]]
<td width=650>[[File:Expansion_of_Alcubierre_drive.png]]
<tr><td>The magnitude of the energy of the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. (The energy density is negative everywhere in the shell, here only the magnitude is shown to aid visual comprehension.)
<td>The expansion of space in the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. Negative expansion means that space is contracting in that direction, positive expansion that it is expanding.
</table>

NOTE: What the? The energy distribution is symmetric! How does the drive know which way to go? There must be significant contributions of other components of the stress-energy tensor, and those have got to be asymmetric along the forward/backward axis. Check this when I get time.

Almost as soon as Alcubierre proposed his warp drive, others began picking it apart.
<ol>
<li>Pfenning and Ford<ref name="PfenningFord_2001">M. J. Pfenning and L. H. Ford, "The unphysical nature of 'Warp Drive'", Classical and Quantum Gravity. 14 (7): 1743–1751 (1997). arXiv:[https://arxiv.org/abs/gr-qc/9702026 gr-qc/9702026]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1997CQGra..14.1743P 1997CQGra..14.1743P]. doi:[https://doi.org/10.1088%2F0264-9381%2F14%2F7%2F011 10.1088/0264-9381/14/7/011]. S2CID [https://api.semanticscholar.org/CorpusID:15279207 15279207].</ref> found that if the negative energy density wall around the bubble obeys [[Wormholes#Quantum_energy_inequalities|quantum energy inequalities]], then for warp speeds of around light speed the shell thickness would be on the order of a hundred Planck lengths; a distance far smaller than any other known physical phenomenon or object.
<li>In the same work, Pfenning and Ford also found that for a bubble a hundred meters in radius, a warp speed of around the speed of light, and the required thickness of about 100 Planck lengths the energy in the bubble shell would be E ≈ -1063 kg c2. The magnitude of this latter value is ten orders of magnitude larger than the energy of the entire visible universe. They do note, however, that if quantum energy inequalities can be ignored then a warp drive with a hundred meters radius but a shell thickness of one meter would "only" have an energy magnitude of about a quarter solar mass.
<li>As with any method of faster than light travel, the warp drive could be used to make a time machine.
<li>Perhaps most seriously, if the warp drive is going faster than light speed, the negative energy regions on the outside of the shell won't be in the warp parts of the bubble. They will have to be moving through space at faster than light speed if the bubble is to maintain its integrity, which is the very problem that the warp drive was designed to avoid. The outside of the warp bubble shell that maintains the warp bubble would fall away and the warp bubble would quickly erode to sub-luminal speeds.<ref name="Van Den Broeck 2000">Chris Van Den Broeck, "On the (im)possibility of warp bubbles", arXiv:gr-qc/9906050 https://doi.org/10.48550/arXiv.gr-qc/9906050</ref>
<li>A more general problem that includes the previous one is that the geometry of the warp drive determines the distribution of energy, momentum, and stress (pressure, shear, and tension) by Einstein's field equation, which states that the stress energy tensor is directly proportional to a reduced order curvature tensor of space-time. However, the stress-energy tensor must also obey the continuity equation – basically, for the energy to change in a given volume of space that energy must come in or go out of the boundary of that volume and similarly for the momentum. So a warp drive can have the warp condition for one moment in time. But after that, the stress-energy that maintains the warp will start to move according to the forces on it and according to the natural paths that it takes through the warped space-time. There is no guarantee that the configuration of the stress-energy at later times will be sufficient to maintain the warp.
<blockquote>
From the perspective of simulating the warp drive dynamically, the key challenge is stability. One can straightforwardly generate the initial metric that describes the Alcubierre metric and infer the instantaneous matter configuration that must support it using the Einstein Equation, but a time-domain evolution of the coupled matter-gravity system necessitates that we specify the equation of state of the matter, which in turn determines how it evolves. In a perfect fluid, this involves specifying the relation between the fluid pressure and density, but in a more general case, the full relation between the
stress-energy components must be given. There is (to our knowledge) no known equation of state that would maintain the warp drive metric in a stable configuration over time - therefore, whilst one can require that initially, the warp bubble is constant, it will quickly evolve away from that state and, in most cases, the warp fluid and spacetime deformations will disperse or collapse into a central point.
 
– Clough, Dietrich, and Khan<ref>Katy Clough, Tim Dietrich, and Sebastian Khan, "What no one has seen before: gravitational waveforms from warp drive collapse", arXiv:2406.02466 [gr-qc] https://doi.org/10.48550/arXiv.2406.02466</ref>
</blockquote>
<li>In order to get the matter (or more generally, stress-energy) that forms the warp shell up to speeds that could allow the shell to exist, you need to be able to, well, accelerate that matter to warp speeds. Which gives a chicken-and-egg problem - in order to achieve superluminal travel you need to be able to get matter or energy moving superluminally in the first place<ref name="Bobrick and Martire">Alexey Bobrick and Gianni Martire, "Introducing Physical Warp Drives", arXiv:2102.06824 [gr-qc] https://doi.org/10.48550/arXiv.2102.06824</ref><ref>D. H. Coule, "No warp drive", Class. Quantum Grav. 15 (1998) 2523–2527. https://doi.org/10.1088/0264-9381/15/8/026</ref>, which rather makes warp drives unnecessary from the get-go.
<li>A warp bubble moving faster than light will not be able to get any signals in front of it as it will overtake its own signal. In fact, anyone in the interior of the bubble would not be able to send signals to the front surface of the bubble. In science-speak, a horizon forms at the front of the bubble<ref name="Natario">José Natário, "Warp drive with zero expansion", Classical and Quantum Gravity. 19 (6): 1157–1166 (2002). arXiv:[https://arxiv.org/abs/gr-qc/0110086 gr-qc/0110086]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2002CQGra..19.1157N 2002CQGra..19.1157N]. doi:[https://doi.org/10.1088%2F0264-9381%2F19%2F6%2F308 10.1088/0264-9381/19/6/308]. S2CID [https://api.semanticscholar.org/CorpusID:15859984 15859984].</ref>. As a consequence, the bubble cannot be controlled, steered, nor stopped from within the bubble itself.
<li>A Horizon will naturally emit Hawking radiation. It is estimated that the radiation will raise the temperature of the interior of the bubble to approximately 1032 K<ref name="Finazzi 2009">Stefano Finazzi, Stefano Liberati, and Carlos Barceló, "Semiclassical instability of dynamical warp drives", Phys. Rev. D 79, 124017 (12 June, 2009) https://doi.org/10.1103/PhysRevD.79.124017 , arXiv:0904.0141 [gr-qc] https://doi.org/10.48550/arXiv.0904.0141</ref> (at this level of accuracy, the result is the same if you replace temperatures in K by the same number in °C or °F). This temperature, however, may be lower if quantum inequalities do not have to be fulfilled.
<li>Semi-classical instabilities at the horizon are expected to lead to a divergence of the stress-energy tensor at the boundary in a short period of time, thus leading to the destruction of the warp bubble<ref name="Finazzi 2009"></ref>.
</ol>
Helpfully, Van Den Broeck<ref name="VanDenBroeck_2000">C. Van Den Broeck, "Alcubierre’s Warp Drive: Problems and Prospects" AIP Conference Proceedings. 504: 1105–1110 (2000). Bibcode:[https://ui.adsabs.harvard.edu/abs/2000AIPC..504.1105V 2000AIPC..504.1105V]. doi:[https://doi.org/10.1063%2F1.1290913 10.1063/1.1290913].</ref> suggested several solutions – or at least mitigations – for some of these problems
<ol>
<li>Quantum inequalities had not been shown to be true in general for highly curved space-times.
<li>A proposed warp drive geometry<ref name="VanDenBroeck_1999">C. Van Den Broeck, "A 'warp drive' with more reasonable total energy requirements". Classical and Quantum Gravity. 16 (12): 3973–3979 (1999). arXiv:[https://arxiv.org/abs/gr-qc/9905084 gr-qc/9905084]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1999CQGra..16.3973V 1999CQGra..16.3973V]. doi:[https://doi.org/10.1088%2F0264-9381%2F16%2F12%2F314 10.1088/0264-9381/16/12/314]. S2CID [https://api.semanticscholar.org/CorpusID:15466313 15466313]. </ref> (see the van Den Broeck drive, below) would be able to minimize the magnitude of the energy required to about that of the mass of our sun. While still large, it is not unphysically large.
<li>Time travel is always going to be a worry with faster than light travel. There's no neat solution to this one.
<li>It may be possible for the negative energy outside of the superluminal region to compress into the superluminal region to form a shock in space-time. The front surface of the bubble would then be a singularity. It is not clear if such a sharp jump in physical properties is possible, but neither is sure that it is impossible, either. However, while this might help with the negative energy in front of the bubble getting swept up because it cannot move faster than light, it does little to help with the material at the back of the bubble on the outside getting left behind because it cannot keep up.
</ol>
Other mitigating research has also shown
<ol start=9>
<li>Semi-classical divergence only occurs in warp drives with only one spatial dimensions. Those with two or more spatial dimensions (as in our universe) only contain points where divergences occur, and these divergences can be managed. In fact, their effects can be reduced by making the warp shape more elongated<ref>Carlos Barceló, Valentin Boyanov, Luis J. Garay, Eduardo Martín-Martínez,
and Jose M. Sánchez Velázquez, "Warp drive aerodynamics", JHEP08 (2022) 288 https://doi.org/10.1007/JHEP08(2022)288 , arXiv:2207.06458 [gr-qc] https://doi.org/10.48550/arXiv.2207.06458</ref>.
</ol>

(And what's with the ring in all the artwork, anyway?)

=== So what is going on? ===

What is it that makes a warp drive a warp drive, and how does it work?

A warp drive is a concept that comes out of the science of general relativity. General relativity collects space and time together into a single "structure" called spacetime. This structure is curved, and its curvature is determined by the material that is in it. Absent forces acting between matter, particles of matter move through spacetime on the shortest possible paths in this curved environment.
<blockquote>
"Spacetime tells matter how to move; matter tells spacetime how to curve." - John Archibald Wheeler
</blockquote>
All of this is encompassed in the Einstein field equation:
<div align=center>
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 8πG
<td rowspan="2" nowrap="nowrap" align="right"> T = R -  
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right"> R g
<tr>
<td style="border-top:solid 1px black;"> c4
<td style="border-top:solid 1px black;"> 2
</table>
</div>
Unfortunately, by convention, both the symbols G and R get used for a lot of different things. Here, symbols in bold are second rank tensors – constructs of arrays of 4×4 numbers where 4 is necessary because that is the number of dimensions of spacetime. Conventionally, these numbers are indexed by a value going from 0 to 3, with 0 being the part corresponding to the coordinate of time and 1 to 3 being the Cartesian coordinates that you might know as x, y, and z.

T is the stress-energy tensor: the (0,0) component represents the energy density at a given point; the (0,i) and (i, 0) components (with i going from 1 to 3) represent the momentum density; and the (i,j) components represent the stresses present, such as pressure or tension or shear.

The other term, R - R g/2, involves a partial description of the curvature. A full description of the curvature requires a construct of 4×4×4×4 components; too much for the stress energy tensor to fully constrain. Which is okay, because this allows curvature to exist away from where matter is, allowing gravity to have long range effects across empty space as well as for phenomena like gravitational waves. But it is enough that a given distribution of matter T gives enough constraints (along with specifying how the geometry behaves far away) that you can generally figure out what the actual gravity is.

This is the usual story that we are told. But there is an alternate way of looking at the same geometry such that the math works out the same, but it is described conceptually in a very different way. This divides spacetime up into "layers", called foliations, that represent slices of space and going from one foliation to the next is increasing in time. This is useful for computation because you can assign coordinates to your system. There are two additional constructs – the lapse function and the shift vector field. Starting at an event on one foliation, the lapse function at that position tells you that after a certain "proper time" (time perceived by an observer at that event) the time coordinate of the observer will be the proper time times the lapse function at that event. And the new spatial coordinates of the observer will be the proper time times the shift vector at that event. (Strictly speaking, this is only exactly true in the limit of the proper time approaching zero or when the lapse function and shift vectors are constant. But hopefully this gives a good enough insight to what these entities are so that we can go on to describe warp drives.)

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Lapse_shift.png]]
<tr><td>A folliation of space-time into slices of constant time coordinates (grey). At an event (black circle), the lapse function (yellow) shows how far into the future the point at that event will be projected in a unit amount of proper time experienced by an observer initially at the event. The shift vector (red) shows how that point changes coordinates after the unit amount of proper time. The cyan vector shows the new coordinates the point is projected onto. The lower black surface indicates a foliation of spacetime at the same time coordinate; after unit proper time, the events on the lower foliation get projected to the curved upper black foliation according to the shift vectors and lapse functions at each of their respective points.
</table>
</table>

Warp drives are generally restricted to spacetime geometries with a constant lapse function - all observers go forward by the same time coordinate after experiencing the same amount of proper time. Thus, all warp drives have no time dilation.
Far from the warp bubble, the shift vector vanishes – each spatial coordinate is mapped to the same spatial coordinate after any amount of proper time so long as the warp drive is not near. But the inside of the warp bubble is an extended region where the shift vector field is non-zero and constant. In this bubble, all spatial coordinates are mapped to a coordinate that shifts linearly in time with the proper time of the observer. Astute readers will see that this linear change in position with time is very similar to a velocity. This region of constant shift vector then moves with the shift vector velocity to keep stuff originally in the bubble so that it stays in the bubble. In between the inside of the bubble and the far away regions with zero shift vector, the shift vector field falls away to match one region to the other.

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Warp_shift.png]]
<tr><td>The shift vector of a one-dimensional slice through warp geometry, in red. At the edges there is no shift. At the center (inside the warp bubble) there is a region of uniform shift so that at a later time the inside of the warp bubble is rigidly translated to different spatial coordinates (in blue). The two foliations remain the same distance apart, because warp drives maintain a unit lapse function.
</table>
</table>

Traditionally, scientists would start with an interesting distribution of matter given by T and then find out what kind of curvature it created. This works well for planets and stars and black holes and cosmic strings and other things which are observed in the universe and you want to see what sort of effect they had. But starting in the 1980's, some physicists started looking at Einstein's field equations the other way around. They start with a desired geometry, find the curvature part of the field equation R - R g/2, and use that to find out what T has to be in order to get that curvature. This was originally done for [[Wormholes|wormholes]], but in 1994 Miguel Alcubierre used this same trick to figure out what could create a region of constant shift vector that moved with the shift vector - the warp drive. With this demand for a warp geometry, the required distribution of matter and energy and momentum and stress can be determined.

The original Alcubierre shift vector N→ produces a spherical warp bubble with an axi-symmetric distribution of, well, everything along the direction of motion with warp velocity vector v→.
<div align="center">
N→ = v→ f(rs)
</div>
Here, rs is the distance from the center of the warp bubble, and f(rs) can be any function that is 1 near rs = 0 and is 0 for rs well beyond the radius of the warp bubble.

=== Alcubierre warp interactions with light and matter ===

Space is not entirely empty. It is filled with a diffuse plasma in the form of the interstellar medium, as well as cosmic radiation, light from stars, and cosmic microwave background radiation. A warp drive propagating through space will encounter this stuff. What happens when this matter and radiation have a warp bubble pass across them?

The first analysis of matter encountering a warp bubble was performed by Pfenning and Ford<ref name="PfenningFord_2001"></ref>. They looked at the warp bubble interaction with a massive object at rest with the frame of reference of the warp drive (keep in mind that the rest frame is the same inside and outside the warp bubble; in this rest frame objects inside the bubble have no momentum even though they are, in some sense, changing location rapidly with time). When the warp bubble passes, the object experiences an acceleration in the direction of the warp bubble motion. When the warp shell passes and the object is inside the bubble, it will be moving with approximately the speed of the bubble. Pfenning and Ford analyzed this problem with a continuous distribution of shell energy that, strictly speaking, never falls to zero except at the bubble center and at spatial infinity, so unless the object passes through the center in Pfenning and Ford's description it will never quite get up to the bubble's speed. In this case, the object will move almost as fast as the object but will pass through the bubble in a finite time, after which it will again be at rest with respect to the reference frame but displaced along the direction of the warp bubble motion by some distance. With this description of the warp bubble, a spacecraft of finite size will always be moving a little slower than the warp bubble and would have to use rockets to keep up with it. in addition, the spacecraft would experience tidal forces that would cause stress on the spacecraft's structure.

Pfenning and Ford also examined cases where the shell is of a finite (possibly infinitesimal) width and falls to zero both inside and outside the bubble. In this case any matter encountering the bubble would be collected at the bow of the bubble and thereafter move along with it.

McMonigal et al. analyzed the situation for both massive particles and light moving along the axis of travel of the bubble<ref name="McMonigalEtAl_2012">B. McMonigal, G. F. Lewis, and P. O'Byrne, "Alcubierre warp drive: On the matter of matter". Physical Review D. 85 (6) 064024 (20 March 2012). arXiv:[https://arxiv.org/abs/1202.5708 1202.5708]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2012PhRvD..85f4024M 2012PhRvD..85f4024M]. doi:[https://doi.org/10.1103%2FPhysRevD.85.064024 10.1103/PhysRevD.85.064024]. S2CID [https://api.semanticscholar.org/CorpusID:3993148 3993148].</ref>. They found that light moving opposite the warp direction passed through the bubble without incident, being only somewhat delayed by passing through the bubble. A warp bubble that is warping at sub-luminal speeds can have light catch up from behind it. This light is able to pass through the bubble, and is somewhat advanced in its path by the speed of the bubble. For super-luminal warp bubbles, however, the situation is different. The bubble will catch up to light moving it its own direction that is originally in front of it. This light cannot escape forward, the bubble being too fast. Nor can it escape backward, as the light is propagating forward and the interior of the bubble is at rest. Thus, the light gets caught at the bow of the warp bubble, unable to escape for so long as the warp bubble is active. This light is strongly blue shifted to extremely energetic x-rays and gamma rays. The space behind the bubble is swept clear of forward-moving light.

The situation for matter over-run by a super-luminal warp bubble is similar. Matter moving backward passes through the bubble, being only swept a ways along the bubble's path as sign of its passing but otherwise continuing on their way. Matter at rest behaves the same way as Pfenning and Ford discovered. Meanwhile, matter moving in the same direction of the warp is overtaken and collects at the bow of the bubble, unable to leave for so long as the bubble is warping. This matter "surfing" on the bubble bow is highly accelerated to relativistic speeds, experiencing extreme time dilation.

For a sub-luminal bubble, there is a speed faster than the warp speed where particles coming up from behind overtake the bubble and pass through, out the front. Particles below this speed but still faster than the warp speed still overtake the bubble, but then bounce backwards out of the bubble from behind. They are still moving forward, but are now moving more slowly than the warp speed. Particles moving forward but slower than the bubble will be overtaken and then bounced out the front with a speed higher than the warp speed. Those particles moving backward will pass through the bubble and exit with their initial speed.

The spacecraft inside the bubble will observe that light moving backward compared to the warp direction is blue-shifted, and matter moving backward passes through with increased energy. Meanwhile, a sub-luminal bubble will see light that was moving forward as red-shifted and matter catching up to the bubble from behind will pass by with reduced energy (superluminal bubbles, of course, do not have any matter or radiation catching up to them from behind).

When a superluminal warp bubble that has been collecting matter and radiation for the duration of its journey and blue-shifting it to much higher energies is turned off, all that matter and energy is released as a blast of radiation in the direction the bubble was warping.

McMonigal et al. conclude by noting that any spacecraft in the warp bubble would need shielding to protect against the blue-shifted radiation and matter of increased energy. In addition the destination would be "blasted into oblivion" by the release of matter and radiation that had been caught in the bubble during the trip.

Clark et al.<ref>Chad Clark, William A. Hiscock, and Shane L. Larson, "Null geodesics in the Alcubierre warp drive spacetime: the view from the bridge", arXiv:gr-qc/9907019 (2018) https://doi.org/10.48550/arXiv.gr-qc/9907019</ref> examine what it would look like from within an operating warp drive. They find that light moving at 90 degrees to the direction of motion will look like it is still coming from 90 degrees. However, light from the front hemisphere is squished progressively toward the forward point the faster you go. For a warp bubble moving faster than the speed of light, there is a cone behind the warp bubble where no light can reach it. However, you don't see a black spot behind you – light from the regions in the backward hemisphere from outside that cone that can reach you is distorted toward the direction behind you to fill in that spot. Light from the forward direction is blue-shifted and light from behind is red-shifted. At speeds of around 200 times light speed, the cosmic microwave background would be blue shifted to energies similar to that of the solar photosphere, heating anything in the warp bubble as if they were close to the sun.

== Van Den Broeck warp drive ==

If an Alcubierre warp bubble a hundred meters across requires a magnitude of energy greater than the entire energy in the observable universe, one option to reduce the magnitude of energy used is to make the warp bubble smaller. Much smaller. Pfenning and Ford<ref name="PfenningFord_2001"></ref> suggested making the warp bubble smaller than an atom. Of course, that brings up the problem of how to stuff a spacecraft in there. Van Den Broeck proposed a solution<ref name="VanDenBroeck_1999"></ref>: make the warp bubble only about the size of an atomic nucleus but expand the space inside the bubble enormously so that a spacecraft could fit in. This is somewhat like a nuclear sized wormhole that leads to a pocket universe that holds the spacecraft. Unfortunately, the metric doesn't fit neatly into any [[Wormholes#Three_dimensions|embedding diagram]] that I can figure out, but the math works out so that spatial coordinates inside the bubble are expanded by a factor of 1017 and a spacecraft would have a few hundred meters of bubble interior to putz around in. This trick manages to reduce the magnitude of energy to only about the mass energy of a few stars similar to our own. Other than that, it is otherwise a normal Alcubierre drive.

Further work<ref>S. Krasnikov, "The quantum inequalities do not forbid spacetime shortcuts", Physical Review D. 67 (10) 104013. https://doi.org/10.1103%2FPhysRevD.67.104013, arXiv:gr-qc/0207057 (2003) https://doi.org/10.48550/arXiv.gr-qc/0207057 </ref> has managed to use Van Den Broeck's trick to reduce the magnitude of the negative energy to only about a milligram.

== Natário warp drive ==

The Alcubierre drive is not the only way to construct a warp drive. José Natário<ref name="Natario"></ref> showed that it was just one example of an entire class of warp drives. These are space-time geometries with a smooth and localized shift vector field a shift vector is a vector field that describes how one "slice" of space out of space-time connects to the next) and a unit lapse function (the lapse function describes how the time you experience changes when going from a "slice" of space-time to the next).

Natário then went on to find in that class a set of warp drives that do not have any expansion or contraction of space at all. Instead, space encountering the front boundary of the warp bubble instead "slides" around the outside of the bubble until it gets to the corresponding place on the back and is left behind there. Another way to think of it is that space entering the bubble shell is compressed in the radial direction of the shell but is simultaneously expanded in the tangential direction so that there is no net change in volume. When it gets to the back, the opposite occurs. In the language of mathematics, the shift vector field is divergenceless.

<table align=center>
<tr><td width=360>[[File:Natario_drive_flow_of_space.png]]
<td width=445>[[File:Natario_warp_drive_horizons.png]]
<tr><td style="vertical-align: top;">A representation of the motion of space in a zero-expansion Natário drive with a vanishingly thin warp shell around the bubble.
<td>The horizons of the generalized Natário class of warp drives (including the Alcubierre warp drive and all fell-Heisenberg drives) at superluminal speed in the direction of the green arrow. No event inside the bubble can affect events outside the bubble in front of the blue line. No events outside the bubble behind the red line can affect events inside the bubble. The lines have an angle α with respect to the direction of motion with sin(α) = 1/vs for vs the speed of the bubble relative to light speed. This is analogous to the Mach cone of supersonic objects.
</table>

NOTE: There are two different things that can be meant by the term "Natário warp drive". The generalized Natário warp drive is the general theory of all warp drives that includes within it the Alcubierre, Fell-Heisenberg, and Lentz warp drives, among infinitely many others. The zero-expansion Natário warp drive is the specific implementation of the generalized Natário warp drive that has a divergenceless shift vector field.

The generalized class of warp drives developed by Natário (including both the Alcubierre and this zero-expansion warp drive) are shown to always have regions in the warp shell where the energy density is negative to at least some observers. You can't get away from it – to warp, you need negative energy density.

The Natário class of warp drives blueshift light coming in from the front and redshift light catching up from the back. For the case of a super-luminal warp bubble, there will of course be a horizon at the back of the bubble and light will not be able to get through from behind. For a bubble speed vs relative to the speed of light, light coming from straight ahead will be blueshifted up in frequency and photon energy by a factor of 1 + vs. For sub-luminal travel, light coming from behind will be redshifted by a factor 1 - vs. In general for n as the unit direction along which the light is propagating, the blueshift factor will be 1 + vs ⋅ n. If the warp shell is infinitesimally thin, light from outside that reaches the center of the bubble will not be distorted in direction although it will be frequency shifted, but observers still inside the bubble but displaced from the center will see the field of view distorted as well as frequency shifted.

Most of the difficulties that apply to the Alcubierre warp drive also apply to the generalized Natário warp drive (and thus to the zero-expansion Natário drive and Fell-Heisenberg drive as well). In particular, quantum energy inequalities, high magnitudes of the energy required, formation of time machines, instabilities related to the stress-energy making up the warp shell being unable to keep up with the bubble, the chicken-and-egg problem of needing FTL to achieve FTL, the formation of horizons making warp bubbles uncontrollable, and Hawking radiation at the horizons, are all still concerns for generic warp drives.

== Fell-Heisenberg warp drives ==

S. D. B. Fell and L. Heisenberg investigated what sort of warp structure is necessary to produce a warp drive with positive energy density everywhere<ref name="Fell-Heisenberg">S. D. B. Fell and L> Heisenberg, "Positive energy warp drive from hidden geometric structures", Classical and Quantum Gravity 38 155020 (2021) https://doi.org/10.1088/1361-6382/ac0e47 https://arxiv.org/abs/2104.06488</ref>. They found that warp drives of the generic Natário form with irrotational shift vectors had a simple expression for the energy density and vanishing momentum (irrotational means that the shift vector can be described as the gradient of a scalar potential field or, equivalently, that the curl of the shift vector field vanishes everywhere). Fell and Heisenberg proposed several potential fields that either had positive energy density everywhere except at discontinuities, or that were continuous but had regions that violated the weak energy condition (i. e. negative energy densities in some frames of reference). One specific example they provided of a warp field had an energy that is about 10,000 times less than the mass-energy of our sun. Or only about half the mass-energy of Jupiter. This is a significant improvement over many other proposed drives.

Despite the introduction discussing warp drive configurations that satisfy the various energy conditions, the configurations described in the paper are shown to locally violate the weak and strong energy conditions. Nonetheless, the energy density is still mostly positive. Santiago, Schuster, and Visser's<ref name="Santiago Schuster Visser 2022">J. Santiago, S. Schuster, and M. Visser, "Generic warp drives violate the null energy condition", Physical Review D 105, 064038 (2022) https://doi.org/10.1103/PhysRevD.105.064038</ref> work shows that the various energy conditions must still be violated by this warp drive; to not violate these, energy density must be positive in all reference frames not just those of the co-moving observer.

Any vector field can be described by its divergence, its curl, and a constant. The constant part of a warp drive's shift vector can be discarded because then the warp drive would not be localized. It is interesting, therefore, that the Natário zero-expansion drive has zero divergence and can be described only by its curl, while the Fell-Heisenberg drive has zero curl and can be described only by its divergence. These two classes of drives thus rely on opposite choices of setting one of the two different parts describing a vector field to zero. As might be expected, the Fell-Heisenberg drive is not zero expansion – space-time will change in volume as it passes through the warp bubble (although after going all the way through it will return to its original volume). However, objects in space-time will retain their original orientation as they pass through the warp bubble.

Previous examples of warp drives, such as the Alcubierre and zero-expansion Natário drives, tend toward a flat space-time as you go far enough away faster than would occur if there were any mass present in the warp drive. That is, the ADM mass of such drives is zero. However, one of the Fell-Heisenberg examples described a spacetime where the geometry approaches that of what you would get for a massive object at the warp location, giving one of the first examples of a warp drive with mass.

In the Fell-Heisenberg drive the momentum vanishes everywhere. Yet the energy occupies regions where the shift vector is varying rapidly. The lack of momentum means that the energy will not move to keep up with the differential expansion and movement of the spacetime. As a consequence, at later times the energy will have a different distribution than what is necessary to maintain the given warp configuration. The exact time evolution is not solved but this likely leads to collapse of warp bubble. Note that this is not a problem unique to the Fell-Heisenberg drive. Assuming a geometry that describes a warp produces a specific distribution of the stress-energy tensor that satisfies Einstein's equation. But there is no guarantee that this stress energy tensor satisfies the continuity equation, which describes how the distribution of matter and momentum change with time. The issue of stability of any warp drive is not well understood, but most warp drive geometries will be unstable in the sense that their matter that generates the warp fields will not be able to keep up with the motion of the warp. The Fell-Heisenberg drive is unique only in that, with vanishing momentum, there is no configuration that allows stable propagation.

(n. b. The Heisenberg here is Lavinia Heisenberg, not the Werner Heisenberg of quantum physics and uncertainty principle fame.)

== Lentz warp drive ==

E. Lentz<ref name="Lentz">E. W. Lentz, "Breaking the warp barrier: hyper-fast solitons in Einstein–Maxwell-plasma theory", Classical and Quantum Gravity. 38 075015 (2021). arXiv:[https://arxiv.org/abs/2006.07125 2006.07125]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2021CQGra..38g5015L 2021CQGra..38g5015L]. doi:[https://doi.org/10.1088%2F1361-6382%2Fabe692 10.1088/1361-6382/abe692]. ISSN [https://search.worldcat.org/issn/0264-9381 0264-9381]. S2CID [https://api.semanticscholar.org/CorpusID:219635854 219635854].</ref> investigated a particular configuration of the Fell-Heisenberg class of warp drives, where the scalar field has a hyperbolic shape. The necessary conditions were found to be satisfied by a selection of rhomboid regions of the scalar field, resulting in an energy density that is everywhere positive and a shift vector with vanishing tidal force at the center for a comfortable ride.

However, Santiago, Schuster, and Visser<ref name="Santiago Schuster Visser 2022"></ref> dispute claims that the Lentz drive satisfies the energy conditions, noting that everywhere positive energy density in one frame of reference is insufficient to establish that the energy density is positive in all reference frames; knowledge of the Cauchy stress tensor is also needed. They show that any generic warp drive will violate the strong energy condition, null energy condition, and weak energy condition.

== Conservation laws ==

(Discussion of asymptotic flatness: what is it? Where does it apply?)

(The "big four" conservation laws and how they relate to asymptotic flatness.)

(ADM mass -> 0; conservation of energy/mass issues.)<ref>Sebastian Schuster, Jessica Santiago,and Matt Visser, "ADM mass in warp drive spacetimes" General Relativity and Gravitation (2023) 55:14 https://doi.org/10.1007/s10714-022-03061-9</ref>

(Discuss issues of conservation of angular momentum.)

(If angular momentum conservation is ignored, discuss how momentum & energy are affected by outside forces. In a gravitational field equivalent to inertial frame ... like being in an accelerated elevator; will acquire momentum buildup while staying "at rest".)

== Warp Railroads ==

Van Den Broeck<ref name="VanDenBroeck_2000"></ref> suggested a warp railroad,

You can set up devices ahead of time along the path of the bubble that produce the negative energy regions for the warp bubble at the appropriate time without needing the negative energy to ever move with the bubble at all. At least for bubbles on a predictable schedule and route.

helps with:

-stability (elaborate)

-conservation laws (elaborate)

Mention Krasnikov<ref name="Krasnikov 1998">S. V. Krasnikov, "Hyperfast travel in general relativity" Phys. Rev. D 57 4760</ref>

Hyperwave<ref name="Pieri 2024">Lorenzo Pieri, "Hyperwave: hyper-Fast Communication within General Relativity", arXiv:2311.12069 [gr-qc] https://doi.org/10.48550/arXiv.2311.12069</ref> makes use of this warp railroad idea.

== Quantum effects ==

(quantum stuff here<ref name="Hiscock1997">W. A. Hiscock, "Quantum effects in the Alcubierre warp drive spacetime", Classical and Quantum Gravity 14 L183 https://doi.org/10.1088/0264-9381/14/11/002 https://arxiv.org/abs/gr-qc/9707024</ref><ref name="Finazzi et al 2009">S. Finazzi, S. Liberati, C. Barceló, "Semiclassical instability of dynamical warp drives", Physical Review D 79, 124017 (2009)https://doi.org/10.1103/PhysRevD.79.124017 https://arxiv.org/abs/0904.0141</ref>)

From<ref name="Pieri 2024"></ref>
<blockquote>
It should be emphasised that Quantum Inequalities have not been proven for realistic interacting quantum fields, where the they are expected to be weaker if they exist [Cadamuro, 2019], and they often require Minkowskian conditions at infinity, a condition which is not appropriate for fields inside bounded regions, as in a Casimir devices.
</blockquote>

== Warp drives and black holes ==

What happens if you drive your warp drive into a black hole?

The pop-sci reason that nothing can escape a black hole is that at the event horizon the escape velocity is faster than light and nothing can go faster than light. A warp drive can go faster than light.

The more detailed reason from general relativity is that at the event horizon space and time are rotated sufficiently that "inward" becomes "forward in time." You can no more go farther away from the singularity in a black hole once you are inside the event horizon than you can go backward in time. Any super-luminal travel can go backward in time, and warp drives can engage in super-luminal travel.

The Hawking area theorem that says that black holes can only grow, never shrink relies on the condition that the energy is everywhere positive. Warp drives have negative energy density.

So in principle, there's nothing that prevents a warp drive from taking a dip into a black hole and coming back out again.

Garattini and Zatrimaylov looked into the problem of a warp drive near (and in) a black hole<ref name="Garattini and Zatrimaylov">R. Garattini and K. Zatrimaylov, "Black holes, warp drives, and energy conditions", Physics Letters B 856 138910 (2024) https://doi.org/10.1016/j.physletb.2024.138910</ref>. The found that the black hole reduces the negative energy requirement for the warp bubble to move inward toward the black hole singularity, but increases the negative energy needed to move away from the center of the black hole. In addition, a warp drive parked at the event horizon would allow light from inside the horizon to pass through the bubble and escape back outside.

== Credit ==
Author: Luke Campbell

== References ==

[[Category:Physics]][[Category:Infrastructure]][[Category:Physics & Engineering‏‎]][[Category:Physics & Math & Engineering]][[Category:Transportation & Infrastructure‏‎]][[Category:Metric Engineering]]

Warp Drives

2026-07-27T00:02:00Z

Lwcamp: /* So what is going on? */

{{PageConstructionNotice}}

Science fiction often features spacecraft that can seemingly move across space and get between the place of departure and the destination much faster than light could have done. This appears to contradict the theory of relativity, which predicts unequivocally that nothing can move through space faster than light. Because relativity has been incredibly successful at describing nature, with its many other predictions regularly being confirmed to extraordinary accuracy and within the bounds of uncertainty of all the experiments that tested them, it gives confidence that relativity is a correct description of reality. Which seems to rather throw a wet towel on our hopes for rapid travel between stars.

However, while relativity does not allow things to move through space faster than light, it places no such restrictions on how fast space-time itself can expand, contract, or move around. This leads to the idea of a warp drive – the spacecraft remains stationary within a region of highly curved space-time, and that region moves at super-luminal speeds rather than the spacecraft.

== The Alcubierre warp drive ==

The first warp drive geometry that satisfied the Einstein field equations of relativity was proposed by Miguel Alcubierre<ref>M. Alcubierre, "The warp drive: hyper-fast travel within general relativity." Classical and Quantum Gravity. 11 (5): L73–L77 (1994). [https://arxiv.org/abs/gr-qc/0009013 arXiv:gr-qc/0009013]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1994CQGra..11L..73A 1994CQGra..11L..73A]. doi:[https://doi.org/10.1088%2F0264-9381%2F11%2F5%2F001 10.1088/0264-9381/11/5/001]. S2CID [https://api.semanticscholar.org/CorpusID:4797900 4797900].</ref>. In this geometry, a sphere of space-time moves at an arbitrary speed (potentially but not necessarily a speed much faster than light). Objects within the sphere are moved along with the sphere; an object at rest within the sphere would be moved along with the sphere indefinitely. Space is expanding at the rear boundary and contracting at the front boundary in order to keep the sphere moving. In order to satisfy the Einstein field equations, the boundary of the sphere must have a negative energy density. The challenges of space-time geometries with negative energy densities are described in our page on [[Wormholes#Exotic_energy_conditions|wormholes]], for our purposes it is enough to note that negative energy densities can pose problems if not handled carefully, there may be limits on how much negative energy you can have without nearby positive energy density, and it may not be possible to get enough negative energy to support a warp drive; although none of this is ruled out by physics – yet!

<table align=center>
<tr><td width=650>[[File:Energy_magnitude_of_Alcubierre_drive.png]]
<td width=650>[[File:Expansion_of_Alcubierre_drive.png]]
<tr><td>The magnitude of the energy of the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. (The energy density is negative everywhere in the shell, here only the magnitude is shown to aid visual comprehension.)
<td>The expansion of space in the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. Negative expansion means that space is contracting in that direction, positive expansion that it is expanding.
</table>

NOTE: What the? The energy distribution is symmetric! How does the drive know which way to go? There must be significant contributions of other components of the stress-energy tensor, and those have got to be asymmetric along the forward/backward axis. Check this when I get time.

Almost as soon as Alcubierre proposed his warp drive, others began picking it apart.
<ol>
<li>Pfenning and Ford<ref name="PfenningFord_2001">M. J. Pfenning and L. H. Ford, "The unphysical nature of 'Warp Drive'", Classical and Quantum Gravity. 14 (7): 1743–1751 (1997). arXiv:[https://arxiv.org/abs/gr-qc/9702026 gr-qc/9702026]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1997CQGra..14.1743P 1997CQGra..14.1743P]. doi:[https://doi.org/10.1088%2F0264-9381%2F14%2F7%2F011 10.1088/0264-9381/14/7/011]. S2CID [https://api.semanticscholar.org/CorpusID:15279207 15279207].</ref> found that if the negative energy density wall around the bubble obeys [[Wormholes#Quantum_energy_inequalities|quantum energy inequalities]], then for warp speeds of around light speed the shell thickness would be on the order of a hundred Planck lengths; a distance far smaller than any other known physical phenomenon or object.
<li>In the same work, Pfenning and Ford also found that for a bubble a hundred meters in radius, a warp speed of around the speed of light, and the required thickness of about 100 Planck lengths the energy in the bubble shell would be E ≈ -1063 kg c2. The magnitude of this latter value is ten orders of magnitude larger than the energy of the entire visible universe. They do note, however, that if quantum energy inequalities can be ignored then a warp drive with a hundred meters radius but a shell thickness of one meter would "only" have an energy magnitude of about a quarter solar mass.
<li>As with any method of faster than light travel, the warp drive could be used to make a time machine.
<li>Perhaps most seriously, if the warp drive is going faster than light speed, the negative energy regions on the outside of the shell won't be in the warp parts of the bubble. They will have to be moving through space at faster than light speed if the bubble is to maintain its integrity, which is the very problem that the warp drive was designed to avoid. The outside of the warp bubble shell that maintains the warp bubble would fall away and the warp bubble would quickly erode to sub-luminal speeds.<ref name="Van Den Broeck 2000">Chris Van Den Broeck, "On the (im)possibility of warp bubbles", arXiv:gr-qc/9906050 https://doi.org/10.48550/arXiv.gr-qc/9906050</ref>
<li>A more general problem that includes the previous one is that the geometry of the warp drive determines the distribution of energy, momentum, and stress (pressure, shear, and tension) by Einstein's field equation, which states that the stress energy tensor is directly proportional to a reduced order curvature tensor of space-time. However, the stress-energy tensor must also obey the continuity equation – basically, for the energy to change in a given volume of space that energy must come in or go out of the boundary of that volume and similarly for the momentum. So a warp drive can have the warp condition for one moment in time. But after that, the stress-energy that maintains the warp will start to move according to the forces on it and according to the natural paths that it takes through the warped space-time. There is no guarantee that the configuration of the stress-energy at later times will be sufficient to maintain the warp.
<blockquote>
From the perspective of simulating the warp drive dynamically, the key challenge is stability. One can straightforwardly generate the initial metric that describes the Alcubierre metric and infer the instantaneous matter configuration that must support it using the Einstein Equation, but a time-domain evolution of the coupled matter-gravity system necessitates that we specify the equation of state of the matter, which in turn determines how it evolves. In a perfect fluid, this involves specifying the relation between the fluid pressure and density, but in a more general case, the full relation between the
stress-energy components must be given. There is (to our knowledge) no known equation of state that would maintain the warp drive metric in a stable configuration over time - therefore, whilst one can require that initially, the warp bubble is constant, it will quickly evolve away from that state and, in most cases, the warp fluid and spacetime deformations will disperse or collapse into a central point.
 
– Clough, Dietrich, and Khan<ref>Katy Clough, Tim Dietrich, and Sebastian Khan, "What no one has seen before: gravitational waveforms from warp drive collapse", arXiv:2406.02466 [gr-qc] https://doi.org/10.48550/arXiv.2406.02466</ref>
</blockquote>
<li>In order to get the matter (or more generally, stress-energy) that forms the warp shell up to speeds that could allow the shell to exist, you need to be able to, well, accelerate that matter to warp speeds. Which gives a chicken-and-egg problem - in order to achieve superluminal travel you need to be able to get matter or energy moving superluminally in the first place<ref name="Bobrick and Martire">Alexey Bobrick and Gianni Martire, "Introducing Physical Warp Drives", arXiv:2102.06824 [gr-qc] https://doi.org/10.48550/arXiv.2102.06824</ref><ref>D. H. Coule, "No warp drive", Class. Quantum Grav. 15 (1998) 2523–2527. https://doi.org/10.1088/0264-9381/15/8/026</ref>, which rather makes warp drives unnecessary from the get-go.
<li>A warp bubble moving faster than light will not be able to get any signals in front of it as it will overtake its own signal. In fact, anyone in the interior of the bubble would not be able to send signals to the front surface of the bubble. In science-speak, a horizon forms at the front of the bubble<ref name="Natario">José Natário, "Warp drive with zero expansion", Classical and Quantum Gravity. 19 (6): 1157–1166 (2002). arXiv:[https://arxiv.org/abs/gr-qc/0110086 gr-qc/0110086]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2002CQGra..19.1157N 2002CQGra..19.1157N]. doi:[https://doi.org/10.1088%2F0264-9381%2F19%2F6%2F308 10.1088/0264-9381/19/6/308]. S2CID [https://api.semanticscholar.org/CorpusID:15859984 15859984].</ref>. As a consequence, the bubble cannot be controlled, steered, nor stopped from within the bubble itself.
<li>A Horizon will naturally emit Hawking radiation. It is estimated that the radiation will raise the temperature of the interior of the bubble to approximately 1032 K<ref name="Finazzi 2009">Stefano Finazzi, Stefano Liberati, and Carlos Barceló, "Semiclassical instability of dynamical warp drives", Phys. Rev. D 79, 124017 (12 June, 2009) https://doi.org/10.1103/PhysRevD.79.124017 , arXiv:0904.0141 [gr-qc] https://doi.org/10.48550/arXiv.0904.0141</ref> (at this level of accuracy, the result is the same if you replace temperatures in K by the same number in °C or °F). This temperature, however, may be lower if quantum inequalities do not have to be fulfilled.
<li>Semi-classical instabilities at the horizon are expected to lead to a divergence of the stress-energy tensor at the boundary in a short period of time, thus leading to the destruction of the warp bubble<ref name="Finazzi 2009"></ref>.
</ol>
Helpfully, Van Den Broeck<ref name="VanDenBroeck_2000">C. Van Den Broeck, "Alcubierre’s Warp Drive: Problems and Prospects" AIP Conference Proceedings. 504: 1105–1110 (2000). Bibcode:[https://ui.adsabs.harvard.edu/abs/2000AIPC..504.1105V 2000AIPC..504.1105V]. doi:[https://doi.org/10.1063%2F1.1290913 10.1063/1.1290913].</ref> suggested several solutions – or at least mitigations – for some of these problems
<ol>
<li>Quantum inequalities had not been shown to be true in general for highly curved space-times.
<li>A proposed warp drive geometry<ref name="VanDenBroeck_1999">C. Van Den Broeck, "A 'warp drive' with more reasonable total energy requirements". Classical and Quantum Gravity. 16 (12): 3973–3979 (1999). arXiv:[https://arxiv.org/abs/gr-qc/9905084 gr-qc/9905084]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1999CQGra..16.3973V 1999CQGra..16.3973V]. doi:[https://doi.org/10.1088%2F0264-9381%2F16%2F12%2F314 10.1088/0264-9381/16/12/314]. S2CID [https://api.semanticscholar.org/CorpusID:15466313 15466313]. </ref> (see the van Den Broeck drive, below) would be able to minimize the magnitude of the energy required to about that of the mass of our sun. While still large, it is not unphysically large.
<li>Time travel is always going to be a worry with faster than light travel. There's no neat solution to this one.
<li>It may be possible for the negative energy outside of the superluminal region to compress into the superluminal region to form a shock in space-time. The front surface of the bubble would then be a singularity. It is not clear if such a sharp jump in physical properties is possible, but neither is sure that it is impossible, either. However, while this might help with the negative energy in front of the bubble getting swept up because it cannot move faster than light, it does little to help with the material at the back of the bubble on the outside getting left behind because it cannot keep up.
</ol>
Other mitigating research has also shown
<ol start=9>
<li>Semi-classical divergence only occurs in warp drives with only one spatial dimensions. Those with two or more spatial dimensions (as in our universe) only contain points where divergences occur, and these divergences can be managed. In fact, their effects can be reduced by making the warp shape more elongated<ref>Carlos Barceló, Valentin Boyanov, Luis J. Garay, Eduardo Martín-Martínez,
and Jose M. Sánchez Velázquez, "Warp drive aerodynamics", JHEP08 (2022) 288 https://doi.org/10.1007/JHEP08(2022)288 , arXiv:2207.06458 [gr-qc] https://doi.org/10.48550/arXiv.2207.06458</ref>.
</ol>

(And what's with the ring in all the artwork, anyway?)

=== So what is going on? ===

What is it that makes a warp drive a warp drive, and how does it work?

A warp drive is a concept that comes out of the science of general relativity. General relativity collects space and time together into a single "structure" called spacetime. This structure is curved, and its curvature is determined by the material that is in it. Absent forces acting between matter, particles of matter move through spacetime on the shortest possible paths in this curved environment.
<blockquote>
"Spacetime tells matter how to move; matter tells spacetime how to curve." - John Archibald Wheeler
</blockquote>
All of this is encompassed in the Einstein field equation:
<div align=center>
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 8πG
<td rowspan="2" nowrap="nowrap" align="right"> T = R -  
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right"> R g
<tr>
<td style="border-top:solid 1px black;"> c4
<td style="border-top:solid 1px black;"> 2
</table>
</div>
Unfortunately, by convention, both the symbols G and R get used for a lot of different things. Here, symbols in bold are second rank tensors – constructs of arrays of 4×4 numbers where 4 is necessary because that is the number of dimensions of spacetime. Conventionally, these numbers are indexed by a value going from 0 to 3, with 0 being the part corresponding to the coordinate of time and 1 to 3 being the Cartesian coordinates that you might know as x, y, and z.

T is the stress-energy tensor: the (0,0) component represents the energy density at a given point; the (0,i) and (i, 0) components (with i going from 1 to 3) represent the momentum density; and the (i,j) components represent the stresses present, such as pressure or tension or shear.

The other term, R - R g/2, involves a partial description of the curvature. A full description of the curvature requires a construct of 4×4×4×4 components; too much for the stress energy tensor to fully constrain. Which is okay, because this allows curvature to exist away from where matter is, allowing gravity to have long range effects across empty space as well as for phenomena like gravitational waves. But it is enough that a given distribution of matter T gives enough constraints (along with specifying how the geometry behaves far away) that you can generally figure out what the actual gravity is.

This is the usual story that we are told. But there is an alternate way of looking at the same geometry such that the math works out the same, but it is described conceptually in a very different way. This divides spacetime up into "layers", called foliations, that represent slices of space and going from one foliation to the next is increasing in time. This is useful for computation because you can assign coordinates to your system. There are two additional constructs – the lapse function and the shift vector field. Starting at an event on one foliation, the lapse function at that position tells you that after a certain "proper time" (time perceived by an observer at that event) the time coordinate of the observer will be the proper time times the lapse function at that event. And the new spatial coordinates of the observer will be the proper time times the shift vector at that event. (Strictly speaking, this is only exactly true in the limit of the proper time approaching zero or when the lapse function and shift vectors are constant. But hopefully this gives a good enough insight to what these entities are so that we can go on to describe warp drives.)

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Lapse_shift.png]]
<tr><td>A folliation of space-time into slices of constant time coordinates (grey). At an event (black circle), the lapse function (yellow) shows how far into the future the point at that event will be projected in a unit amount of proper time experienced by an observer initially at the event. The shift vector (red) shows how that point changes coordinates after the unit amount of proper time. The cyan vector shows the new coordinates the point is projected onto. The lower black surface indicates a foliation of spacetime at the same time coordinate; after unit proper time, the events on the lower foliation get projected to the curved upper black foliation according to the shift vectors and lapse functions at each of their respective points.
</table>
</table>

Warp drives are generally restricted to spacetime geometries with a constant lapse function - all observers go forward by the same time coordinate after experiencing the same amount of proper time. Thus, all warp drives have no time dilation.
Far from the warp bubble, the shift vector vanishes – each spatial coordinate is mapped to the same spatial coordinate after any amount of proper time so long as the warp drive is not near. But the inside of the warp bubble is an extended region where the shift vector field is non-zero and constant. In this bubble, all spatial coordinates are mapped to a coordinate that shifts linearly in time with the proper time of the observer. Astute readers will see that this linear change in position with time is very similar to a velocity. This region of constant shift vector then moves with the shift vector velocity to keep stuff originally in the bubble so that it stays in the bubble. In between the inside of the bubble and the far away regions with zero shift vector, the shift vector field falls away to match one region to the other.

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Warp_shift.png]]
<tr><td>The shift vector of a one-dimensional slice through warp geometry, in red. At the edges there is no shift. At the center (inside the warp bubble) there is a region of uniform shift so that at a later time the inside of the warp bubble is rigidly translated to different spatial coordinates (in blue). The two foliations remain the same distance apart, because warp drives maintain a unit lapse function.
</table>
</table>

Traditionally, scientists would start with an interesting distribution of matter given by T and then find out what kind of curvature it created. This works well for planets and stars and black holes and cosmic strings and other things which are observed in the universe and you want to see what sort of effect they had. But starting in the 1980's, some physicists started looking at Einstein's field equations the other way around. They start with a desired geometry, find the curvature part of the field equation R - R g/2, and use that to find out what T has to be in order to get that curvature. This was originally done for [[Wormholes|wormholes]], but in 1994 Miguel Alcubierre used this same trick to figure out what could create a region of constant shift vector that moved with the shift vector - the warp drive. With this demand for a warp geometry, the required distribution of matter and energy and momentum and stress can be determined.

The original Alcubierre shift vector N→ produces a spherical warp bubble with an axi-symmetric distribution of, well, everything along the direction of motion with warp velocity vector v→.

<div align="center">
N→ = v→ f(rs)
</div>

=== Alcubierre warp interactions with light and matter ===

Space is not entirely empty. It is filled with a diffuse plasma in the form of the interstellar medium, as well as cosmic radiation, light from stars, and cosmic microwave background radiation. A warp drive propagating through space will encounter this stuff. What happens when this matter and radiation have a warp bubble pass across them?

The first analysis of matter encountering a warp bubble was performed by Pfenning and Ford<ref name="PfenningFord_2001"></ref>. They looked at the warp bubble interaction with a massive object at rest with the frame of reference of the warp drive (keep in mind that the rest frame is the same inside and outside the warp bubble; in this rest frame objects inside the bubble have no momentum even though they are, in some sense, changing location rapidly with time). When the warp bubble passes, the object experiences an acceleration in the direction of the warp bubble motion. When the warp shell passes and the object is inside the bubble, it will be moving with approximately the speed of the bubble. Pfenning and Ford analyzed this problem with a continuous distribution of shell energy that, strictly speaking, never falls to zero except at the bubble center and at spatial infinity, so unless the object passes through the center in Pfenning and Ford's description it will never quite get up to the bubble's speed. In this case, the object will move almost as fast as the object but will pass through the bubble in a finite time, after which it will again be at rest with respect to the reference frame but displaced along the direction of the warp bubble motion by some distance. With this description of the warp bubble, a spacecraft of finite size will always be moving a little slower than the warp bubble and would have to use rockets to keep up with it. in addition, the spacecraft would experience tidal forces that would cause stress on the spacecraft's structure.

Pfenning and Ford also examined cases where the shell is of a finite (possibly infinitesimal) width and falls to zero both inside and outside the bubble. In this case any matter encountering the bubble would be collected at the bow of the bubble and thereafter move along with it.

McMonigal et al. analyzed the situation for both massive particles and light moving along the axis of travel of the bubble<ref name="McMonigalEtAl_2012">B. McMonigal, G. F. Lewis, and P. O'Byrne, "Alcubierre warp drive: On the matter of matter". Physical Review D. 85 (6) 064024 (20 March 2012). arXiv:[https://arxiv.org/abs/1202.5708 1202.5708]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2012PhRvD..85f4024M 2012PhRvD..85f4024M]. doi:[https://doi.org/10.1103%2FPhysRevD.85.064024 10.1103/PhysRevD.85.064024]. S2CID [https://api.semanticscholar.org/CorpusID:3993148 3993148].</ref>. They found that light moving opposite the warp direction passed through the bubble without incident, being only somewhat delayed by passing through the bubble. A warp bubble that is warping at sub-luminal speeds can have light catch up from behind it. This light is able to pass through the bubble, and is somewhat advanced in its path by the speed of the bubble. For super-luminal warp bubbles, however, the situation is different. The bubble will catch up to light moving it its own direction that is originally in front of it. This light cannot escape forward, the bubble being too fast. Nor can it escape backward, as the light is propagating forward and the interior of the bubble is at rest. Thus, the light gets caught at the bow of the warp bubble, unable to escape for so long as the warp bubble is active. This light is strongly blue shifted to extremely energetic x-rays and gamma rays. The space behind the bubble is swept clear of forward-moving light.

The situation for matter over-run by a super-luminal warp bubble is similar. Matter moving backward passes through the bubble, being only swept a ways along the bubble's path as sign of its passing but otherwise continuing on their way. Matter at rest behaves the same way as Pfenning and Ford discovered. Meanwhile, matter moving in the same direction of the warp is overtaken and collects at the bow of the bubble, unable to leave for so long as the bubble is warping. This matter "surfing" on the bubble bow is highly accelerated to relativistic speeds, experiencing extreme time dilation.

For a sub-luminal bubble, there is a speed faster than the warp speed where particles coming up from behind overtake the bubble and pass through, out the front. Particles below this speed but still faster than the warp speed still overtake the bubble, but then bounce backwards out of the bubble from behind. They are still moving forward, but are now moving more slowly than the warp speed. Particles moving forward but slower than the bubble will be overtaken and then bounced out the front with a speed higher than the warp speed. Those particles moving backward will pass through the bubble and exit with their initial speed.

The spacecraft inside the bubble will observe that light moving backward compared to the warp direction is blue-shifted, and matter moving backward passes through with increased energy. Meanwhile, a sub-luminal bubble will see light that was moving forward as red-shifted and matter catching up to the bubble from behind will pass by with reduced energy (superluminal bubbles, of course, do not have any matter or radiation catching up to them from behind).

When a superluminal warp bubble that has been collecting matter and radiation for the duration of its journey and blue-shifting it to much higher energies is turned off, all that matter and energy is released as a blast of radiation in the direction the bubble was warping.

McMonigal et al. conclude by noting that any spacecraft in the warp bubble would need shielding to protect against the blue-shifted radiation and matter of increased energy. In addition the destination would be "blasted into oblivion" by the release of matter and radiation that had been caught in the bubble during the trip.

Clark et al.<ref>Chad Clark, William A. Hiscock, and Shane L. Larson, "Null geodesics in the Alcubierre warp drive spacetime: the view from the bridge", arXiv:gr-qc/9907019 (2018) https://doi.org/10.48550/arXiv.gr-qc/9907019</ref> examine what it would look like from within an operating warp drive. They find that light moving at 90 degrees to the direction of motion will look like it is still coming from 90 degrees. However, light from the front hemisphere is squished progressively toward the forward point the faster you go. For a warp bubble moving faster than the speed of light, there is a cone behind the warp bubble where no light can reach it. However, you don't see a black spot behind you – light from the regions in the backward hemisphere from outside that cone that can reach you is distorted toward the direction behind you to fill in that spot. Light from the forward direction is blue-shifted and light from behind is red-shifted. At speeds of around 200 times light speed, the cosmic microwave background would be blue shifted to energies similar to that of the solar photosphere, heating anything in the warp bubble as if they were close to the sun.

== Van Den Broeck warp drive ==

If an Alcubierre warp bubble a hundred meters across requires a magnitude of energy greater than the entire energy in the observable universe, one option to reduce the magnitude of energy used is to make the warp bubble smaller. Much smaller. Pfenning and Ford<ref name="PfenningFord_2001"></ref> suggested making the warp bubble smaller than an atom. Of course, that brings up the problem of how to stuff a spacecraft in there. Van Den Broeck proposed a solution<ref name="VanDenBroeck_1999"></ref>: make the warp bubble only about the size of an atomic nucleus but expand the space inside the bubble enormously so that a spacecraft could fit in. This is somewhat like a nuclear sized wormhole that leads to a pocket universe that holds the spacecraft. Unfortunately, the metric doesn't fit neatly into any [[Wormholes#Three_dimensions|embedding diagram]] that I can figure out, but the math works out so that spatial coordinates inside the bubble are expanded by a factor of 1017 and a spacecraft would have a few hundred meters of bubble interior to putz around in. This trick manages to reduce the magnitude of energy to only about the mass energy of a few stars similar to our own. Other than that, it is otherwise a normal Alcubierre drive.

Further work<ref>S. Krasnikov, "The quantum inequalities do not forbid spacetime shortcuts", Physical Review D. 67 (10) 104013. https://doi.org/10.1103%2FPhysRevD.67.104013, arXiv:gr-qc/0207057 (2003) https://doi.org/10.48550/arXiv.gr-qc/0207057 </ref> has managed to use Van Den Broeck's trick to reduce the magnitude of the negative energy to only about a milligram.

== Natário warp drive ==

The Alcubierre drive is not the only way to construct a warp drive. José Natário<ref name="Natario"></ref> showed that it was just one example of an entire class of warp drives. These are space-time geometries with a smooth and localized shift vector field a shift vector is a vector field that describes how one "slice" of space out of space-time connects to the next) and a unit lapse function (the lapse function describes how the time you experience changes when going from a "slice" of space-time to the next).

Natário then went on to find in that class a set of warp drives that do not have any expansion or contraction of space at all. Instead, space encountering the front boundary of the warp bubble instead "slides" around the outside of the bubble until it gets to the corresponding place on the back and is left behind there. Another way to think of it is that space entering the bubble shell is compressed in the radial direction of the shell but is simultaneously expanded in the tangential direction so that there is no net change in volume. When it gets to the back, the opposite occurs. In the language of mathematics, the shift vector field is divergenceless.

<table align=center>
<tr><td width=360>[[File:Natario_drive_flow_of_space.png]]
<td width=445>[[File:Natario_warp_drive_horizons.png]]
<tr><td style="vertical-align: top;">A representation of the motion of space in a zero-expansion Natário drive with a vanishingly thin warp shell around the bubble.
<td>The horizons of the generalized Natário class of warp drives (including the Alcubierre warp drive and all fell-Heisenberg drives) at superluminal speed in the direction of the green arrow. No event inside the bubble can affect events outside the bubble in front of the blue line. No events outside the bubble behind the red line can affect events inside the bubble. The lines have an angle α with respect to the direction of motion with sin(α) = 1/vs for vs the speed of the bubble relative to light speed. This is analogous to the Mach cone of supersonic objects.
</table>

NOTE: There are two different things that can be meant by the term "Natário warp drive". The generalized Natário warp drive is the general theory of all warp drives that includes within it the Alcubierre, Fell-Heisenberg, and Lentz warp drives, among infinitely many others. The zero-expansion Natário warp drive is the specific implementation of the generalized Natário warp drive that has a divergenceless shift vector field.

The generalized class of warp drives developed by Natário (including both the Alcubierre and this zero-expansion warp drive) are shown to always have regions in the warp shell where the energy density is negative to at least some observers. You can't get away from it – to warp, you need negative energy density.

The Natário class of warp drives blueshift light coming in from the front and redshift light catching up from the back. For the case of a super-luminal warp bubble, there will of course be a horizon at the back of the bubble and light will not be able to get through from behind. For a bubble speed vs relative to the speed of light, light coming from straight ahead will be blueshifted up in frequency and photon energy by a factor of 1 + vs. For sub-luminal travel, light coming from behind will be redshifted by a factor 1 - vs. In general for n as the unit direction along which the light is propagating, the blueshift factor will be 1 + vs ⋅ n. If the warp shell is infinitesimally thin, light from outside that reaches the center of the bubble will not be distorted in direction although it will be frequency shifted, but observers still inside the bubble but displaced from the center will see the field of view distorted as well as frequency shifted.

Most of the difficulties that apply to the Alcubierre warp drive also apply to the generalized Natário warp drive (and thus to the zero-expansion Natário drive and Fell-Heisenberg drive as well). In particular, quantum energy inequalities, high magnitudes of the energy required, formation of time machines, instabilities related to the stress-energy making up the warp shell being unable to keep up with the bubble, the chicken-and-egg problem of needing FTL to achieve FTL, the formation of horizons making warp bubbles uncontrollable, and Hawking radiation at the horizons, are all still concerns for generic warp drives.

== Fell-Heisenberg warp drives ==

S. D. B. Fell and L. Heisenberg investigated what sort of warp structure is necessary to produce a warp drive with positive energy density everywhere<ref name="Fell-Heisenberg">S. D. B. Fell and L> Heisenberg, "Positive energy warp drive from hidden geometric structures", Classical and Quantum Gravity 38 155020 (2021) https://doi.org/10.1088/1361-6382/ac0e47 https://arxiv.org/abs/2104.06488</ref>. They found that warp drives of the generic Natário form with irrotational shift vectors had a simple expression for the energy density and vanishing momentum (irrotational means that the shift vector can be described as the gradient of a scalar potential field or, equivalently, that the curl of the shift vector field vanishes everywhere). Fell and Heisenberg proposed several potential fields that either had positive energy density everywhere except at discontinuities, or that were continuous but had regions that violated the weak energy condition (i. e. negative energy densities in some frames of reference). One specific example they provided of a warp field had an energy that is about 10,000 times less than the mass-energy of our sun. Or only about half the mass-energy of Jupiter. This is a significant improvement over many other proposed drives.

Despite the introduction discussing warp drive configurations that satisfy the various energy conditions, the configurations described in the paper are shown to locally violate the weak and strong energy conditions. Nonetheless, the energy density is still mostly positive. Santiago, Schuster, and Visser's<ref name="Santiago Schuster Visser 2022">J. Santiago, S. Schuster, and M. Visser, "Generic warp drives violate the null energy condition", Physical Review D 105, 064038 (2022) https://doi.org/10.1103/PhysRevD.105.064038</ref> work shows that the various energy conditions must still be violated by this warp drive; to not violate these, energy density must be positive in all reference frames not just those of the co-moving observer.

Any vector field can be described by its divergence, its curl, and a constant. The constant part of a warp drive's shift vector can be discarded because then the warp drive would not be localized. It is interesting, therefore, that the Natário zero-expansion drive has zero divergence and can be described only by its curl, while the Fell-Heisenberg drive has zero curl and can be described only by its divergence. These two classes of drives thus rely on opposite choices of setting one of the two different parts describing a vector field to zero. As might be expected, the Fell-Heisenberg drive is not zero expansion – space-time will change in volume as it passes through the warp bubble (although after going all the way through it will return to its original volume). However, objects in space-time will retain their original orientation as they pass through the warp bubble.

Previous examples of warp drives, such as the Alcubierre and zero-expansion Natário drives, tend toward a flat space-time as you go far enough away faster than would occur if there were any mass present in the warp drive. That is, the ADM mass of such drives is zero. However, one of the Fell-Heisenberg examples described a spacetime where the geometry approaches that of what you would get for a massive object at the warp location, giving one of the first examples of a warp drive with mass.

In the Fell-Heisenberg drive the momentum vanishes everywhere. Yet the energy occupies regions where the shift vector is varying rapidly. The lack of momentum means that the energy will not move to keep up with the differential expansion and movement of the spacetime. As a consequence, at later times the energy will have a different distribution than what is necessary to maintain the given warp configuration. The exact time evolution is not solved but this likely leads to collapse of warp bubble. Note that this is not a problem unique to the Fell-Heisenberg drive. Assuming a geometry that describes a warp produces a specific distribution of the stress-energy tensor that satisfies Einstein's equation. But there is no guarantee that this stress energy tensor satisfies the continuity equation, which describes how the distribution of matter and momentum change with time. The issue of stability of any warp drive is not well understood, but most warp drive geometries will be unstable in the sense that their matter that generates the warp fields will not be able to keep up with the motion of the warp. The Fell-Heisenberg drive is unique only in that, with vanishing momentum, there is no configuration that allows stable propagation.

(n. b. The Heisenberg here is Lavinia Heisenberg, not the Werner Heisenberg of quantum physics and uncertainty principle fame.)

== Lentz warp drive ==

E. Lentz<ref name="Lentz">E. W. Lentz, "Breaking the warp barrier: hyper-fast solitons in Einstein–Maxwell-plasma theory", Classical and Quantum Gravity. 38 075015 (2021). arXiv:[https://arxiv.org/abs/2006.07125 2006.07125]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2021CQGra..38g5015L 2021CQGra..38g5015L]. doi:[https://doi.org/10.1088%2F1361-6382%2Fabe692 10.1088/1361-6382/abe692]. ISSN [https://search.worldcat.org/issn/0264-9381 0264-9381]. S2CID [https://api.semanticscholar.org/CorpusID:219635854 219635854].</ref> investigated a particular configuration of the Fell-Heisenberg class of warp drives, where the scalar field has a hyperbolic shape. The necessary conditions were found to be satisfied by a selection of rhomboid regions of the scalar field, resulting in an energy density that is everywhere positive and a shift vector with vanishing tidal force at the center for a comfortable ride.

However, Santiago, Schuster, and Visser<ref name="Santiago Schuster Visser 2022"></ref> dispute claims that the Lentz drive satisfies the energy conditions, noting that everywhere positive energy density in one frame of reference is insufficient to establish that the energy density is positive in all reference frames; knowledge of the Cauchy stress tensor is also needed. They show that any generic warp drive will violate the strong energy condition, null energy condition, and weak energy condition.

== Conservation laws ==

(Discussion of asymptotic flatness: what is it? Where does it apply?)

(The "big four" conservation laws and how they relate to asymptotic flatness.)

(ADM mass -> 0; conservation of energy/mass issues.)<ref>Sebastian Schuster, Jessica Santiago,and Matt Visser, "ADM mass in warp drive spacetimes" General Relativity and Gravitation (2023) 55:14 https://doi.org/10.1007/s10714-022-03061-9</ref>

(Discuss issues of conservation of angular momentum.)

(If angular momentum conservation is ignored, discuss how momentum & energy are affected by outside forces. In a gravitational field equivalent to inertial frame ... like being in an accelerated elevator; will acquire momentum buildup while staying "at rest".)

== Warp Railroads ==

Van Den Broeck<ref name="VanDenBroeck_2000"></ref> suggested a warp railroad,

You can set up devices ahead of time along the path of the bubble that produce the negative energy regions for the warp bubble at the appropriate time without needing the negative energy to ever move with the bubble at all. At least for bubbles on a predictable schedule and route.

helps with:

-stability (elaborate)

-conservation laws (elaborate)

Mention Krasnikov<ref name="Krasnikov 1998">S. V. Krasnikov, "Hyperfast travel in general relativity" Phys. Rev. D 57 4760</ref>

Hyperwave<ref name="Pieri 2024">Lorenzo Pieri, "Hyperwave: hyper-Fast Communication within General Relativity", arXiv:2311.12069 [gr-qc] https://doi.org/10.48550/arXiv.2311.12069</ref> makes use of this warp railroad idea.

== Quantum effects ==

(quantum stuff here<ref name="Hiscock1997">W. A. Hiscock, "Quantum effects in the Alcubierre warp drive spacetime", Classical and Quantum Gravity 14 L183 https://doi.org/10.1088/0264-9381/14/11/002 https://arxiv.org/abs/gr-qc/9707024</ref><ref name="Finazzi et al 2009">S. Finazzi, S. Liberati, C. Barceló, "Semiclassical instability of dynamical warp drives", Physical Review D 79, 124017 (2009)https://doi.org/10.1103/PhysRevD.79.124017 https://arxiv.org/abs/0904.0141</ref>)

From<ref name="Pieri 2024"></ref>
<blockquote>
It should be emphasised that Quantum Inequalities have not been proven for realistic interacting quantum fields, where the they are expected to be weaker if they exist [Cadamuro, 2019], and they often require Minkowskian conditions at infinity, a condition which is not appropriate for fields inside bounded regions, as in a Casimir devices.
</blockquote>

== Warp drives and black holes ==

What happens if you drive your warp drive into a black hole?

The pop-sci reason that nothing can escape a black hole is that at the event horizon the escape velocity is faster than light and nothing can go faster than light. A warp drive can go faster than light.

The more detailed reason from general relativity is that at the event horizon space and time are rotated sufficiently that "inward" becomes "forward in time." You can no more go farther away from the singularity in a black hole once you are inside the event horizon than you can go backward in time. Any super-luminal travel can go backward in time, and warp drives can engage in super-luminal travel.

The Hawking area theorem that says that black holes can only grow, never shrink relies on the condition that the energy is everywhere positive. Warp drives have negative energy density.

So in principle, there's nothing that prevents a warp drive from taking a dip into a black hole and coming back out again.

Garattini and Zatrimaylov looked into the problem of a warp drive near (and in) a black hole<ref name="Garattini and Zatrimaylov">R. Garattini and K. Zatrimaylov, "Black holes, warp drives, and energy conditions", Physics Letters B 856 138910 (2024) https://doi.org/10.1016/j.physletb.2024.138910</ref>. The found that the black hole reduces the negative energy requirement for the warp bubble to move inward toward the black hole singularity, but increases the negative energy needed to move away from the center of the black hole. In addition, a warp drive parked at the event horizon would allow light from inside the horizon to pass through the bubble and escape back outside.

== Credit ==
Author: Luke Campbell

== References ==

[[Category:Physics]][[Category:Infrastructure]][[Category:Physics & Engineering‏‎]][[Category:Physics & Math & Engineering]][[Category:Transportation & Infrastructure‏‎]][[Category:Metric Engineering]]

Warp Drives

2026-07-27T00:00:31Z

Lwcamp: /* So what is going on? */

{{PageConstructionNotice}}

Science fiction often features spacecraft that can seemingly move across space and get between the place of departure and the destination much faster than light could have done. This appears to contradict the theory of relativity, which predicts unequivocally that nothing can move through space faster than light. Because relativity has been incredibly successful at describing nature, with its many other predictions regularly being confirmed to extraordinary accuracy and within the bounds of uncertainty of all the experiments that tested them, it gives confidence that relativity is a correct description of reality. Which seems to rather throw a wet towel on our hopes for rapid travel between stars.

However, while relativity does not allow things to move through space faster than light, it places no such restrictions on how fast space-time itself can expand, contract, or move around. This leads to the idea of a warp drive – the spacecraft remains stationary within a region of highly curved space-time, and that region moves at super-luminal speeds rather than the spacecraft.

== The Alcubierre warp drive ==

The first warp drive geometry that satisfied the Einstein field equations of relativity was proposed by Miguel Alcubierre<ref>M. Alcubierre, "The warp drive: hyper-fast travel within general relativity." Classical and Quantum Gravity. 11 (5): L73–L77 (1994). [https://arxiv.org/abs/gr-qc/0009013 arXiv:gr-qc/0009013]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1994CQGra..11L..73A 1994CQGra..11L..73A]. doi:[https://doi.org/10.1088%2F0264-9381%2F11%2F5%2F001 10.1088/0264-9381/11/5/001]. S2CID [https://api.semanticscholar.org/CorpusID:4797900 4797900].</ref>. In this geometry, a sphere of space-time moves at an arbitrary speed (potentially but not necessarily a speed much faster than light). Objects within the sphere are moved along with the sphere; an object at rest within the sphere would be moved along with the sphere indefinitely. Space is expanding at the rear boundary and contracting at the front boundary in order to keep the sphere moving. In order to satisfy the Einstein field equations, the boundary of the sphere must have a negative energy density. The challenges of space-time geometries with negative energy densities are described in our page on [[Wormholes#Exotic_energy_conditions|wormholes]], for our purposes it is enough to note that negative energy densities can pose problems if not handled carefully, there may be limits on how much negative energy you can have without nearby positive energy density, and it may not be possible to get enough negative energy to support a warp drive; although none of this is ruled out by physics – yet!

<table align=center>
<tr><td width=650>[[File:Energy_magnitude_of_Alcubierre_drive.png]]
<td width=650>[[File:Expansion_of_Alcubierre_drive.png]]
<tr><td>The magnitude of the energy of the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. (The energy density is negative everywhere in the shell, here only the magnitude is shown to aid visual comprehension.)
<td>The expansion of space in the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. Negative expansion means that space is contracting in that direction, positive expansion that it is expanding.
</table>

NOTE: What the? The energy distribution is symmetric! How does the drive know which way to go? There must be significant contributions of other components of the stress-energy tensor, and those have got to be asymmetric along the forward/backward axis. Check this when I get time.

Almost as soon as Alcubierre proposed his warp drive, others began picking it apart.
<ol>
<li>Pfenning and Ford<ref name="PfenningFord_2001">M. J. Pfenning and L. H. Ford, "The unphysical nature of 'Warp Drive'", Classical and Quantum Gravity. 14 (7): 1743–1751 (1997). arXiv:[https://arxiv.org/abs/gr-qc/9702026 gr-qc/9702026]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1997CQGra..14.1743P 1997CQGra..14.1743P]. doi:[https://doi.org/10.1088%2F0264-9381%2F14%2F7%2F011 10.1088/0264-9381/14/7/011]. S2CID [https://api.semanticscholar.org/CorpusID:15279207 15279207].</ref> found that if the negative energy density wall around the bubble obeys [[Wormholes#Quantum_energy_inequalities|quantum energy inequalities]], then for warp speeds of around light speed the shell thickness would be on the order of a hundred Planck lengths; a distance far smaller than any other known physical phenomenon or object.
<li>In the same work, Pfenning and Ford also found that for a bubble a hundred meters in radius, a warp speed of around the speed of light, and the required thickness of about 100 Planck lengths the energy in the bubble shell would be E ≈ -1063 kg c2. The magnitude of this latter value is ten orders of magnitude larger than the energy of the entire visible universe. They do note, however, that if quantum energy inequalities can be ignored then a warp drive with a hundred meters radius but a shell thickness of one meter would "only" have an energy magnitude of about a quarter solar mass.
<li>As with any method of faster than light travel, the warp drive could be used to make a time machine.
<li>Perhaps most seriously, if the warp drive is going faster than light speed, the negative energy regions on the outside of the shell won't be in the warp parts of the bubble. They will have to be moving through space at faster than light speed if the bubble is to maintain its integrity, which is the very problem that the warp drive was designed to avoid. The outside of the warp bubble shell that maintains the warp bubble would fall away and the warp bubble would quickly erode to sub-luminal speeds.<ref name="Van Den Broeck 2000">Chris Van Den Broeck, "On the (im)possibility of warp bubbles", arXiv:gr-qc/9906050 https://doi.org/10.48550/arXiv.gr-qc/9906050</ref>
<li>A more general problem that includes the previous one is that the geometry of the warp drive determines the distribution of energy, momentum, and stress (pressure, shear, and tension) by Einstein's field equation, which states that the stress energy tensor is directly proportional to a reduced order curvature tensor of space-time. However, the stress-energy tensor must also obey the continuity equation – basically, for the energy to change in a given volume of space that energy must come in or go out of the boundary of that volume and similarly for the momentum. So a warp drive can have the warp condition for one moment in time. But after that, the stress-energy that maintains the warp will start to move according to the forces on it and according to the natural paths that it takes through the warped space-time. There is no guarantee that the configuration of the stress-energy at later times will be sufficient to maintain the warp.
<blockquote>
From the perspective of simulating the warp drive dynamically, the key challenge is stability. One can straightforwardly generate the initial metric that describes the Alcubierre metric and infer the instantaneous matter configuration that must support it using the Einstein Equation, but a time-domain evolution of the coupled matter-gravity system necessitates that we specify the equation of state of the matter, which in turn determines how it evolves. In a perfect fluid, this involves specifying the relation between the fluid pressure and density, but in a more general case, the full relation between the
stress-energy components must be given. There is (to our knowledge) no known equation of state that would maintain the warp drive metric in a stable configuration over time - therefore, whilst one can require that initially, the warp bubble is constant, it will quickly evolve away from that state and, in most cases, the warp fluid and spacetime deformations will disperse or collapse into a central point.
 
– Clough, Dietrich, and Khan<ref>Katy Clough, Tim Dietrich, and Sebastian Khan, "What no one has seen before: gravitational waveforms from warp drive collapse", arXiv:2406.02466 [gr-qc] https://doi.org/10.48550/arXiv.2406.02466</ref>
</blockquote>
<li>In order to get the matter (or more generally, stress-energy) that forms the warp shell up to speeds that could allow the shell to exist, you need to be able to, well, accelerate that matter to warp speeds. Which gives a chicken-and-egg problem - in order to achieve superluminal travel you need to be able to get matter or energy moving superluminally in the first place<ref name="Bobrick and Martire">Alexey Bobrick and Gianni Martire, "Introducing Physical Warp Drives", arXiv:2102.06824 [gr-qc] https://doi.org/10.48550/arXiv.2102.06824</ref><ref>D. H. Coule, "No warp drive", Class. Quantum Grav. 15 (1998) 2523–2527. https://doi.org/10.1088/0264-9381/15/8/026</ref>, which rather makes warp drives unnecessary from the get-go.
<li>A warp bubble moving faster than light will not be able to get any signals in front of it as it will overtake its own signal. In fact, anyone in the interior of the bubble would not be able to send signals to the front surface of the bubble. In science-speak, a horizon forms at the front of the bubble<ref name="Natario">José Natário, "Warp drive with zero expansion", Classical and Quantum Gravity. 19 (6): 1157–1166 (2002). arXiv:[https://arxiv.org/abs/gr-qc/0110086 gr-qc/0110086]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2002CQGra..19.1157N 2002CQGra..19.1157N]. doi:[https://doi.org/10.1088%2F0264-9381%2F19%2F6%2F308 10.1088/0264-9381/19/6/308]. S2CID [https://api.semanticscholar.org/CorpusID:15859984 15859984].</ref>. As a consequence, the bubble cannot be controlled, steered, nor stopped from within the bubble itself.
<li>A Horizon will naturally emit Hawking radiation. It is estimated that the radiation will raise the temperature of the interior of the bubble to approximately 1032 K<ref name="Finazzi 2009">Stefano Finazzi, Stefano Liberati, and Carlos Barceló, "Semiclassical instability of dynamical warp drives", Phys. Rev. D 79, 124017 (12 June, 2009) https://doi.org/10.1103/PhysRevD.79.124017 , arXiv:0904.0141 [gr-qc] https://doi.org/10.48550/arXiv.0904.0141</ref> (at this level of accuracy, the result is the same if you replace temperatures in K by the same number in °C or °F). This temperature, however, may be lower if quantum inequalities do not have to be fulfilled.
<li>Semi-classical instabilities at the horizon are expected to lead to a divergence of the stress-energy tensor at the boundary in a short period of time, thus leading to the destruction of the warp bubble<ref name="Finazzi 2009"></ref>.
</ol>
Helpfully, Van Den Broeck<ref name="VanDenBroeck_2000">C. Van Den Broeck, "Alcubierre’s Warp Drive: Problems and Prospects" AIP Conference Proceedings. 504: 1105–1110 (2000). Bibcode:[https://ui.adsabs.harvard.edu/abs/2000AIPC..504.1105V 2000AIPC..504.1105V]. doi:[https://doi.org/10.1063%2F1.1290913 10.1063/1.1290913].</ref> suggested several solutions – or at least mitigations – for some of these problems
<ol>
<li>Quantum inequalities had not been shown to be true in general for highly curved space-times.
<li>A proposed warp drive geometry<ref name="VanDenBroeck_1999">C. Van Den Broeck, "A 'warp drive' with more reasonable total energy requirements". Classical and Quantum Gravity. 16 (12): 3973–3979 (1999). arXiv:[https://arxiv.org/abs/gr-qc/9905084 gr-qc/9905084]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1999CQGra..16.3973V 1999CQGra..16.3973V]. doi:[https://doi.org/10.1088%2F0264-9381%2F16%2F12%2F314 10.1088/0264-9381/16/12/314]. S2CID [https://api.semanticscholar.org/CorpusID:15466313 15466313]. </ref> (see the van Den Broeck drive, below) would be able to minimize the magnitude of the energy required to about that of the mass of our sun. While still large, it is not unphysically large.
<li>Time travel is always going to be a worry with faster than light travel. There's no neat solution to this one.
<li>It may be possible for the negative energy outside of the superluminal region to compress into the superluminal region to form a shock in space-time. The front surface of the bubble would then be a singularity. It is not clear if such a sharp jump in physical properties is possible, but neither is sure that it is impossible, either. However, while this might help with the negative energy in front of the bubble getting swept up because it cannot move faster than light, it does little to help with the material at the back of the bubble on the outside getting left behind because it cannot keep up.
</ol>
Other mitigating research has also shown
<ol start=9>
<li>Semi-classical divergence only occurs in warp drives with only one spatial dimensions. Those with two or more spatial dimensions (as in our universe) only contain points where divergences occur, and these divergences can be managed. In fact, their effects can be reduced by making the warp shape more elongated<ref>Carlos Barceló, Valentin Boyanov, Luis J. Garay, Eduardo Martín-Martínez,
and Jose M. Sánchez Velázquez, "Warp drive aerodynamics", JHEP08 (2022) 288 https://doi.org/10.1007/JHEP08(2022)288 , arXiv:2207.06458 [gr-qc] https://doi.org/10.48550/arXiv.2207.06458</ref>.
</ol>

(And what's with the ring in all the artwork, anyway?)

=== So what is going on? ===

What is it that makes a warp drive a warp drive, and how does it work?

A warp drive is a concept that comes out of the science of general relativity. General relativity collects space and time together into a single "structure" called spacetime. This structure is curved, and its curvature is determined by the material that is in it. Absent forces acting between matter, particles of matter move through spacetime on the shortest possible paths in this curved environment.
<blockquote>
"Spacetime tells matter how to move; matter tells spacetime how to curve." - John Archibald Wheeler
</blockquote>
All of this is encompassed in the Einstein field equation:
<div align=center>
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 8πG
<td rowspan="2" nowrap="nowrap" align="right"> T = R -  
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right"> R g
<tr>
<td style="border-top:solid 1px black;"> c4
<td style="border-top:solid 1px black;"> 2
</table>
</div>
Unfortunately, by convention, both the symbols G and R get used for a lot of different things. Here, symbols in bold are second rank tensors – constructs of arrays of 4×4 numbers where 4 is necessary because that is the number of dimensions of spacetime. Conventionally, these numbers are indexed by a value going from 0 to 3, with 0 being the part corresponding to the coordinate of time and 1 to 3 being the Cartesian coordinates that you might know as x, y, and z.

T is the stress-energy tensor: the (0,0) component represents the energy density at a given point; the (0,i) and (i, 0) components (with i going from 1 to 3) represent the momentum density; and the (i,j) components represent the stresses present, such as pressure or tension or shear.

The other term, R - R g/2, involves a partial description of the curvature. A full description of the curvature requires a construct of 4×4×4×4 components; too much for the stress energy tensor to fully constrain. Which is okay, because this allows curvature to exist away from where matter is, allowing gravity to have long range effects across empty space as well as for phenomena like gravitational waves. But it is enough that a given distribution of matter T gives enough constraints (along with specifying how the geometry behaves far away) that you can generally figure out what the actual gravity is.

This is the usual story that we are told. But there is an alternate way of looking at the same geometry such that the math works out the same, but it is described conceptually in a very different way. This divides spacetime up into "layers", called foliations, that represent slices of space and going from one foliation to the next is increasing in time. This is useful for computation because you can assign coordinates to your system. There are two additional constructs – the lapse function and the shift vector field. Starting at an event on one foliation, the lapse function at that position tells you that after a certain "proper time" (time perceived by an observer at that event) the time coordinate of the observer will be the proper time times the lapse function at that event. And the new spatial coordinates of the observer will be the proper time times the shift vector at that event. (Strictly speaking, this is only exactly true in the limit of the proper time approaching zero or when the lapse function and shift vectors are constant. But hopefully this gives a good enough insight to what these entities are so that we can go on to describe warp drives.)

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Lapse_shift.png]]
<tr><td>A folliation of space-time into slices of constant time coordinates (grey). At an event (black circle), the lapse function (yellow) shows how far into the future the point at that event will be projected in a unit amount of proper time experienced by an observer initially at the event. The shift vector (red) shows how that point changes coordinates after the unit amount of proper time. The cyan vector shows the new coordinates the point is projected onto. The lower black surface indicates a foliation of spacetime at the same time coordinate; after unit proper time, the events on the lower foliation get projected to the curved upper black foliation according to the shift vectors and lapse functions at each of their respective points.
</table>
</table>

Warp drives are generally restricted to spacetime geometries with a constant lapse function - all observers go forward by the same time coordinate after experiencing the same amount of proper time. Thus, all warp drives have no time dilation.
Far from the warp bubble, the shift vector vanishes – each spatial coordinate is mapped to the same spatial coordinate after any amount of proper time so long as the warp drive is not near. But the inside of the warp bubble is an extended region where the shift vector field is non-zero and constant. In this bubble, all spatial coordinates are mapped to a coordinate that shifts linearly in time with the proper time of the observer. Astute readers will see that this linear change in position with time is very similar to a velocity. This region of constant shift vector then moves with the shift vector velocity to keep stuff originally in the bubble so that it stays in the bubble. In between the inside of the bubble and the far away regions with zero shift vector, the shift vector field falls away to match one region to the other.

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Warp_shift.png]]
<tr><td>The shift vector of a one-dimensional slice through warp geometry, in red. At the edges there is no shift. At the center (inside the warp bubble) there is a region of uniform shift so that at a later time the inside of the warp bubble is rigidly translated to different spatial coordinates (in blue). The two foliations remain the same distance apart, because warp drives maintain a unit lapse function.
</table>
</table>

Traditionally, scientists would start with an interesting distribution of matter given by T and then find out what kind of curvature it created. This works well for planets and stars and black holes and cosmic strings and other things which are observed in the universe and you want to see what sort of effect they had. But starting in the 1980's, some physicists started looking at Einstein's field equations the other way around. They start with a desired geometry, find the curvature part of the field equation R - R g/2, and use that to find out what T has to be in order to get that curvature. This was originally done for [[Wormholes|wormholes]], but in 1994 Miguel Alcubierre used this same trick to figure out what could create a region of constant shift vector that moved with the shift vector - the warp drive. With this demand for a warp geometry, the required distribution of matter and energy and momentum and stress can be determined.

The original Alcubierre shift vector N→ produces a spherical warp bubble with an axi-symmetric distribution of, well, everything along the direction of motion with warp velocity vector v→.

N→ = v→ f(rs)

=== Alcubierre warp interactions with light and matter ===

Space is not entirely empty. It is filled with a diffuse plasma in the form of the interstellar medium, as well as cosmic radiation, light from stars, and cosmic microwave background radiation. A warp drive propagating through space will encounter this stuff. What happens when this matter and radiation have a warp bubble pass across them?

The first analysis of matter encountering a warp bubble was performed by Pfenning and Ford<ref name="PfenningFord_2001"></ref>. They looked at the warp bubble interaction with a massive object at rest with the frame of reference of the warp drive (keep in mind that the rest frame is the same inside and outside the warp bubble; in this rest frame objects inside the bubble have no momentum even though they are, in some sense, changing location rapidly with time). When the warp bubble passes, the object experiences an acceleration in the direction of the warp bubble motion. When the warp shell passes and the object is inside the bubble, it will be moving with approximately the speed of the bubble. Pfenning and Ford analyzed this problem with a continuous distribution of shell energy that, strictly speaking, never falls to zero except at the bubble center and at spatial infinity, so unless the object passes through the center in Pfenning and Ford's description it will never quite get up to the bubble's speed. In this case, the object will move almost as fast as the object but will pass through the bubble in a finite time, after which it will again be at rest with respect to the reference frame but displaced along the direction of the warp bubble motion by some distance. With this description of the warp bubble, a spacecraft of finite size will always be moving a little slower than the warp bubble and would have to use rockets to keep up with it. in addition, the spacecraft would experience tidal forces that would cause stress on the spacecraft's structure.

Pfenning and Ford also examined cases where the shell is of a finite (possibly infinitesimal) width and falls to zero both inside and outside the bubble. In this case any matter encountering the bubble would be collected at the bow of the bubble and thereafter move along with it.

McMonigal et al. analyzed the situation for both massive particles and light moving along the axis of travel of the bubble<ref name="McMonigalEtAl_2012">B. McMonigal, G. F. Lewis, and P. O'Byrne, "Alcubierre warp drive: On the matter of matter". Physical Review D. 85 (6) 064024 (20 March 2012). arXiv:[https://arxiv.org/abs/1202.5708 1202.5708]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2012PhRvD..85f4024M 2012PhRvD..85f4024M]. doi:[https://doi.org/10.1103%2FPhysRevD.85.064024 10.1103/PhysRevD.85.064024]. S2CID [https://api.semanticscholar.org/CorpusID:3993148 3993148].</ref>. They found that light moving opposite the warp direction passed through the bubble without incident, being only somewhat delayed by passing through the bubble. A warp bubble that is warping at sub-luminal speeds can have light catch up from behind it. This light is able to pass through the bubble, and is somewhat advanced in its path by the speed of the bubble. For super-luminal warp bubbles, however, the situation is different. The bubble will catch up to light moving it its own direction that is originally in front of it. This light cannot escape forward, the bubble being too fast. Nor can it escape backward, as the light is propagating forward and the interior of the bubble is at rest. Thus, the light gets caught at the bow of the warp bubble, unable to escape for so long as the warp bubble is active. This light is strongly blue shifted to extremely energetic x-rays and gamma rays. The space behind the bubble is swept clear of forward-moving light.

The situation for matter over-run by a super-luminal warp bubble is similar. Matter moving backward passes through the bubble, being only swept a ways along the bubble's path as sign of its passing but otherwise continuing on their way. Matter at rest behaves the same way as Pfenning and Ford discovered. Meanwhile, matter moving in the same direction of the warp is overtaken and collects at the bow of the bubble, unable to leave for so long as the bubble is warping. This matter "surfing" on the bubble bow is highly accelerated to relativistic speeds, experiencing extreme time dilation.

For a sub-luminal bubble, there is a speed faster than the warp speed where particles coming up from behind overtake the bubble and pass through, out the front. Particles below this speed but still faster than the warp speed still overtake the bubble, but then bounce backwards out of the bubble from behind. They are still moving forward, but are now moving more slowly than the warp speed. Particles moving forward but slower than the bubble will be overtaken and then bounced out the front with a speed higher than the warp speed. Those particles moving backward will pass through the bubble and exit with their initial speed.

The spacecraft inside the bubble will observe that light moving backward compared to the warp direction is blue-shifted, and matter moving backward passes through with increased energy. Meanwhile, a sub-luminal bubble will see light that was moving forward as red-shifted and matter catching up to the bubble from behind will pass by with reduced energy (superluminal bubbles, of course, do not have any matter or radiation catching up to them from behind).

When a superluminal warp bubble that has been collecting matter and radiation for the duration of its journey and blue-shifting it to much higher energies is turned off, all that matter and energy is released as a blast of radiation in the direction the bubble was warping.

McMonigal et al. conclude by noting that any spacecraft in the warp bubble would need shielding to protect against the blue-shifted radiation and matter of increased energy. In addition the destination would be "blasted into oblivion" by the release of matter and radiation that had been caught in the bubble during the trip.

Clark et al.<ref>Chad Clark, William A. Hiscock, and Shane L. Larson, "Null geodesics in the Alcubierre warp drive spacetime: the view from the bridge", arXiv:gr-qc/9907019 (2018) https://doi.org/10.48550/arXiv.gr-qc/9907019</ref> examine what it would look like from within an operating warp drive. They find that light moving at 90 degrees to the direction of motion will look like it is still coming from 90 degrees. However, light from the front hemisphere is squished progressively toward the forward point the faster you go. For a warp bubble moving faster than the speed of light, there is a cone behind the warp bubble where no light can reach it. However, you don't see a black spot behind you – light from the regions in the backward hemisphere from outside that cone that can reach you is distorted toward the direction behind you to fill in that spot. Light from the forward direction is blue-shifted and light from behind is red-shifted. At speeds of around 200 times light speed, the cosmic microwave background would be blue shifted to energies similar to that of the solar photosphere, heating anything in the warp bubble as if they were close to the sun.

== Van Den Broeck warp drive ==

If an Alcubierre warp bubble a hundred meters across requires a magnitude of energy greater than the entire energy in the observable universe, one option to reduce the magnitude of energy used is to make the warp bubble smaller. Much smaller. Pfenning and Ford<ref name="PfenningFord_2001"></ref> suggested making the warp bubble smaller than an atom. Of course, that brings up the problem of how to stuff a spacecraft in there. Van Den Broeck proposed a solution<ref name="VanDenBroeck_1999"></ref>: make the warp bubble only about the size of an atomic nucleus but expand the space inside the bubble enormously so that a spacecraft could fit in. This is somewhat like a nuclear sized wormhole that leads to a pocket universe that holds the spacecraft. Unfortunately, the metric doesn't fit neatly into any [[Wormholes#Three_dimensions|embedding diagram]] that I can figure out, but the math works out so that spatial coordinates inside the bubble are expanded by a factor of 1017 and a spacecraft would have a few hundred meters of bubble interior to putz around in. This trick manages to reduce the magnitude of energy to only about the mass energy of a few stars similar to our own. Other than that, it is otherwise a normal Alcubierre drive.

Further work<ref>S. Krasnikov, "The quantum inequalities do not forbid spacetime shortcuts", Physical Review D. 67 (10) 104013. https://doi.org/10.1103%2FPhysRevD.67.104013, arXiv:gr-qc/0207057 (2003) https://doi.org/10.48550/arXiv.gr-qc/0207057 </ref> has managed to use Van Den Broeck's trick to reduce the magnitude of the negative energy to only about a milligram.

== Natário warp drive ==

The Alcubierre drive is not the only way to construct a warp drive. José Natário<ref name="Natario"></ref> showed that it was just one example of an entire class of warp drives. These are space-time geometries with a smooth and localized shift vector field a shift vector is a vector field that describes how one "slice" of space out of space-time connects to the next) and a unit lapse function (the lapse function describes how the time you experience changes when going from a "slice" of space-time to the next).

Natário then went on to find in that class a set of warp drives that do not have any expansion or contraction of space at all. Instead, space encountering the front boundary of the warp bubble instead "slides" around the outside of the bubble until it gets to the corresponding place on the back and is left behind there. Another way to think of it is that space entering the bubble shell is compressed in the radial direction of the shell but is simultaneously expanded in the tangential direction so that there is no net change in volume. When it gets to the back, the opposite occurs. In the language of mathematics, the shift vector field is divergenceless.

<table align=center>
<tr><td width=360>[[File:Natario_drive_flow_of_space.png]]
<td width=445>[[File:Natario_warp_drive_horizons.png]]
<tr><td style="vertical-align: top;">A representation of the motion of space in a zero-expansion Natário drive with a vanishingly thin warp shell around the bubble.
<td>The horizons of the generalized Natário class of warp drives (including the Alcubierre warp drive and all fell-Heisenberg drives) at superluminal speed in the direction of the green arrow. No event inside the bubble can affect events outside the bubble in front of the blue line. No events outside the bubble behind the red line can affect events inside the bubble. The lines have an angle α with respect to the direction of motion with sin(α) = 1/vs for vs the speed of the bubble relative to light speed. This is analogous to the Mach cone of supersonic objects.
</table>

NOTE: There are two different things that can be meant by the term "Natário warp drive". The generalized Natário warp drive is the general theory of all warp drives that includes within it the Alcubierre, Fell-Heisenberg, and Lentz warp drives, among infinitely many others. The zero-expansion Natário warp drive is the specific implementation of the generalized Natário warp drive that has a divergenceless shift vector field.

The generalized class of warp drives developed by Natário (including both the Alcubierre and this zero-expansion warp drive) are shown to always have regions in the warp shell where the energy density is negative to at least some observers. You can't get away from it – to warp, you need negative energy density.

The Natário class of warp drives blueshift light coming in from the front and redshift light catching up from the back. For the case of a super-luminal warp bubble, there will of course be a horizon at the back of the bubble and light will not be able to get through from behind. For a bubble speed vs relative to the speed of light, light coming from straight ahead will be blueshifted up in frequency and photon energy by a factor of 1 + vs. For sub-luminal travel, light coming from behind will be redshifted by a factor 1 - vs. In general for n as the unit direction along which the light is propagating, the blueshift factor will be 1 + vs ⋅ n. If the warp shell is infinitesimally thin, light from outside that reaches the center of the bubble will not be distorted in direction although it will be frequency shifted, but observers still inside the bubble but displaced from the center will see the field of view distorted as well as frequency shifted.

Most of the difficulties that apply to the Alcubierre warp drive also apply to the generalized Natário warp drive (and thus to the zero-expansion Natário drive and Fell-Heisenberg drive as well). In particular, quantum energy inequalities, high magnitudes of the energy required, formation of time machines, instabilities related to the stress-energy making up the warp shell being unable to keep up with the bubble, the chicken-and-egg problem of needing FTL to achieve FTL, the formation of horizons making warp bubbles uncontrollable, and Hawking radiation at the horizons, are all still concerns for generic warp drives.

== Fell-Heisenberg warp drives ==

S. D. B. Fell and L. Heisenberg investigated what sort of warp structure is necessary to produce a warp drive with positive energy density everywhere<ref name="Fell-Heisenberg">S. D. B. Fell and L> Heisenberg, "Positive energy warp drive from hidden geometric structures", Classical and Quantum Gravity 38 155020 (2021) https://doi.org/10.1088/1361-6382/ac0e47 https://arxiv.org/abs/2104.06488</ref>. They found that warp drives of the generic Natário form with irrotational shift vectors had a simple expression for the energy density and vanishing momentum (irrotational means that the shift vector can be described as the gradient of a scalar potential field or, equivalently, that the curl of the shift vector field vanishes everywhere). Fell and Heisenberg proposed several potential fields that either had positive energy density everywhere except at discontinuities, or that were continuous but had regions that violated the weak energy condition (i. e. negative energy densities in some frames of reference). One specific example they provided of a warp field had an energy that is about 10,000 times less than the mass-energy of our sun. Or only about half the mass-energy of Jupiter. This is a significant improvement over many other proposed drives.

Despite the introduction discussing warp drive configurations that satisfy the various energy conditions, the configurations described in the paper are shown to locally violate the weak and strong energy conditions. Nonetheless, the energy density is still mostly positive. Santiago, Schuster, and Visser's<ref name="Santiago Schuster Visser 2022">J. Santiago, S. Schuster, and M. Visser, "Generic warp drives violate the null energy condition", Physical Review D 105, 064038 (2022) https://doi.org/10.1103/PhysRevD.105.064038</ref> work shows that the various energy conditions must still be violated by this warp drive; to not violate these, energy density must be positive in all reference frames not just those of the co-moving observer.

Any vector field can be described by its divergence, its curl, and a constant. The constant part of a warp drive's shift vector can be discarded because then the warp drive would not be localized. It is interesting, therefore, that the Natário zero-expansion drive has zero divergence and can be described only by its curl, while the Fell-Heisenberg drive has zero curl and can be described only by its divergence. These two classes of drives thus rely on opposite choices of setting one of the two different parts describing a vector field to zero. As might be expected, the Fell-Heisenberg drive is not zero expansion – space-time will change in volume as it passes through the warp bubble (although after going all the way through it will return to its original volume). However, objects in space-time will retain their original orientation as they pass through the warp bubble.

Previous examples of warp drives, such as the Alcubierre and zero-expansion Natário drives, tend toward a flat space-time as you go far enough away faster than would occur if there were any mass present in the warp drive. That is, the ADM mass of such drives is zero. However, one of the Fell-Heisenberg examples described a spacetime where the geometry approaches that of what you would get for a massive object at the warp location, giving one of the first examples of a warp drive with mass.

In the Fell-Heisenberg drive the momentum vanishes everywhere. Yet the energy occupies regions where the shift vector is varying rapidly. The lack of momentum means that the energy will not move to keep up with the differential expansion and movement of the spacetime. As a consequence, at later times the energy will have a different distribution than what is necessary to maintain the given warp configuration. The exact time evolution is not solved but this likely leads to collapse of warp bubble. Note that this is not a problem unique to the Fell-Heisenberg drive. Assuming a geometry that describes a warp produces a specific distribution of the stress-energy tensor that satisfies Einstein's equation. But there is no guarantee that this stress energy tensor satisfies the continuity equation, which describes how the distribution of matter and momentum change with time. The issue of stability of any warp drive is not well understood, but most warp drive geometries will be unstable in the sense that their matter that generates the warp fields will not be able to keep up with the motion of the warp. The Fell-Heisenberg drive is unique only in that, with vanishing momentum, there is no configuration that allows stable propagation.

(n. b. The Heisenberg here is Lavinia Heisenberg, not the Werner Heisenberg of quantum physics and uncertainty principle fame.)

== Lentz warp drive ==

E. Lentz<ref name="Lentz">E. W. Lentz, "Breaking the warp barrier: hyper-fast solitons in Einstein–Maxwell-plasma theory", Classical and Quantum Gravity. 38 075015 (2021). arXiv:[https://arxiv.org/abs/2006.07125 2006.07125]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2021CQGra..38g5015L 2021CQGra..38g5015L]. doi:[https://doi.org/10.1088%2F1361-6382%2Fabe692 10.1088/1361-6382/abe692]. ISSN [https://search.worldcat.org/issn/0264-9381 0264-9381]. S2CID [https://api.semanticscholar.org/CorpusID:219635854 219635854].</ref> investigated a particular configuration of the Fell-Heisenberg class of warp drives, where the scalar field has a hyperbolic shape. The necessary conditions were found to be satisfied by a selection of rhomboid regions of the scalar field, resulting in an energy density that is everywhere positive and a shift vector with vanishing tidal force at the center for a comfortable ride.

However, Santiago, Schuster, and Visser<ref name="Santiago Schuster Visser 2022"></ref> dispute claims that the Lentz drive satisfies the energy conditions, noting that everywhere positive energy density in one frame of reference is insufficient to establish that the energy density is positive in all reference frames; knowledge of the Cauchy stress tensor is also needed. They show that any generic warp drive will violate the strong energy condition, null energy condition, and weak energy condition.

== Conservation laws ==

(Discussion of asymptotic flatness: what is it? Where does it apply?)

(The "big four" conservation laws and how they relate to asymptotic flatness.)

(ADM mass -> 0; conservation of energy/mass issues.)<ref>Sebastian Schuster, Jessica Santiago,and Matt Visser, "ADM mass in warp drive spacetimes" General Relativity and Gravitation (2023) 55:14 https://doi.org/10.1007/s10714-022-03061-9</ref>

(Discuss issues of conservation of angular momentum.)

(If angular momentum conservation is ignored, discuss how momentum & energy are affected by outside forces. In a gravitational field equivalent to inertial frame ... like being in an accelerated elevator; will acquire momentum buildup while staying "at rest".)

== Warp Railroads ==

Van Den Broeck<ref name="VanDenBroeck_2000"></ref> suggested a warp railroad,

You can set up devices ahead of time along the path of the bubble that produce the negative energy regions for the warp bubble at the appropriate time without needing the negative energy to ever move with the bubble at all. At least for bubbles on a predictable schedule and route.

helps with:

-stability (elaborate)

-conservation laws (elaborate)

Mention Krasnikov<ref name="Krasnikov 1998">S. V. Krasnikov, "Hyperfast travel in general relativity" Phys. Rev. D 57 4760</ref>

Hyperwave<ref name="Pieri 2024">Lorenzo Pieri, "Hyperwave: hyper-Fast Communication within General Relativity", arXiv:2311.12069 [gr-qc] https://doi.org/10.48550/arXiv.2311.12069</ref> makes use of this warp railroad idea.

== Quantum effects ==

(quantum stuff here<ref name="Hiscock1997">W. A. Hiscock, "Quantum effects in the Alcubierre warp drive spacetime", Classical and Quantum Gravity 14 L183 https://doi.org/10.1088/0264-9381/14/11/002 https://arxiv.org/abs/gr-qc/9707024</ref><ref name="Finazzi et al 2009">S. Finazzi, S. Liberati, C. Barceló, "Semiclassical instability of dynamical warp drives", Physical Review D 79, 124017 (2009)https://doi.org/10.1103/PhysRevD.79.124017 https://arxiv.org/abs/0904.0141</ref>)

From<ref name="Pieri 2024"></ref>
<blockquote>
It should be emphasised that Quantum Inequalities have not been proven for realistic interacting quantum fields, where the they are expected to be weaker if they exist [Cadamuro, 2019], and they often require Minkowskian conditions at infinity, a condition which is not appropriate for fields inside bounded regions, as in a Casimir devices.
</blockquote>

== Warp drives and black holes ==

What happens if you drive your warp drive into a black hole?

The pop-sci reason that nothing can escape a black hole is that at the event horizon the escape velocity is faster than light and nothing can go faster than light. A warp drive can go faster than light.

The more detailed reason from general relativity is that at the event horizon space and time are rotated sufficiently that "inward" becomes "forward in time." You can no more go farther away from the singularity in a black hole once you are inside the event horizon than you can go backward in time. Any super-luminal travel can go backward in time, and warp drives can engage in super-luminal travel.

The Hawking area theorem that says that black holes can only grow, never shrink relies on the condition that the energy is everywhere positive. Warp drives have negative energy density.

So in principle, there's nothing that prevents a warp drive from taking a dip into a black hole and coming back out again.

Garattini and Zatrimaylov looked into the problem of a warp drive near (and in) a black hole<ref name="Garattini and Zatrimaylov">R. Garattini and K. Zatrimaylov, "Black holes, warp drives, and energy conditions", Physics Letters B 856 138910 (2024) https://doi.org/10.1016/j.physletb.2024.138910</ref>. The found that the black hole reduces the negative energy requirement for the warp bubble to move inward toward the black hole singularity, but increases the negative energy needed to move away from the center of the black hole. In addition, a warp drive parked at the event horizon would allow light from inside the horizon to pass through the bubble and escape back outside.

== Credit ==
Author: Luke Campbell

== References ==

[[Category:Physics]][[Category:Infrastructure]][[Category:Physics & Engineering‏‎]][[Category:Physics & Math & Engineering]][[Category:Transportation & Infrastructure‏‎]][[Category:Metric Engineering]]

Warp Drives

2026-07-26T23:25:56Z

Lwcamp: /* So what is going on? */

{{PageConstructionNotice}}

Science fiction often features spacecraft that can seemingly move across space and get between the place of departure and the destination much faster than light could have done. This appears to contradict the theory of relativity, which predicts unequivocally that nothing can move through space faster than light. Because relativity has been incredibly successful at describing nature, with its many other predictions regularly being confirmed to extraordinary accuracy and within the bounds of uncertainty of all the experiments that tested them, it gives confidence that relativity is a correct description of reality. Which seems to rather throw a wet towel on our hopes for rapid travel between stars.

However, while relativity does not allow things to move through space faster than light, it places no such restrictions on how fast space-time itself can expand, contract, or move around. This leads to the idea of a warp drive – the spacecraft remains stationary within a region of highly curved space-time, and that region moves at super-luminal speeds rather than the spacecraft.

== The Alcubierre warp drive ==

The first warp drive geometry that satisfied the Einstein field equations of relativity was proposed by Miguel Alcubierre<ref>M. Alcubierre, "The warp drive: hyper-fast travel within general relativity." Classical and Quantum Gravity. 11 (5): L73–L77 (1994). [https://arxiv.org/abs/gr-qc/0009013 arXiv:gr-qc/0009013]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1994CQGra..11L..73A 1994CQGra..11L..73A]. doi:[https://doi.org/10.1088%2F0264-9381%2F11%2F5%2F001 10.1088/0264-9381/11/5/001]. S2CID [https://api.semanticscholar.org/CorpusID:4797900 4797900].</ref>. In this geometry, a sphere of space-time moves at an arbitrary speed (potentially but not necessarily a speed much faster than light). Objects within the sphere are moved along with the sphere; an object at rest within the sphere would be moved along with the sphere indefinitely. Space is expanding at the rear boundary and contracting at the front boundary in order to keep the sphere moving. In order to satisfy the Einstein field equations, the boundary of the sphere must have a negative energy density. The challenges of space-time geometries with negative energy densities are described in our page on [[Wormholes#Exotic_energy_conditions|wormholes]], for our purposes it is enough to note that negative energy densities can pose problems if not handled carefully, there may be limits on how much negative energy you can have without nearby positive energy density, and it may not be possible to get enough negative energy to support a warp drive; although none of this is ruled out by physics – yet!

<table align=center>
<tr><td width=650>[[File:Energy_magnitude_of_Alcubierre_drive.png]]
<td width=650>[[File:Expansion_of_Alcubierre_drive.png]]
<tr><td>The magnitude of the energy of the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. (The energy density is negative everywhere in the shell, here only the magnitude is shown to aid visual comprehension.)
<td>The expansion of space in the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. Negative expansion means that space is contracting in that direction, positive expansion that it is expanding.
</table>

NOTE: What the? The energy distribution is symmetric! How does the drive know which way to go? There must be significant contributions of other components of the stress-energy tensor, and those have got to be asymmetric along the forward/backward axis. Check this when I get time.

Almost as soon as Alcubierre proposed his warp drive, others began picking it apart.
<ol>
<li>Pfenning and Ford<ref name="PfenningFord_2001">M. J. Pfenning and L. H. Ford, "The unphysical nature of 'Warp Drive'", Classical and Quantum Gravity. 14 (7): 1743–1751 (1997). arXiv:[https://arxiv.org/abs/gr-qc/9702026 gr-qc/9702026]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1997CQGra..14.1743P 1997CQGra..14.1743P]. doi:[https://doi.org/10.1088%2F0264-9381%2F14%2F7%2F011 10.1088/0264-9381/14/7/011]. S2CID [https://api.semanticscholar.org/CorpusID:15279207 15279207].</ref> found that if the negative energy density wall around the bubble obeys [[Wormholes#Quantum_energy_inequalities|quantum energy inequalities]], then for warp speeds of around light speed the shell thickness would be on the order of a hundred Planck lengths; a distance far smaller than any other known physical phenomenon or object.
<li>In the same work, Pfenning and Ford also found that for a bubble a hundred meters in radius, a warp speed of around the speed of light, and the required thickness of about 100 Planck lengths the energy in the bubble shell would be E ≈ -1063 kg c2. The magnitude of this latter value is ten orders of magnitude larger than the energy of the entire visible universe. They do note, however, that if quantum energy inequalities can be ignored then a warp drive with a hundred meters radius but a shell thickness of one meter would "only" have an energy magnitude of about a quarter solar mass.
<li>As with any method of faster than light travel, the warp drive could be used to make a time machine.
<li>Perhaps most seriously, if the warp drive is going faster than light speed, the negative energy regions on the outside of the shell won't be in the warp parts of the bubble. They will have to be moving through space at faster than light speed if the bubble is to maintain its integrity, which is the very problem that the warp drive was designed to avoid. The outside of the warp bubble shell that maintains the warp bubble would fall away and the warp bubble would quickly erode to sub-luminal speeds.<ref name="Van Den Broeck 2000">Chris Van Den Broeck, "On the (im)possibility of warp bubbles", arXiv:gr-qc/9906050 https://doi.org/10.48550/arXiv.gr-qc/9906050</ref>
<li>A more general problem that includes the previous one is that the geometry of the warp drive determines the distribution of energy, momentum, and stress (pressure, shear, and tension) by Einstein's field equation, which states that the stress energy tensor is directly proportional to a reduced order curvature tensor of space-time. However, the stress-energy tensor must also obey the continuity equation – basically, for the energy to change in a given volume of space that energy must come in or go out of the boundary of that volume and similarly for the momentum. So a warp drive can have the warp condition for one moment in time. But after that, the stress-energy that maintains the warp will start to move according to the forces on it and according to the natural paths that it takes through the warped space-time. There is no guarantee that the configuration of the stress-energy at later times will be sufficient to maintain the warp.
<blockquote>
From the perspective of simulating the warp drive dynamically, the key challenge is stability. One can straightforwardly generate the initial metric that describes the Alcubierre metric and infer the instantaneous matter configuration that must support it using the Einstein Equation, but a time-domain evolution of the coupled matter-gravity system necessitates that we specify the equation of state of the matter, which in turn determines how it evolves. In a perfect fluid, this involves specifying the relation between the fluid pressure and density, but in a more general case, the full relation between the
stress-energy components must be given. There is (to our knowledge) no known equation of state that would maintain the warp drive metric in a stable configuration over time - therefore, whilst one can require that initially, the warp bubble is constant, it will quickly evolve away from that state and, in most cases, the warp fluid and spacetime deformations will disperse or collapse into a central point.
 
– Clough, Dietrich, and Khan<ref>Katy Clough, Tim Dietrich, and Sebastian Khan, "What no one has seen before: gravitational waveforms from warp drive collapse", arXiv:2406.02466 [gr-qc] https://doi.org/10.48550/arXiv.2406.02466</ref>
</blockquote>
<li>In order to get the matter (or more generally, stress-energy) that forms the warp shell up to speeds that could allow the shell to exist, you need to be able to, well, accelerate that matter to warp speeds. Which gives a chicken-and-egg problem - in order to achieve superluminal travel you need to be able to get matter or energy moving superluminally in the first place<ref name="Bobrick and Martire">Alexey Bobrick and Gianni Martire, "Introducing Physical Warp Drives", arXiv:2102.06824 [gr-qc] https://doi.org/10.48550/arXiv.2102.06824</ref><ref>D. H. Coule, "No warp drive", Class. Quantum Grav. 15 (1998) 2523–2527. https://doi.org/10.1088/0264-9381/15/8/026</ref>, which rather makes warp drives unnecessary from the get-go.
<li>A warp bubble moving faster than light will not be able to get any signals in front of it as it will overtake its own signal. In fact, anyone in the interior of the bubble would not be able to send signals to the front surface of the bubble. In science-speak, a horizon forms at the front of the bubble<ref name="Natario">José Natário, "Warp drive with zero expansion", Classical and Quantum Gravity. 19 (6): 1157–1166 (2002). arXiv:[https://arxiv.org/abs/gr-qc/0110086 gr-qc/0110086]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2002CQGra..19.1157N 2002CQGra..19.1157N]. doi:[https://doi.org/10.1088%2F0264-9381%2F19%2F6%2F308 10.1088/0264-9381/19/6/308]. S2CID [https://api.semanticscholar.org/CorpusID:15859984 15859984].</ref>. As a consequence, the bubble cannot be controlled, steered, nor stopped from within the bubble itself.
<li>A Horizon will naturally emit Hawking radiation. It is estimated that the radiation will raise the temperature of the interior of the bubble to approximately 1032 K<ref name="Finazzi 2009">Stefano Finazzi, Stefano Liberati, and Carlos Barceló, "Semiclassical instability of dynamical warp drives", Phys. Rev. D 79, 124017 (12 June, 2009) https://doi.org/10.1103/PhysRevD.79.124017 , arXiv:0904.0141 [gr-qc] https://doi.org/10.48550/arXiv.0904.0141</ref> (at this level of accuracy, the result is the same if you replace temperatures in K by the same number in °C or °F). This temperature, however, may be lower if quantum inequalities do not have to be fulfilled.
<li>Semi-classical instabilities at the horizon are expected to lead to a divergence of the stress-energy tensor at the boundary in a short period of time, thus leading to the destruction of the warp bubble<ref name="Finazzi 2009"></ref>.
</ol>
Helpfully, Van Den Broeck<ref name="VanDenBroeck_2000">C. Van Den Broeck, "Alcubierre’s Warp Drive: Problems and Prospects" AIP Conference Proceedings. 504: 1105–1110 (2000). Bibcode:[https://ui.adsabs.harvard.edu/abs/2000AIPC..504.1105V 2000AIPC..504.1105V]. doi:[https://doi.org/10.1063%2F1.1290913 10.1063/1.1290913].</ref> suggested several solutions – or at least mitigations – for some of these problems
<ol>
<li>Quantum inequalities had not been shown to be true in general for highly curved space-times.
<li>A proposed warp drive geometry<ref name="VanDenBroeck_1999">C. Van Den Broeck, "A 'warp drive' with more reasonable total energy requirements". Classical and Quantum Gravity. 16 (12): 3973–3979 (1999). arXiv:[https://arxiv.org/abs/gr-qc/9905084 gr-qc/9905084]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1999CQGra..16.3973V 1999CQGra..16.3973V]. doi:[https://doi.org/10.1088%2F0264-9381%2F16%2F12%2F314 10.1088/0264-9381/16/12/314]. S2CID [https://api.semanticscholar.org/CorpusID:15466313 15466313]. </ref> (see the van Den Broeck drive, below) would be able to minimize the magnitude of the energy required to about that of the mass of our sun. While still large, it is not unphysically large.
<li>Time travel is always going to be a worry with faster than light travel. There's no neat solution to this one.
<li>It may be possible for the negative energy outside of the superluminal region to compress into the superluminal region to form a shock in space-time. The front surface of the bubble would then be a singularity. It is not clear if such a sharp jump in physical properties is possible, but neither is sure that it is impossible, either. However, while this might help with the negative energy in front of the bubble getting swept up because it cannot move faster than light, it does little to help with the material at the back of the bubble on the outside getting left behind because it cannot keep up.
</ol>
Other mitigating research has also shown
<ol start=9>
<li>Semi-classical divergence only occurs in warp drives with only one spatial dimensions. Those with two or more spatial dimensions (as in our universe) only contain points where divergences occur, and these divergences can be managed. In fact, their effects can be reduced by making the warp shape more elongated<ref>Carlos Barceló, Valentin Boyanov, Luis J. Garay, Eduardo Martín-Martínez,
and Jose M. Sánchez Velázquez, "Warp drive aerodynamics", JHEP08 (2022) 288 https://doi.org/10.1007/JHEP08(2022)288 , arXiv:2207.06458 [gr-qc] https://doi.org/10.48550/arXiv.2207.06458</ref>.
</ol>

(And what's with the ring in all the artwork, anyway?)

=== So what is going on? ===

What is it that makes a warp drive a warp drive, and how does it work?

A warp drive is a concept that comes out of the science of general relativity. General relativity collects space and time together into a single "structure" called spacetime. This structure is curved, and its curvature is determined by the material that is in it. Absent forces acting between matter, particles of matter move through spacetime on the shortest possible paths in this curved environment.
<blockquote>
"Spacetime tells matter how to move; matter tells spacetime how to curve." - John Archibald Wheeler
</blockquote>
All of this is encompassed in the Einstein field equation:
<div align=center>
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 8πG
<td rowspan="2" nowrap="nowrap" align="right"> T = R -  
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right"> R g
<tr>
<td style="border-top:solid 1px black;"> c4
<td style="border-top:solid 1px black;"> 2
</table>
</div>
Unfortunately, by convention, both the symbols G and R get used for a lot of different things. Here, symbols in bold are second rank tensors – constructs of arrays of 4×4 numbers where 4 is necessary because that is the number of dimensions of spacetime. Conventionally, these numbers are indexed by a value going from 0 to 3, with 0 being the part corresponding to the coordinate of time and 1 to 3 being the Cartesian coordinates that you might know as x, y, and z.

T is the stress-energy tensor: the (0,0) component represents the energy density at a given point; the (0,i) and (i, 0) components (with i going from 1 to 3) represent the momentum density; and the (i,j) components represent the stresses present, such as pressure or tension or shear.

The other term, R - R g, involves a partial description of the curvature. A full description of the curvature requires a construct of 4×4×4×4 components; too much for the stress energy tensor to fully constrain. Which is okay, because this allows curvature to exist away from where matter is, allowing gravity to have long range effects across empty space as well as for phenomena like gravitational waves. But it is enough that a given distribution of matter T gives enough constraints (along with specifying how the geometry behaves far away) that you can generally figure out what the actual gravity is.

This is the usual story that we are told. But there is an alternate way of looking at the same geometry such that the math works out the same, but it is described conceptually in a very different way. This divides spacetime up into "layers", called foliations, that represent slices of space and going from one foliation to the next is increasing in time. This is useful for computation because you can assign coordinates to your system. There are two additional constructs – the lapse function and the shift vector field. Starting at an event on one foliation, the lapse function at that position tells you that after a certain "proper time" (time perceived by an observer at that event) the time coordinate of the observer will be the proper time times the lapse function at that event. And the new spatial coordinates of the observer will be the proper time times the shift vector at that event. (Strictly speaking, this is only exactly true in the limit of the proper time approaching zero or when the lapse function and shift vectors are constant. But hopefully this gives a good enough insight to what these entities are so that we can go on to describe warp drives.)

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Lapse_shift.png]]
<tr><td>A folliation of space-time into slices of constant time coordinates (grey). At an event (black circle), the lapse function (yellow) shows how far into the future the point at that event will be projected in a unit amount of proper time experienced by an observer initially at the event. The shift vector (red) shows how that point changes coordinates after the unit amount of proper time. The cyan vector shows the new coordinates the point is projected onto. The lower black surface indicates a foliation of spacetime at the same time coordinate; after unit proper time, the events on the lower foliation get projected to the curved upper black foliation according to the shift vectors and lapse functions at each of their respective points.
</table>
</table>

Warp drives are generally restricted to spacetime geometries with a constant lapse function - all observers go forward by the same time coordinate after experiencing the same amount of proper time.
Far from the warp bubble, the shift vector vanishes – each spatial coordinate is mapped to the same spatial coordinate after any amount of proper time so long as the warp drive is not near. But the inside of the warp bubble is an extended region where the shift vector field is non-zero and constant. In this bubble, all spatial coordinates are mapped to a coordinate that shifts linearly in time with the proper time of the observer. Astute readers will see that this linear change in position with time is very similar to a velocity.

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Warp_shift.png]]
<tr><td>The shift vector of a one-dimensional slice through warp geometry, in red. At the edges there is no shift. At the center (inside the warp bubble) there is a region of uniform shift so that at a later time the inside of the warp bubble is rigidly translated to different spatial coordinates (in blue). The two foliations remain the same distance apart, because warp drives maintain a unit lapse function.
</table>
</table>

Solve backwards - assume a structure of space-time, figure out what distribution of matter is required for this from the EFE.

What sort of spacetime geometry produces a warp?

Unit lapse function <-> no time dilation

Region of near constant shift vector - that region is "moving" spacetime, the warp bubble.

Shift vector falls off to zero away from warp bubble -> normal flat spacetime.

=== Alcubierre warp interactions with light and matter ===

Space is not entirely empty. It is filled with a diffuse plasma in the form of the interstellar medium, as well as cosmic radiation, light from stars, and cosmic microwave background radiation. A warp drive propagating through space will encounter this stuff. What happens when this matter and radiation have a warp bubble pass across them?

The first analysis of matter encountering a warp bubble was performed by Pfenning and Ford<ref name="PfenningFord_2001"></ref>. They looked at the warp bubble interaction with a massive object at rest with the frame of reference of the warp drive (keep in mind that the rest frame is the same inside and outside the warp bubble; in this rest frame objects inside the bubble have no momentum even though they are, in some sense, changing location rapidly with time). When the warp bubble passes, the object experiences an acceleration in the direction of the warp bubble motion. When the warp shell passes and the object is inside the bubble, it will be moving with approximately the speed of the bubble. Pfenning and Ford analyzed this problem with a continuous distribution of shell energy that, strictly speaking, never falls to zero except at the bubble center and at spatial infinity, so unless the object passes through the center in Pfenning and Ford's description it will never quite get up to the bubble's speed. In this case, the object will move almost as fast as the object but will pass through the bubble in a finite time, after which it will again be at rest with respect to the reference frame but displaced along the direction of the warp bubble motion by some distance. With this description of the warp bubble, a spacecraft of finite size will always be moving a little slower than the warp bubble and would have to use rockets to keep up with it. in addition, the spacecraft would experience tidal forces that would cause stress on the spacecraft's structure.

Pfenning and Ford also examined cases where the shell is of a finite (possibly infinitesimal) width and falls to zero both inside and outside the bubble. In this case any matter encountering the bubble would be collected at the bow of the bubble and thereafter move along with it.

McMonigal et al. analyzed the situation for both massive particles and light moving along the axis of travel of the bubble<ref name="McMonigalEtAl_2012">B. McMonigal, G. F. Lewis, and P. O'Byrne, "Alcubierre warp drive: On the matter of matter". Physical Review D. 85 (6) 064024 (20 March 2012). arXiv:[https://arxiv.org/abs/1202.5708 1202.5708]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2012PhRvD..85f4024M 2012PhRvD..85f4024M]. doi:[https://doi.org/10.1103%2FPhysRevD.85.064024 10.1103/PhysRevD.85.064024]. S2CID [https://api.semanticscholar.org/CorpusID:3993148 3993148].</ref>. They found that light moving opposite the warp direction passed through the bubble without incident, being only somewhat delayed by passing through the bubble. A warp bubble that is warping at sub-luminal speeds can have light catch up from behind it. This light is able to pass through the bubble, and is somewhat advanced in its path by the speed of the bubble. For super-luminal warp bubbles, however, the situation is different. The bubble will catch up to light moving it its own direction that is originally in front of it. This light cannot escape forward, the bubble being too fast. Nor can it escape backward, as the light is propagating forward and the interior of the bubble is at rest. Thus, the light gets caught at the bow of the warp bubble, unable to escape for so long as the warp bubble is active. This light is strongly blue shifted to extremely energetic x-rays and gamma rays. The space behind the bubble is swept clear of forward-moving light.

The situation for matter over-run by a super-luminal warp bubble is similar. Matter moving backward passes through the bubble, being only swept a ways along the bubble's path as sign of its passing but otherwise continuing on their way. Matter at rest behaves the same way as Pfenning and Ford discovered. Meanwhile, matter moving in the same direction of the warp is overtaken and collects at the bow of the bubble, unable to leave for so long as the bubble is warping. This matter "surfing" on the bubble bow is highly accelerated to relativistic speeds, experiencing extreme time dilation.

For a sub-luminal bubble, there is a speed faster than the warp speed where particles coming up from behind overtake the bubble and pass through, out the front. Particles below this speed but still faster than the warp speed still overtake the bubble, but then bounce backwards out of the bubble from behind. They are still moving forward, but are now moving more slowly than the warp speed. Particles moving forward but slower than the bubble will be overtaken and then bounced out the front with a speed higher than the warp speed. Those particles moving backward will pass through the bubble and exit with their initial speed.

The spacecraft inside the bubble will observe that light moving backward compared to the warp direction is blue-shifted, and matter moving backward passes through with increased energy. Meanwhile, a sub-luminal bubble will see light that was moving forward as red-shifted and matter catching up to the bubble from behind will pass by with reduced energy (superluminal bubbles, of course, do not have any matter or radiation catching up to them from behind).

When a superluminal warp bubble that has been collecting matter and radiation for the duration of its journey and blue-shifting it to much higher energies is turned off, all that matter and energy is released as a blast of radiation in the direction the bubble was warping.

McMonigal et al. conclude by noting that any spacecraft in the warp bubble would need shielding to protect against the blue-shifted radiation and matter of increased energy. In addition the destination would be "blasted into oblivion" by the release of matter and radiation that had been caught in the bubble during the trip.

Clark et al.<ref>Chad Clark, William A. Hiscock, and Shane L. Larson, "Null geodesics in the Alcubierre warp drive spacetime: the view from the bridge", arXiv:gr-qc/9907019 (2018) https://doi.org/10.48550/arXiv.gr-qc/9907019</ref> examine what it would look like from within an operating warp drive. They find that light moving at 90 degrees to the direction of motion will look like it is still coming from 90 degrees. However, light from the front hemisphere is squished progressively toward the forward point the faster you go. For a warp bubble moving faster than the speed of light, there is a cone behind the warp bubble where no light can reach it. However, you don't see a black spot behind you – light from the regions in the backward hemisphere from outside that cone that can reach you is distorted toward the direction behind you to fill in that spot. Light from the forward direction is blue-shifted and light from behind is red-shifted. At speeds of around 200 times light speed, the cosmic microwave background would be blue shifted to energies similar to that of the solar photosphere, heating anything in the warp bubble as if they were close to the sun.

== Van Den Broeck warp drive ==

If an Alcubierre warp bubble a hundred meters across requires a magnitude of energy greater than the entire energy in the observable universe, one option to reduce the magnitude of energy used is to make the warp bubble smaller. Much smaller. Pfenning and Ford<ref name="PfenningFord_2001"></ref> suggested making the warp bubble smaller than an atom. Of course, that brings up the problem of how to stuff a spacecraft in there. Van Den Broeck proposed a solution<ref name="VanDenBroeck_1999"></ref>: make the warp bubble only about the size of an atomic nucleus but expand the space inside the bubble enormously so that a spacecraft could fit in. This is somewhat like a nuclear sized wormhole that leads to a pocket universe that holds the spacecraft. Unfortunately, the metric doesn't fit neatly into any [[Wormholes#Three_dimensions|embedding diagram]] that I can figure out, but the math works out so that spatial coordinates inside the bubble are expanded by a factor of 1017 and a spacecraft would have a few hundred meters of bubble interior to putz around in. This trick manages to reduce the magnitude of energy to only about the mass energy of a few stars similar to our own. Other than that, it is otherwise a normal Alcubierre drive.

Further work<ref>S. Krasnikov, "The quantum inequalities do not forbid spacetime shortcuts", Physical Review D. 67 (10) 104013. https://doi.org/10.1103%2FPhysRevD.67.104013, arXiv:gr-qc/0207057 (2003) https://doi.org/10.48550/arXiv.gr-qc/0207057 </ref> has managed to use Van Den Broeck's trick to reduce the magnitude of the negative energy to only about a milligram.

== Natário warp drive ==

The Alcubierre drive is not the only way to construct a warp drive. José Natário<ref name="Natario"></ref> showed that it was just one example of an entire class of warp drives. These are space-time geometries with a smooth and localized shift vector field a shift vector is a vector field that describes how one "slice" of space out of space-time connects to the next) and a unit lapse function (the lapse function describes how the time you experience changes when going from a "slice" of space-time to the next).

Natário then went on to find in that class a set of warp drives that do not have any expansion or contraction of space at all. Instead, space encountering the front boundary of the warp bubble instead "slides" around the outside of the bubble until it gets to the corresponding place on the back and is left behind there. Another way to think of it is that space entering the bubble shell is compressed in the radial direction of the shell but is simultaneously expanded in the tangential direction so that there is no net change in volume. When it gets to the back, the opposite occurs. In the language of mathematics, the shift vector field is divergenceless.

<table align=center>
<tr><td width=360>[[File:Natario_drive_flow_of_space.png]]
<td width=445>[[File:Natario_warp_drive_horizons.png]]
<tr><td style="vertical-align: top;">A representation of the motion of space in a zero-expansion Natário drive with a vanishingly thin warp shell around the bubble.
<td>The horizons of the generalized Natário class of warp drives (including the Alcubierre warp drive and all fell-Heisenberg drives) at superluminal speed in the direction of the green arrow. No event inside the bubble can affect events outside the bubble in front of the blue line. No events outside the bubble behind the red line can affect events inside the bubble. The lines have an angle α with respect to the direction of motion with sin(α) = 1/vs for vs the speed of the bubble relative to light speed. This is analogous to the Mach cone of supersonic objects.
</table>

NOTE: There are two different things that can be meant by the term "Natário warp drive". The generalized Natário warp drive is the general theory of all warp drives that includes within it the Alcubierre, Fell-Heisenberg, and Lentz warp drives, among infinitely many others. The zero-expansion Natário warp drive is the specific implementation of the generalized Natário warp drive that has a divergenceless shift vector field.

The generalized class of warp drives developed by Natário (including both the Alcubierre and this zero-expansion warp drive) are shown to always have regions in the warp shell where the energy density is negative to at least some observers. You can't get away from it – to warp, you need negative energy density.

The Natário class of warp drives blueshift light coming in from the front and redshift light catching up from the back. For the case of a super-luminal warp bubble, there will of course be a horizon at the back of the bubble and light will not be able to get through from behind. For a bubble speed vs relative to the speed of light, light coming from straight ahead will be blueshifted up in frequency and photon energy by a factor of 1 + vs. For sub-luminal travel, light coming from behind will be redshifted by a factor 1 - vs. In general for n as the unit direction along which the light is propagating, the blueshift factor will be 1 + vs ⋅ n. If the warp shell is infinitesimally thin, light from outside that reaches the center of the bubble will not be distorted in direction although it will be frequency shifted, but observers still inside the bubble but displaced from the center will see the field of view distorted as well as frequency shifted.

Most of the difficulties that apply to the Alcubierre warp drive also apply to the generalized Natário warp drive (and thus to the zero-expansion Natário drive and Fell-Heisenberg drive as well). In particular, quantum energy inequalities, high magnitudes of the energy required, formation of time machines, instabilities related to the stress-energy making up the warp shell being unable to keep up with the bubble, the chicken-and-egg problem of needing FTL to achieve FTL, the formation of horizons making warp bubbles uncontrollable, and Hawking radiation at the horizons, are all still concerns for generic warp drives.

== Fell-Heisenberg warp drives ==

S. D. B. Fell and L. Heisenberg investigated what sort of warp structure is necessary to produce a warp drive with positive energy density everywhere<ref name="Fell-Heisenberg">S. D. B. Fell and L> Heisenberg, "Positive energy warp drive from hidden geometric structures", Classical and Quantum Gravity 38 155020 (2021) https://doi.org/10.1088/1361-6382/ac0e47 https://arxiv.org/abs/2104.06488</ref>. They found that warp drives of the generic Natário form with irrotational shift vectors had a simple expression for the energy density and vanishing momentum (irrotational means that the shift vector can be described as the gradient of a scalar potential field or, equivalently, that the curl of the shift vector field vanishes everywhere). Fell and Heisenberg proposed several potential fields that either had positive energy density everywhere except at discontinuities, or that were continuous but had regions that violated the weak energy condition (i. e. negative energy densities in some frames of reference). One specific example they provided of a warp field had an energy that is about 10,000 times less than the mass-energy of our sun. Or only about half the mass-energy of Jupiter. This is a significant improvement over many other proposed drives.

Despite the introduction discussing warp drive configurations that satisfy the various energy conditions, the configurations described in the paper are shown to locally violate the weak and strong energy conditions. Nonetheless, the energy density is still mostly positive. Santiago, Schuster, and Visser's<ref name="Santiago Schuster Visser 2022">J. Santiago, S. Schuster, and M. Visser, "Generic warp drives violate the null energy condition", Physical Review D 105, 064038 (2022) https://doi.org/10.1103/PhysRevD.105.064038</ref> work shows that the various energy conditions must still be violated by this warp drive; to not violate these, energy density must be positive in all reference frames not just those of the co-moving observer.

Any vector field can be described by its divergence, its curl, and a constant. The constant part of a warp drive's shift vector can be discarded because then the warp drive would not be localized. It is interesting, therefore, that the Natário zero-expansion drive has zero divergence and can be described only by its curl, while the Fell-Heisenberg drive has zero curl and can be described only by its divergence. These two classes of drives thus rely on opposite choices of setting one of the two different parts describing a vector field to zero. As might be expected, the Fell-Heisenberg drive is not zero expansion – space-time will change in volume as it passes through the warp bubble (although after going all the way through it will return to its original volume). However, objects in space-time will retain their original orientation as they pass through the warp bubble.

Previous examples of warp drives, such as the Alcubierre and zero-expansion Natário drives, tend toward a flat space-time as you go far enough away faster than would occur if there were any mass present in the warp drive. That is, the ADM mass of such drives is zero. However, one of the Fell-Heisenberg examples described a spacetime where the geometry approaches that of what you would get for a massive object at the warp location, giving one of the first examples of a warp drive with mass.

In the Fell-Heisenberg drive the momentum vanishes everywhere. Yet the energy occupies regions where the shift vector is varying rapidly. The lack of momentum means that the energy will not move to keep up with the differential expansion and movement of the spacetime. As a consequence, at later times the energy will have a different distribution than what is necessary to maintain the given warp configuration. The exact time evolution is not solved but this likely leads to collapse of warp bubble. Note that this is not a problem unique to the Fell-Heisenberg drive. Assuming a geometry that describes a warp produces a specific distribution of the stress-energy tensor that satisfies Einstein's equation. But there is no guarantee that this stress energy tensor satisfies the continuity equation, which describes how the distribution of matter and momentum change with time. The issue of stability of any warp drive is not well understood, but most warp drive geometries will be unstable in the sense that their matter that generates the warp fields will not be able to keep up with the motion of the warp. The Fell-Heisenberg drive is unique only in that, with vanishing momentum, there is no configuration that allows stable propagation.

(n. b. The Heisenberg here is Lavinia Heisenberg, not the Werner Heisenberg of quantum physics and uncertainty principle fame.)

== Lentz warp drive ==

E. Lentz<ref name="Lentz">E. W. Lentz, "Breaking the warp barrier: hyper-fast solitons in Einstein–Maxwell-plasma theory", Classical and Quantum Gravity. 38 075015 (2021). arXiv:[https://arxiv.org/abs/2006.07125 2006.07125]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2021CQGra..38g5015L 2021CQGra..38g5015L]. doi:[https://doi.org/10.1088%2F1361-6382%2Fabe692 10.1088/1361-6382/abe692]. ISSN [https://search.worldcat.org/issn/0264-9381 0264-9381]. S2CID [https://api.semanticscholar.org/CorpusID:219635854 219635854].</ref> investigated a particular configuration of the Fell-Heisenberg class of warp drives, where the scalar field has a hyperbolic shape. The necessary conditions were found to be satisfied by a selection of rhomboid regions of the scalar field, resulting in an energy density that is everywhere positive and a shift vector with vanishing tidal force at the center for a comfortable ride.

However, Santiago, Schuster, and Visser<ref name="Santiago Schuster Visser 2022"></ref> dispute claims that the Lentz drive satisfies the energy conditions, noting that everywhere positive energy density in one frame of reference is insufficient to establish that the energy density is positive in all reference frames; knowledge of the Cauchy stress tensor is also needed. They show that any generic warp drive will violate the strong energy condition, null energy condition, and weak energy condition.

== Conservation laws ==

(Discussion of asymptotic flatness: what is it? Where does it apply?)

(The "big four" conservation laws and how they relate to asymptotic flatness.)

(ADM mass -> 0; conservation of energy/mass issues.)<ref>Sebastian Schuster, Jessica Santiago,and Matt Visser, "ADM mass in warp drive spacetimes" General Relativity and Gravitation (2023) 55:14 https://doi.org/10.1007/s10714-022-03061-9</ref>

(Discuss issues of conservation of angular momentum.)

(If angular momentum conservation is ignored, discuss how momentum & energy are affected by outside forces. In a gravitational field equivalent to inertial frame ... like being in an accelerated elevator; will acquire momentum buildup while staying "at rest".)

== Warp Railroads ==

Van Den Broeck<ref name="VanDenBroeck_2000"></ref> suggested a warp railroad,

You can set up devices ahead of time along the path of the bubble that produce the negative energy regions for the warp bubble at the appropriate time without needing the negative energy to ever move with the bubble at all. At least for bubbles on a predictable schedule and route.

helps with:

-stability (elaborate)

-conservation laws (elaborate)

Mention Krasnikov<ref name="Krasnikov 1998">S. V. Krasnikov, "Hyperfast travel in general relativity" Phys. Rev. D 57 4760</ref>

Hyperwave<ref name="Pieri 2024">Lorenzo Pieri, "Hyperwave: hyper-Fast Communication within General Relativity", arXiv:2311.12069 [gr-qc] https://doi.org/10.48550/arXiv.2311.12069</ref> makes use of this warp railroad idea.

== Quantum effects ==

(quantum stuff here<ref name="Hiscock1997">W. A. Hiscock, "Quantum effects in the Alcubierre warp drive spacetime", Classical and Quantum Gravity 14 L183 https://doi.org/10.1088/0264-9381/14/11/002 https://arxiv.org/abs/gr-qc/9707024</ref><ref name="Finazzi et al 2009">S. Finazzi, S. Liberati, C. Barceló, "Semiclassical instability of dynamical warp drives", Physical Review D 79, 124017 (2009)https://doi.org/10.1103/PhysRevD.79.124017 https://arxiv.org/abs/0904.0141</ref>)

From<ref name="Pieri 2024"></ref>
<blockquote>
It should be emphasised that Quantum Inequalities have not been proven for realistic interacting quantum fields, where the they are expected to be weaker if they exist [Cadamuro, 2019], and they often require Minkowskian conditions at infinity, a condition which is not appropriate for fields inside bounded regions, as in a Casimir devices.
</blockquote>

== Warp drives and black holes ==

What happens if you drive your warp drive into a black hole?

The pop-sci reason that nothing can escape a black hole is that at the event horizon the escape velocity is faster than light and nothing can go faster than light. A warp drive can go faster than light.

The more detailed reason from general relativity is that at the event horizon space and time are rotated sufficiently that "inward" becomes "forward in time." You can no more go farther away from the singularity in a black hole once you are inside the event horizon than you can go backward in time. Any super-luminal travel can go backward in time, and warp drives can engage in super-luminal travel.

The Hawking area theorem that says that black holes can only grow, never shrink relies on the condition that the energy is everywhere positive. Warp drives have negative energy density.

So in principle, there's nothing that prevents a warp drive from taking a dip into a black hole and coming back out again.

Garattini and Zatrimaylov looked into the problem of a warp drive near (and in) a black hole<ref name="Garattini and Zatrimaylov">R. Garattini and K. Zatrimaylov, "Black holes, warp drives, and energy conditions", Physics Letters B 856 138910 (2024) https://doi.org/10.1016/j.physletb.2024.138910</ref>. The found that the black hole reduces the negative energy requirement for the warp bubble to move inward toward the black hole singularity, but increases the negative energy needed to move away from the center of the black hole. In addition, a warp drive parked at the event horizon would allow light from inside the horizon to pass through the bubble and escape back outside.

== Credit ==
Author: Luke Campbell

== References ==

[[Category:Physics]][[Category:Infrastructure]][[Category:Physics & Engineering‏‎]][[Category:Physics & Math & Engineering]][[Category:Transportation & Infrastructure‏‎]][[Category:Metric Engineering]]

Warp Drives

2026-07-26T23:19:16Z

Lwcamp: /* So what is going on? */

{{PageConstructionNotice}}

Science fiction often features spacecraft that can seemingly move across space and get between the place of departure and the destination much faster than light could have done. This appears to contradict the theory of relativity, which predicts unequivocally that nothing can move through space faster than light. Because relativity has been incredibly successful at describing nature, with its many other predictions regularly being confirmed to extraordinary accuracy and within the bounds of uncertainty of all the experiments that tested them, it gives confidence that relativity is a correct description of reality. Which seems to rather throw a wet towel on our hopes for rapid travel between stars.

However, while relativity does not allow things to move through space faster than light, it places no such restrictions on how fast space-time itself can expand, contract, or move around. This leads to the idea of a warp drive – the spacecraft remains stationary within a region of highly curved space-time, and that region moves at super-luminal speeds rather than the spacecraft.

== The Alcubierre warp drive ==

The first warp drive geometry that satisfied the Einstein field equations of relativity was proposed by Miguel Alcubierre<ref>M. Alcubierre, "The warp drive: hyper-fast travel within general relativity." Classical and Quantum Gravity. 11 (5): L73–L77 (1994). [https://arxiv.org/abs/gr-qc/0009013 arXiv:gr-qc/0009013]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1994CQGra..11L..73A 1994CQGra..11L..73A]. doi:[https://doi.org/10.1088%2F0264-9381%2F11%2F5%2F001 10.1088/0264-9381/11/5/001]. S2CID [https://api.semanticscholar.org/CorpusID:4797900 4797900].</ref>. In this geometry, a sphere of space-time moves at an arbitrary speed (potentially but not necessarily a speed much faster than light). Objects within the sphere are moved along with the sphere; an object at rest within the sphere would be moved along with the sphere indefinitely. Space is expanding at the rear boundary and contracting at the front boundary in order to keep the sphere moving. In order to satisfy the Einstein field equations, the boundary of the sphere must have a negative energy density. The challenges of space-time geometries with negative energy densities are described in our page on [[Wormholes#Exotic_energy_conditions|wormholes]], for our purposes it is enough to note that negative energy densities can pose problems if not handled carefully, there may be limits on how much negative energy you can have without nearby positive energy density, and it may not be possible to get enough negative energy to support a warp drive; although none of this is ruled out by physics – yet!

<table align=center>
<tr><td width=650>[[File:Energy_magnitude_of_Alcubierre_drive.png]]
<td width=650>[[File:Expansion_of_Alcubierre_drive.png]]
<tr><td>The magnitude of the energy of the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. (The energy density is negative everywhere in the shell, here only the magnitude is shown to aid visual comprehension.)
<td>The expansion of space in the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. Negative expansion means that space is contracting in that direction, positive expansion that it is expanding.
</table>

NOTE: What the? The energy distribution is symmetric! How does the drive know which way to go? There must be significant contributions of other components of the stress-energy tensor, and those have got to be asymmetric along the forward/backward axis. Check this when I get time.

Almost as soon as Alcubierre proposed his warp drive, others began picking it apart.
<ol>
<li>Pfenning and Ford<ref name="PfenningFord_2001">M. J. Pfenning and L. H. Ford, "The unphysical nature of 'Warp Drive'", Classical and Quantum Gravity. 14 (7): 1743–1751 (1997). arXiv:[https://arxiv.org/abs/gr-qc/9702026 gr-qc/9702026]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1997CQGra..14.1743P 1997CQGra..14.1743P]. doi:[https://doi.org/10.1088%2F0264-9381%2F14%2F7%2F011 10.1088/0264-9381/14/7/011]. S2CID [https://api.semanticscholar.org/CorpusID:15279207 15279207].</ref> found that if the negative energy density wall around the bubble obeys [[Wormholes#Quantum_energy_inequalities|quantum energy inequalities]], then for warp speeds of around light speed the shell thickness would be on the order of a hundred Planck lengths; a distance far smaller than any other known physical phenomenon or object.
<li>In the same work, Pfenning and Ford also found that for a bubble a hundred meters in radius, a warp speed of around the speed of light, and the required thickness of about 100 Planck lengths the energy in the bubble shell would be E ≈ -1063 kg c2. The magnitude of this latter value is ten orders of magnitude larger than the energy of the entire visible universe. They do note, however, that if quantum energy inequalities can be ignored then a warp drive with a hundred meters radius but a shell thickness of one meter would "only" have an energy magnitude of about a quarter solar mass.
<li>As with any method of faster than light travel, the warp drive could be used to make a time machine.
<li>Perhaps most seriously, if the warp drive is going faster than light speed, the negative energy regions on the outside of the shell won't be in the warp parts of the bubble. They will have to be moving through space at faster than light speed if the bubble is to maintain its integrity, which is the very problem that the warp drive was designed to avoid. The outside of the warp bubble shell that maintains the warp bubble would fall away and the warp bubble would quickly erode to sub-luminal speeds.<ref name="Van Den Broeck 2000">Chris Van Den Broeck, "On the (im)possibility of warp bubbles", arXiv:gr-qc/9906050 https://doi.org/10.48550/arXiv.gr-qc/9906050</ref>
<li>A more general problem that includes the previous one is that the geometry of the warp drive determines the distribution of energy, momentum, and stress (pressure, shear, and tension) by Einstein's field equation, which states that the stress energy tensor is directly proportional to a reduced order curvature tensor of space-time. However, the stress-energy tensor must also obey the continuity equation – basically, for the energy to change in a given volume of space that energy must come in or go out of the boundary of that volume and similarly for the momentum. So a warp drive can have the warp condition for one moment in time. But after that, the stress-energy that maintains the warp will start to move according to the forces on it and according to the natural paths that it takes through the warped space-time. There is no guarantee that the configuration of the stress-energy at later times will be sufficient to maintain the warp.
<blockquote>
From the perspective of simulating the warp drive dynamically, the key challenge is stability. One can straightforwardly generate the initial metric that describes the Alcubierre metric and infer the instantaneous matter configuration that must support it using the Einstein Equation, but a time-domain evolution of the coupled matter-gravity system necessitates that we specify the equation of state of the matter, which in turn determines how it evolves. In a perfect fluid, this involves specifying the relation between the fluid pressure and density, but in a more general case, the full relation between the
stress-energy components must be given. There is (to our knowledge) no known equation of state that would maintain the warp drive metric in a stable configuration over time - therefore, whilst one can require that initially, the warp bubble is constant, it will quickly evolve away from that state and, in most cases, the warp fluid and spacetime deformations will disperse or collapse into a central point.
 
– Clough, Dietrich, and Khan<ref>Katy Clough, Tim Dietrich, and Sebastian Khan, "What no one has seen before: gravitational waveforms from warp drive collapse", arXiv:2406.02466 [gr-qc] https://doi.org/10.48550/arXiv.2406.02466</ref>
</blockquote>
<li>In order to get the matter (or more generally, stress-energy) that forms the warp shell up to speeds that could allow the shell to exist, you need to be able to, well, accelerate that matter to warp speeds. Which gives a chicken-and-egg problem - in order to achieve superluminal travel you need to be able to get matter or energy moving superluminally in the first place<ref name="Bobrick and Martire">Alexey Bobrick and Gianni Martire, "Introducing Physical Warp Drives", arXiv:2102.06824 [gr-qc] https://doi.org/10.48550/arXiv.2102.06824</ref><ref>D. H. Coule, "No warp drive", Class. Quantum Grav. 15 (1998) 2523–2527. https://doi.org/10.1088/0264-9381/15/8/026</ref>, which rather makes warp drives unnecessary from the get-go.
<li>A warp bubble moving faster than light will not be able to get any signals in front of it as it will overtake its own signal. In fact, anyone in the interior of the bubble would not be able to send signals to the front surface of the bubble. In science-speak, a horizon forms at the front of the bubble<ref name="Natario">José Natário, "Warp drive with zero expansion", Classical and Quantum Gravity. 19 (6): 1157–1166 (2002). arXiv:[https://arxiv.org/abs/gr-qc/0110086 gr-qc/0110086]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2002CQGra..19.1157N 2002CQGra..19.1157N]. doi:[https://doi.org/10.1088%2F0264-9381%2F19%2F6%2F308 10.1088/0264-9381/19/6/308]. S2CID [https://api.semanticscholar.org/CorpusID:15859984 15859984].</ref>. As a consequence, the bubble cannot be controlled, steered, nor stopped from within the bubble itself.
<li>A Horizon will naturally emit Hawking radiation. It is estimated that the radiation will raise the temperature of the interior of the bubble to approximately 1032 K<ref name="Finazzi 2009">Stefano Finazzi, Stefano Liberati, and Carlos Barceló, "Semiclassical instability of dynamical warp drives", Phys. Rev. D 79, 124017 (12 June, 2009) https://doi.org/10.1103/PhysRevD.79.124017 , arXiv:0904.0141 [gr-qc] https://doi.org/10.48550/arXiv.0904.0141</ref> (at this level of accuracy, the result is the same if you replace temperatures in K by the same number in °C or °F). This temperature, however, may be lower if quantum inequalities do not have to be fulfilled.
<li>Semi-classical instabilities at the horizon are expected to lead to a divergence of the stress-energy tensor at the boundary in a short period of time, thus leading to the destruction of the warp bubble<ref name="Finazzi 2009"></ref>.
</ol>
Helpfully, Van Den Broeck<ref name="VanDenBroeck_2000">C. Van Den Broeck, "Alcubierre’s Warp Drive: Problems and Prospects" AIP Conference Proceedings. 504: 1105–1110 (2000). Bibcode:[https://ui.adsabs.harvard.edu/abs/2000AIPC..504.1105V 2000AIPC..504.1105V]. doi:[https://doi.org/10.1063%2F1.1290913 10.1063/1.1290913].</ref> suggested several solutions – or at least mitigations – for some of these problems
<ol>
<li>Quantum inequalities had not been shown to be true in general for highly curved space-times.
<li>A proposed warp drive geometry<ref name="VanDenBroeck_1999">C. Van Den Broeck, "A 'warp drive' with more reasonable total energy requirements". Classical and Quantum Gravity. 16 (12): 3973–3979 (1999). arXiv:[https://arxiv.org/abs/gr-qc/9905084 gr-qc/9905084]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1999CQGra..16.3973V 1999CQGra..16.3973V]. doi:[https://doi.org/10.1088%2F0264-9381%2F16%2F12%2F314 10.1088/0264-9381/16/12/314]. S2CID [https://api.semanticscholar.org/CorpusID:15466313 15466313]. </ref> (see the van Den Broeck drive, below) would be able to minimize the magnitude of the energy required to about that of the mass of our sun. While still large, it is not unphysically large.
<li>Time travel is always going to be a worry with faster than light travel. There's no neat solution to this one.
<li>It may be possible for the negative energy outside of the superluminal region to compress into the superluminal region to form a shock in space-time. The front surface of the bubble would then be a singularity. It is not clear if such a sharp jump in physical properties is possible, but neither is sure that it is impossible, either. However, while this might help with the negative energy in front of the bubble getting swept up because it cannot move faster than light, it does little to help with the material at the back of the bubble on the outside getting left behind because it cannot keep up.
</ol>
Other mitigating research has also shown
<ol start=9>
<li>Semi-classical divergence only occurs in warp drives with only one spatial dimensions. Those with two or more spatial dimensions (as in our universe) only contain points where divergences occur, and these divergences can be managed. In fact, their effects can be reduced by making the warp shape more elongated<ref>Carlos Barceló, Valentin Boyanov, Luis J. Garay, Eduardo Martín-Martínez,
and Jose M. Sánchez Velázquez, "Warp drive aerodynamics", JHEP08 (2022) 288 https://doi.org/10.1007/JHEP08(2022)288 , arXiv:2207.06458 [gr-qc] https://doi.org/10.48550/arXiv.2207.06458</ref>.
</ol>

(And what's with the ring in all the artwork, anyway?)

=== So what is going on? ===

What is it that makes a warp drive a warp drive, and how does it work?

A warp drive is a concept that comes out of the science of general relativity. General relativity collects space and time together into a single "structure" called spacetime. This structure is curved, and its curvature is determined by the material that is in it. Absent forces acting between matter, particles of matter move through spacetime on the shortest possible paths in this curved environment.
<blockquote>
"Spacetime tells matter how to move; matter tells spacetime how to curve." - John Archibald Wheeler
</blockquote>
All of this is encompassed in the Einstein field equation:
<div align=center>
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 8πG
<td rowspan="2" nowrap="nowrap" align="right"> T = R -  
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right"> R g
<tr>
<td style="border-top:solid 1px black;"> c4
<td style="border-top:solid 1px black;"> 2
</table>
</div>
Unfortunately, by convention, both the symbols G and R get used for a lot of different things. Here, symbols in bold are second rank tensors – constructs of arrays of 4×4 numbers where 4 is necessary because that is the number of dimensions of spacetime. Conventionally, these numbers are indexed by a value going from 0 to 3, with 0 being the part corresponding to the coordinate of time and 1 to 3 being the Cartesian coordinates that you might know as x, y, and z.

T is the stress-energy tensor: the (0,0) component represents the energy density at a given point; the (0,i) and (i, 0) components (with i going from 1 to 3) represent the momentum density; and the (i,j) components represent the stresses present, such as pressure or tension or shear.

The other term, R - R g, involves a partial description of the curvature. A full description of the curvature requires a construct of 4×4×4×4 components; too much for the stress energy tensor to fully constrain. Which is okay, because this allows curvature to exist away from where matter is, allowing gravity to have long range effects across empty space as well as for phenomena like gravitational waves. But it is enough that a given distribution of matter T gives enough constraints (along with specifying how the geometry behaves far away) that you can generally figure out what the actual gravity is.

This is the usual story that we are told. But there is an alternate way of looking at the same geometry such that the math works out the same, but it is described conceptually in a very different way. This divides spacetime up into "layers", called foliations, that represent slices of space and going from one foliation to the next is increasing in time. This is useful for computation because you can assign coordinates to your system. There are two additional constructs – the lapse function and the shift vector field. Starting at an event on one foliation, the lapse function at that position tells you that after a certain "proper time" (time perceived by an observer at that event) the time coordinate of the observer will be the proper time times the lapse function at that event. And the new spatial coordinates of the observer will be the proper time times the shift vector at that event. (Strictly speaking, this is only exactly true in the limit of the proper time approaching zero or when the lapse function and shift vectors are constant. But hopefully this gives a good enough insight to what these entities are so that we can go on to describe warp drives.)

Describing spacetime - lapse function & shift vectors (foliations, etc.)
Diagram of what the shift vector does.

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Lapse_shift.png]]
<tr><td>A folliation of space-time into slices of constant time coordinates (grey). At an event (black circle), the lapse function (yellow) shows how far into the future the point at that event will be projected in a unit amount of proper time experienced by an observer initially at the event. The shift vector (red) shows how that point changes coordinates after the unit amount of proper time. The cyan vector shows the new coordinates the point is projected onto. The lower black surface indicates a foliation of spacetime at the same time coordinate; after unit proper time, the events on the lower foliation get projected to the curved upper black foliation according to the shift vectors and lapse functions at each of their respective points.
</table>
</table>

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Warp_shift.png]]
<tr><td>The shift vector of a warp geometry, in red. At the edges there is no shift. At the center (inside the warp bubble) there is a region of uniform shift so that at a later time the inside of the warp bubble is rigidly translated to different spatial coordinates (in blue). The two foliations remain the same distance apart, because warp drives maintain a unit lapse function.
</table>
</table>

Solve backwards - assume a structure of space-time, figure out what distribution of matter is required for this from the EFE.

What sort of spacetime geometry produces a warp?

Unit lapse function <-> no time dilation

Region of near constant shift vector - that region is "moving" spacetime, the warp bubble.

Shift vector falls off to zero away from warp bubble -> normal flat spacetime.

=== Alcubierre warp interactions with light and matter ===

Space is not entirely empty. It is filled with a diffuse plasma in the form of the interstellar medium, as well as cosmic radiation, light from stars, and cosmic microwave background radiation. A warp drive propagating through space will encounter this stuff. What happens when this matter and radiation have a warp bubble pass across them?

The first analysis of matter encountering a warp bubble was performed by Pfenning and Ford<ref name="PfenningFord_2001"></ref>. They looked at the warp bubble interaction with a massive object at rest with the frame of reference of the warp drive (keep in mind that the rest frame is the same inside and outside the warp bubble; in this rest frame objects inside the bubble have no momentum even though they are, in some sense, changing location rapidly with time). When the warp bubble passes, the object experiences an acceleration in the direction of the warp bubble motion. When the warp shell passes and the object is inside the bubble, it will be moving with approximately the speed of the bubble. Pfenning and Ford analyzed this problem with a continuous distribution of shell energy that, strictly speaking, never falls to zero except at the bubble center and at spatial infinity, so unless the object passes through the center in Pfenning and Ford's description it will never quite get up to the bubble's speed. In this case, the object will move almost as fast as the object but will pass through the bubble in a finite time, after which it will again be at rest with respect to the reference frame but displaced along the direction of the warp bubble motion by some distance. With this description of the warp bubble, a spacecraft of finite size will always be moving a little slower than the warp bubble and would have to use rockets to keep up with it. in addition, the spacecraft would experience tidal forces that would cause stress on the spacecraft's structure.

Pfenning and Ford also examined cases where the shell is of a finite (possibly infinitesimal) width and falls to zero both inside and outside the bubble. In this case any matter encountering the bubble would be collected at the bow of the bubble and thereafter move along with it.

McMonigal et al. analyzed the situation for both massive particles and light moving along the axis of travel of the bubble<ref name="McMonigalEtAl_2012">B. McMonigal, G. F. Lewis, and P. O'Byrne, "Alcubierre warp drive: On the matter of matter". Physical Review D. 85 (6) 064024 (20 March 2012). arXiv:[https://arxiv.org/abs/1202.5708 1202.5708]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2012PhRvD..85f4024M 2012PhRvD..85f4024M]. doi:[https://doi.org/10.1103%2FPhysRevD.85.064024 10.1103/PhysRevD.85.064024]. S2CID [https://api.semanticscholar.org/CorpusID:3993148 3993148].</ref>. They found that light moving opposite the warp direction passed through the bubble without incident, being only somewhat delayed by passing through the bubble. A warp bubble that is warping at sub-luminal speeds can have light catch up from behind it. This light is able to pass through the bubble, and is somewhat advanced in its path by the speed of the bubble. For super-luminal warp bubbles, however, the situation is different. The bubble will catch up to light moving it its own direction that is originally in front of it. This light cannot escape forward, the bubble being too fast. Nor can it escape backward, as the light is propagating forward and the interior of the bubble is at rest. Thus, the light gets caught at the bow of the warp bubble, unable to escape for so long as the warp bubble is active. This light is strongly blue shifted to extremely energetic x-rays and gamma rays. The space behind the bubble is swept clear of forward-moving light.

The situation for matter over-run by a super-luminal warp bubble is similar. Matter moving backward passes through the bubble, being only swept a ways along the bubble's path as sign of its passing but otherwise continuing on their way. Matter at rest behaves the same way as Pfenning and Ford discovered. Meanwhile, matter moving in the same direction of the warp is overtaken and collects at the bow of the bubble, unable to leave for so long as the bubble is warping. This matter "surfing" on the bubble bow is highly accelerated to relativistic speeds, experiencing extreme time dilation.

For a sub-luminal bubble, there is a speed faster than the warp speed where particles coming up from behind overtake the bubble and pass through, out the front. Particles below this speed but still faster than the warp speed still overtake the bubble, but then bounce backwards out of the bubble from behind. They are still moving forward, but are now moving more slowly than the warp speed. Particles moving forward but slower than the bubble will be overtaken and then bounced out the front with a speed higher than the warp speed. Those particles moving backward will pass through the bubble and exit with their initial speed.

The spacecraft inside the bubble will observe that light moving backward compared to the warp direction is blue-shifted, and matter moving backward passes through with increased energy. Meanwhile, a sub-luminal bubble will see light that was moving forward as red-shifted and matter catching up to the bubble from behind will pass by with reduced energy (superluminal bubbles, of course, do not have any matter or radiation catching up to them from behind).

When a superluminal warp bubble that has been collecting matter and radiation for the duration of its journey and blue-shifting it to much higher energies is turned off, all that matter and energy is released as a blast of radiation in the direction the bubble was warping.

McMonigal et al. conclude by noting that any spacecraft in the warp bubble would need shielding to protect against the blue-shifted radiation and matter of increased energy. In addition the destination would be "blasted into oblivion" by the release of matter and radiation that had been caught in the bubble during the trip.

Clark et al.<ref>Chad Clark, William A. Hiscock, and Shane L. Larson, "Null geodesics in the Alcubierre warp drive spacetime: the view from the bridge", arXiv:gr-qc/9907019 (2018) https://doi.org/10.48550/arXiv.gr-qc/9907019</ref> examine what it would look like from within an operating warp drive. They find that light moving at 90 degrees to the direction of motion will look like it is still coming from 90 degrees. However, light from the front hemisphere is squished progressively toward the forward point the faster you go. For a warp bubble moving faster than the speed of light, there is a cone behind the warp bubble where no light can reach it. However, you don't see a black spot behind you – light from the regions in the backward hemisphere from outside that cone that can reach you is distorted toward the direction behind you to fill in that spot. Light from the forward direction is blue-shifted and light from behind is red-shifted. At speeds of around 200 times light speed, the cosmic microwave background would be blue shifted to energies similar to that of the solar photosphere, heating anything in the warp bubble as if they were close to the sun.

== Van Den Broeck warp drive ==

If an Alcubierre warp bubble a hundred meters across requires a magnitude of energy greater than the entire energy in the observable universe, one option to reduce the magnitude of energy used is to make the warp bubble smaller. Much smaller. Pfenning and Ford<ref name="PfenningFord_2001"></ref> suggested making the warp bubble smaller than an atom. Of course, that brings up the problem of how to stuff a spacecraft in there. Van Den Broeck proposed a solution<ref name="VanDenBroeck_1999"></ref>: make the warp bubble only about the size of an atomic nucleus but expand the space inside the bubble enormously so that a spacecraft could fit in. This is somewhat like a nuclear sized wormhole that leads to a pocket universe that holds the spacecraft. Unfortunately, the metric doesn't fit neatly into any [[Wormholes#Three_dimensions|embedding diagram]] that I can figure out, but the math works out so that spatial coordinates inside the bubble are expanded by a factor of 1017 and a spacecraft would have a few hundred meters of bubble interior to putz around in. This trick manages to reduce the magnitude of energy to only about the mass energy of a few stars similar to our own. Other than that, it is otherwise a normal Alcubierre drive.

Further work<ref>S. Krasnikov, "The quantum inequalities do not forbid spacetime shortcuts", Physical Review D. 67 (10) 104013. https://doi.org/10.1103%2FPhysRevD.67.104013, arXiv:gr-qc/0207057 (2003) https://doi.org/10.48550/arXiv.gr-qc/0207057 </ref> has managed to use Van Den Broeck's trick to reduce the magnitude of the negative energy to only about a milligram.

== Natário warp drive ==

The Alcubierre drive is not the only way to construct a warp drive. José Natário<ref name="Natario"></ref> showed that it was just one example of an entire class of warp drives. These are space-time geometries with a smooth and localized shift vector field a shift vector is a vector field that describes how one "slice" of space out of space-time connects to the next) and a unit lapse function (the lapse function describes how the time you experience changes when going from a "slice" of space-time to the next).

Natário then went on to find in that class a set of warp drives that do not have any expansion or contraction of space at all. Instead, space encountering the front boundary of the warp bubble instead "slides" around the outside of the bubble until it gets to the corresponding place on the back and is left behind there. Another way to think of it is that space entering the bubble shell is compressed in the radial direction of the shell but is simultaneously expanded in the tangential direction so that there is no net change in volume. When it gets to the back, the opposite occurs. In the language of mathematics, the shift vector field is divergenceless.

<table align=center>
<tr><td width=360>[[File:Natario_drive_flow_of_space.png]]
<td width=445>[[File:Natario_warp_drive_horizons.png]]
<tr><td style="vertical-align: top;">A representation of the motion of space in a zero-expansion Natário drive with a vanishingly thin warp shell around the bubble.
<td>The horizons of the generalized Natário class of warp drives (including the Alcubierre warp drive and all fell-Heisenberg drives) at superluminal speed in the direction of the green arrow. No event inside the bubble can affect events outside the bubble in front of the blue line. No events outside the bubble behind the red line can affect events inside the bubble. The lines have an angle α with respect to the direction of motion with sin(α) = 1/vs for vs the speed of the bubble relative to light speed. This is analogous to the Mach cone of supersonic objects.
</table>

NOTE: There are two different things that can be meant by the term "Natário warp drive". The generalized Natário warp drive is the general theory of all warp drives that includes within it the Alcubierre, Fell-Heisenberg, and Lentz warp drives, among infinitely many others. The zero-expansion Natário warp drive is the specific implementation of the generalized Natário warp drive that has a divergenceless shift vector field.

The generalized class of warp drives developed by Natário (including both the Alcubierre and this zero-expansion warp drive) are shown to always have regions in the warp shell where the energy density is negative to at least some observers. You can't get away from it – to warp, you need negative energy density.

The Natário class of warp drives blueshift light coming in from the front and redshift light catching up from the back. For the case of a super-luminal warp bubble, there will of course be a horizon at the back of the bubble and light will not be able to get through from behind. For a bubble speed vs relative to the speed of light, light coming from straight ahead will be blueshifted up in frequency and photon energy by a factor of 1 + vs. For sub-luminal travel, light coming from behind will be redshifted by a factor 1 - vs. In general for n as the unit direction along which the light is propagating, the blueshift factor will be 1 + vs ⋅ n. If the warp shell is infinitesimally thin, light from outside that reaches the center of the bubble will not be distorted in direction although it will be frequency shifted, but observers still inside the bubble but displaced from the center will see the field of view distorted as well as frequency shifted.

Most of the difficulties that apply to the Alcubierre warp drive also apply to the generalized Natário warp drive (and thus to the zero-expansion Natário drive and Fell-Heisenberg drive as well). In particular, quantum energy inequalities, high magnitudes of the energy required, formation of time machines, instabilities related to the stress-energy making up the warp shell being unable to keep up with the bubble, the chicken-and-egg problem of needing FTL to achieve FTL, the formation of horizons making warp bubbles uncontrollable, and Hawking radiation at the horizons, are all still concerns for generic warp drives.

== Fell-Heisenberg warp drives ==

S. D. B. Fell and L. Heisenberg investigated what sort of warp structure is necessary to produce a warp drive with positive energy density everywhere<ref name="Fell-Heisenberg">S. D. B. Fell and L> Heisenberg, "Positive energy warp drive from hidden geometric structures", Classical and Quantum Gravity 38 155020 (2021) https://doi.org/10.1088/1361-6382/ac0e47 https://arxiv.org/abs/2104.06488</ref>. They found that warp drives of the generic Natário form with irrotational shift vectors had a simple expression for the energy density and vanishing momentum (irrotational means that the shift vector can be described as the gradient of a scalar potential field or, equivalently, that the curl of the shift vector field vanishes everywhere). Fell and Heisenberg proposed several potential fields that either had positive energy density everywhere except at discontinuities, or that were continuous but had regions that violated the weak energy condition (i. e. negative energy densities in some frames of reference). One specific example they provided of a warp field had an energy that is about 10,000 times less than the mass-energy of our sun. Or only about half the mass-energy of Jupiter. This is a significant improvement over many other proposed drives.

Despite the introduction discussing warp drive configurations that satisfy the various energy conditions, the configurations described in the paper are shown to locally violate the weak and strong energy conditions. Nonetheless, the energy density is still mostly positive. Santiago, Schuster, and Visser's<ref name="Santiago Schuster Visser 2022">J. Santiago, S. Schuster, and M. Visser, "Generic warp drives violate the null energy condition", Physical Review D 105, 064038 (2022) https://doi.org/10.1103/PhysRevD.105.064038</ref> work shows that the various energy conditions must still be violated by this warp drive; to not violate these, energy density must be positive in all reference frames not just those of the co-moving observer.

Any vector field can be described by its divergence, its curl, and a constant. The constant part of a warp drive's shift vector can be discarded because then the warp drive would not be localized. It is interesting, therefore, that the Natário zero-expansion drive has zero divergence and can be described only by its curl, while the Fell-Heisenberg drive has zero curl and can be described only by its divergence. These two classes of drives thus rely on opposite choices of setting one of the two different parts describing a vector field to zero. As might be expected, the Fell-Heisenberg drive is not zero expansion – space-time will change in volume as it passes through the warp bubble (although after going all the way through it will return to its original volume). However, objects in space-time will retain their original orientation as they pass through the warp bubble.

Previous examples of warp drives, such as the Alcubierre and zero-expansion Natário drives, tend toward a flat space-time as you go far enough away faster than would occur if there were any mass present in the warp drive. That is, the ADM mass of such drives is zero. However, one of the Fell-Heisenberg examples described a spacetime where the geometry approaches that of what you would get for a massive object at the warp location, giving one of the first examples of a warp drive with mass.

In the Fell-Heisenberg drive the momentum vanishes everywhere. Yet the energy occupies regions where the shift vector is varying rapidly. The lack of momentum means that the energy will not move to keep up with the differential expansion and movement of the spacetime. As a consequence, at later times the energy will have a different distribution than what is necessary to maintain the given warp configuration. The exact time evolution is not solved but this likely leads to collapse of warp bubble. Note that this is not a problem unique to the Fell-Heisenberg drive. Assuming a geometry that describes a warp produces a specific distribution of the stress-energy tensor that satisfies Einstein's equation. But there is no guarantee that this stress energy tensor satisfies the continuity equation, which describes how the distribution of matter and momentum change with time. The issue of stability of any warp drive is not well understood, but most warp drive geometries will be unstable in the sense that their matter that generates the warp fields will not be able to keep up with the motion of the warp. The Fell-Heisenberg drive is unique only in that, with vanishing momentum, there is no configuration that allows stable propagation.

(n. b. The Heisenberg here is Lavinia Heisenberg, not the Werner Heisenberg of quantum physics and uncertainty principle fame.)

== Lentz warp drive ==

E. Lentz<ref name="Lentz">E. W. Lentz, "Breaking the warp barrier: hyper-fast solitons in Einstein–Maxwell-plasma theory", Classical and Quantum Gravity. 38 075015 (2021). arXiv:[https://arxiv.org/abs/2006.07125 2006.07125]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2021CQGra..38g5015L 2021CQGra..38g5015L]. doi:[https://doi.org/10.1088%2F1361-6382%2Fabe692 10.1088/1361-6382/abe692]. ISSN [https://search.worldcat.org/issn/0264-9381 0264-9381]. S2CID [https://api.semanticscholar.org/CorpusID:219635854 219635854].</ref> investigated a particular configuration of the Fell-Heisenberg class of warp drives, where the scalar field has a hyperbolic shape. The necessary conditions were found to be satisfied by a selection of rhomboid regions of the scalar field, resulting in an energy density that is everywhere positive and a shift vector with vanishing tidal force at the center for a comfortable ride.

However, Santiago, Schuster, and Visser<ref name="Santiago Schuster Visser 2022"></ref> dispute claims that the Lentz drive satisfies the energy conditions, noting that everywhere positive energy density in one frame of reference is insufficient to establish that the energy density is positive in all reference frames; knowledge of the Cauchy stress tensor is also needed. They show that any generic warp drive will violate the strong energy condition, null energy condition, and weak energy condition.

== Conservation laws ==

(Discussion of asymptotic flatness: what is it? Where does it apply?)

(The "big four" conservation laws and how they relate to asymptotic flatness.)

(ADM mass -> 0; conservation of energy/mass issues.)<ref>Sebastian Schuster, Jessica Santiago,and Matt Visser, "ADM mass in warp drive spacetimes" General Relativity and Gravitation (2023) 55:14 https://doi.org/10.1007/s10714-022-03061-9</ref>

(Discuss issues of conservation of angular momentum.)

(If angular momentum conservation is ignored, discuss how momentum & energy are affected by outside forces. In a gravitational field equivalent to inertial frame ... like being in an accelerated elevator; will acquire momentum buildup while staying "at rest".)

== Warp Railroads ==

Van Den Broeck<ref name="VanDenBroeck_2000"></ref> suggested a warp railroad,

You can set up devices ahead of time along the path of the bubble that produce the negative energy regions for the warp bubble at the appropriate time without needing the negative energy to ever move with the bubble at all. At least for bubbles on a predictable schedule and route.

helps with:

-stability (elaborate)

-conservation laws (elaborate)

Mention Krasnikov<ref name="Krasnikov 1998">S. V. Krasnikov, "Hyperfast travel in general relativity" Phys. Rev. D 57 4760</ref>

Hyperwave<ref name="Pieri 2024">Lorenzo Pieri, "Hyperwave: hyper-Fast Communication within General Relativity", arXiv:2311.12069 [gr-qc] https://doi.org/10.48550/arXiv.2311.12069</ref> makes use of this warp railroad idea.

== Quantum effects ==

(quantum stuff here<ref name="Hiscock1997">W. A. Hiscock, "Quantum effects in the Alcubierre warp drive spacetime", Classical and Quantum Gravity 14 L183 https://doi.org/10.1088/0264-9381/14/11/002 https://arxiv.org/abs/gr-qc/9707024</ref><ref name="Finazzi et al 2009">S. Finazzi, S. Liberati, C. Barceló, "Semiclassical instability of dynamical warp drives", Physical Review D 79, 124017 (2009)https://doi.org/10.1103/PhysRevD.79.124017 https://arxiv.org/abs/0904.0141</ref>)

From<ref name="Pieri 2024"></ref>
<blockquote>
It should be emphasised that Quantum Inequalities have not been proven for realistic interacting quantum fields, where the they are expected to be weaker if they exist [Cadamuro, 2019], and they often require Minkowskian conditions at infinity, a condition which is not appropriate for fields inside bounded regions, as in a Casimir devices.
</blockquote>

== Warp drives and black holes ==

What happens if you drive your warp drive into a black hole?

The pop-sci reason that nothing can escape a black hole is that at the event horizon the escape velocity is faster than light and nothing can go faster than light. A warp drive can go faster than light.

The more detailed reason from general relativity is that at the event horizon space and time are rotated sufficiently that "inward" becomes "forward in time." You can no more go farther away from the singularity in a black hole once you are inside the event horizon than you can go backward in time. Any super-luminal travel can go backward in time, and warp drives can engage in super-luminal travel.

The Hawking area theorem that says that black holes can only grow, never shrink relies on the condition that the energy is everywhere positive. Warp drives have negative energy density.

So in principle, there's nothing that prevents a warp drive from taking a dip into a black hole and coming back out again.

Garattini and Zatrimaylov looked into the problem of a warp drive near (and in) a black hole<ref name="Garattini and Zatrimaylov">R. Garattini and K. Zatrimaylov, "Black holes, warp drives, and energy conditions", Physics Letters B 856 138910 (2024) https://doi.org/10.1016/j.physletb.2024.138910</ref>. The found that the black hole reduces the negative energy requirement for the warp bubble to move inward toward the black hole singularity, but increases the negative energy needed to move away from the center of the black hole. In addition, a warp drive parked at the event horizon would allow light from inside the horizon to pass through the bubble and escape back outside.

== Credit ==
Author: Luke Campbell

== References ==

[[Category:Physics]][[Category:Infrastructure]][[Category:Physics & Engineering‏‎]][[Category:Physics & Math & Engineering]][[Category:Transportation & Infrastructure‏‎]][[Category:Metric Engineering]]

Warp Drives

2026-07-26T23:08:56Z

Lwcamp: /* So what is going on? */

{{PageConstructionNotice}}

Science fiction often features spacecraft that can seemingly move across space and get between the place of departure and the destination much faster than light could have done. This appears to contradict the theory of relativity, which predicts unequivocally that nothing can move through space faster than light. Because relativity has been incredibly successful at describing nature, with its many other predictions regularly being confirmed to extraordinary accuracy and within the bounds of uncertainty of all the experiments that tested them, it gives confidence that relativity is a correct description of reality. Which seems to rather throw a wet towel on our hopes for rapid travel between stars.

However, while relativity does not allow things to move through space faster than light, it places no such restrictions on how fast space-time itself can expand, contract, or move around. This leads to the idea of a warp drive – the spacecraft remains stationary within a region of highly curved space-time, and that region moves at super-luminal speeds rather than the spacecraft.

== The Alcubierre warp drive ==

The first warp drive geometry that satisfied the Einstein field equations of relativity was proposed by Miguel Alcubierre<ref>M. Alcubierre, "The warp drive: hyper-fast travel within general relativity." Classical and Quantum Gravity. 11 (5): L73–L77 (1994). [https://arxiv.org/abs/gr-qc/0009013 arXiv:gr-qc/0009013]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1994CQGra..11L..73A 1994CQGra..11L..73A]. doi:[https://doi.org/10.1088%2F0264-9381%2F11%2F5%2F001 10.1088/0264-9381/11/5/001]. S2CID [https://api.semanticscholar.org/CorpusID:4797900 4797900].</ref>. In this geometry, a sphere of space-time moves at an arbitrary speed (potentially but not necessarily a speed much faster than light). Objects within the sphere are moved along with the sphere; an object at rest within the sphere would be moved along with the sphere indefinitely. Space is expanding at the rear boundary and contracting at the front boundary in order to keep the sphere moving. In order to satisfy the Einstein field equations, the boundary of the sphere must have a negative energy density. The challenges of space-time geometries with negative energy densities are described in our page on [[Wormholes#Exotic_energy_conditions|wormholes]], for our purposes it is enough to note that negative energy densities can pose problems if not handled carefully, there may be limits on how much negative energy you can have without nearby positive energy density, and it may not be possible to get enough negative energy to support a warp drive; although none of this is ruled out by physics – yet!

<table align=center>
<tr><td width=650>[[File:Energy_magnitude_of_Alcubierre_drive.png]]
<td width=650>[[File:Expansion_of_Alcubierre_drive.png]]
<tr><td>The magnitude of the energy of the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. (The energy density is negative everywhere in the shell, here only the magnitude is shown to aid visual comprehension.)
<td>The expansion of space in the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. Negative expansion means that space is contracting in that direction, positive expansion that it is expanding.
</table>

NOTE: What the? The energy distribution is symmetric! How does the drive know which way to go? There must be significant contributions of other components of the stress-energy tensor, and those have got to be asymmetric along the forward/backward axis. Check this when I get time.

Almost as soon as Alcubierre proposed his warp drive, others began picking it apart.
<ol>
<li>Pfenning and Ford<ref name="PfenningFord_2001">M. J. Pfenning and L. H. Ford, "The unphysical nature of 'Warp Drive'", Classical and Quantum Gravity. 14 (7): 1743–1751 (1997). arXiv:[https://arxiv.org/abs/gr-qc/9702026 gr-qc/9702026]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1997CQGra..14.1743P 1997CQGra..14.1743P]. doi:[https://doi.org/10.1088%2F0264-9381%2F14%2F7%2F011 10.1088/0264-9381/14/7/011]. S2CID [https://api.semanticscholar.org/CorpusID:15279207 15279207].</ref> found that if the negative energy density wall around the bubble obeys [[Wormholes#Quantum_energy_inequalities|quantum energy inequalities]], then for warp speeds of around light speed the shell thickness would be on the order of a hundred Planck lengths; a distance far smaller than any other known physical phenomenon or object.
<li>In the same work, Pfenning and Ford also found that for a bubble a hundred meters in radius, a warp speed of around the speed of light, and the required thickness of about 100 Planck lengths the energy in the bubble shell would be E ≈ -1063 kg c2. The magnitude of this latter value is ten orders of magnitude larger than the energy of the entire visible universe. They do note, however, that if quantum energy inequalities can be ignored then a warp drive with a hundred meters radius but a shell thickness of one meter would "only" have an energy magnitude of about a quarter solar mass.
<li>As with any method of faster than light travel, the warp drive could be used to make a time machine.
<li>Perhaps most seriously, if the warp drive is going faster than light speed, the negative energy regions on the outside of the shell won't be in the warp parts of the bubble. They will have to be moving through space at faster than light speed if the bubble is to maintain its integrity, which is the very problem that the warp drive was designed to avoid. The outside of the warp bubble shell that maintains the warp bubble would fall away and the warp bubble would quickly erode to sub-luminal speeds.<ref name="Van Den Broeck 2000">Chris Van Den Broeck, "On the (im)possibility of warp bubbles", arXiv:gr-qc/9906050 https://doi.org/10.48550/arXiv.gr-qc/9906050</ref>
<li>A more general problem that includes the previous one is that the geometry of the warp drive determines the distribution of energy, momentum, and stress (pressure, shear, and tension) by Einstein's field equation, which states that the stress energy tensor is directly proportional to a reduced order curvature tensor of space-time. However, the stress-energy tensor must also obey the continuity equation – basically, for the energy to change in a given volume of space that energy must come in or go out of the boundary of that volume and similarly for the momentum. So a warp drive can have the warp condition for one moment in time. But after that, the stress-energy that maintains the warp will start to move according to the forces on it and according to the natural paths that it takes through the warped space-time. There is no guarantee that the configuration of the stress-energy at later times will be sufficient to maintain the warp.
<blockquote>
From the perspective of simulating the warp drive dynamically, the key challenge is stability. One can straightforwardly generate the initial metric that describes the Alcubierre metric and infer the instantaneous matter configuration that must support it using the Einstein Equation, but a time-domain evolution of the coupled matter-gravity system necessitates that we specify the equation of state of the matter, which in turn determines how it evolves. In a perfect fluid, this involves specifying the relation between the fluid pressure and density, but in a more general case, the full relation between the
stress-energy components must be given. There is (to our knowledge) no known equation of state that would maintain the warp drive metric in a stable configuration over time - therefore, whilst one can require that initially, the warp bubble is constant, it will quickly evolve away from that state and, in most cases, the warp fluid and spacetime deformations will disperse or collapse into a central point.
 
– Clough, Dietrich, and Khan<ref>Katy Clough, Tim Dietrich, and Sebastian Khan, "What no one has seen before: gravitational waveforms from warp drive collapse", arXiv:2406.02466 [gr-qc] https://doi.org/10.48550/arXiv.2406.02466</ref>
</blockquote>
<li>In order to get the matter (or more generally, stress-energy) that forms the warp shell up to speeds that could allow the shell to exist, you need to be able to, well, accelerate that matter to warp speeds. Which gives a chicken-and-egg problem - in order to achieve superluminal travel you need to be able to get matter or energy moving superluminally in the first place<ref name="Bobrick and Martire">Alexey Bobrick and Gianni Martire, "Introducing Physical Warp Drives", arXiv:2102.06824 [gr-qc] https://doi.org/10.48550/arXiv.2102.06824</ref><ref>D. H. Coule, "No warp drive", Class. Quantum Grav. 15 (1998) 2523–2527. https://doi.org/10.1088/0264-9381/15/8/026</ref>, which rather makes warp drives unnecessary from the get-go.
<li>A warp bubble moving faster than light will not be able to get any signals in front of it as it will overtake its own signal. In fact, anyone in the interior of the bubble would not be able to send signals to the front surface of the bubble. In science-speak, a horizon forms at the front of the bubble<ref name="Natario">José Natário, "Warp drive with zero expansion", Classical and Quantum Gravity. 19 (6): 1157–1166 (2002). arXiv:[https://arxiv.org/abs/gr-qc/0110086 gr-qc/0110086]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2002CQGra..19.1157N 2002CQGra..19.1157N]. doi:[https://doi.org/10.1088%2F0264-9381%2F19%2F6%2F308 10.1088/0264-9381/19/6/308]. S2CID [https://api.semanticscholar.org/CorpusID:15859984 15859984].</ref>. As a consequence, the bubble cannot be controlled, steered, nor stopped from within the bubble itself.
<li>A Horizon will naturally emit Hawking radiation. It is estimated that the radiation will raise the temperature of the interior of the bubble to approximately 1032 K<ref name="Finazzi 2009">Stefano Finazzi, Stefano Liberati, and Carlos Barceló, "Semiclassical instability of dynamical warp drives", Phys. Rev. D 79, 124017 (12 June, 2009) https://doi.org/10.1103/PhysRevD.79.124017 , arXiv:0904.0141 [gr-qc] https://doi.org/10.48550/arXiv.0904.0141</ref> (at this level of accuracy, the result is the same if you replace temperatures in K by the same number in °C or °F). This temperature, however, may be lower if quantum inequalities do not have to be fulfilled.
<li>Semi-classical instabilities at the horizon are expected to lead to a divergence of the stress-energy tensor at the boundary in a short period of time, thus leading to the destruction of the warp bubble<ref name="Finazzi 2009"></ref>.
</ol>
Helpfully, Van Den Broeck<ref name="VanDenBroeck_2000">C. Van Den Broeck, "Alcubierre’s Warp Drive: Problems and Prospects" AIP Conference Proceedings. 504: 1105–1110 (2000). Bibcode:[https://ui.adsabs.harvard.edu/abs/2000AIPC..504.1105V 2000AIPC..504.1105V]. doi:[https://doi.org/10.1063%2F1.1290913 10.1063/1.1290913].</ref> suggested several solutions – or at least mitigations – for some of these problems
<ol>
<li>Quantum inequalities had not been shown to be true in general for highly curved space-times.
<li>A proposed warp drive geometry<ref name="VanDenBroeck_1999">C. Van Den Broeck, "A 'warp drive' with more reasonable total energy requirements". Classical and Quantum Gravity. 16 (12): 3973–3979 (1999). arXiv:[https://arxiv.org/abs/gr-qc/9905084 gr-qc/9905084]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1999CQGra..16.3973V 1999CQGra..16.3973V]. doi:[https://doi.org/10.1088%2F0264-9381%2F16%2F12%2F314 10.1088/0264-9381/16/12/314]. S2CID [https://api.semanticscholar.org/CorpusID:15466313 15466313]. </ref> (see the van Den Broeck drive, below) would be able to minimize the magnitude of the energy required to about that of the mass of our sun. While still large, it is not unphysically large.
<li>Time travel is always going to be a worry with faster than light travel. There's no neat solution to this one.
<li>It may be possible for the negative energy outside of the superluminal region to compress into the superluminal region to form a shock in space-time. The front surface of the bubble would then be a singularity. It is not clear if such a sharp jump in physical properties is possible, but neither is sure that it is impossible, either. However, while this might help with the negative energy in front of the bubble getting swept up because it cannot move faster than light, it does little to help with the material at the back of the bubble on the outside getting left behind because it cannot keep up.
</ol>
Other mitigating research has also shown
<ol start=9>
<li>Semi-classical divergence only occurs in warp drives with only one spatial dimensions. Those with two or more spatial dimensions (as in our universe) only contain points where divergences occur, and these divergences can be managed. In fact, their effects can be reduced by making the warp shape more elongated<ref>Carlos Barceló, Valentin Boyanov, Luis J. Garay, Eduardo Martín-Martínez,
and Jose M. Sánchez Velázquez, "Warp drive aerodynamics", JHEP08 (2022) 288 https://doi.org/10.1007/JHEP08(2022)288 , arXiv:2207.06458 [gr-qc] https://doi.org/10.48550/arXiv.2207.06458</ref>.
</ol>

(And what's with the ring in all the artwork, anyway?)

=== So what is going on? ===

What is it that makes a warp drive a warp drive, and how does it work?

A warp drive is a concept that comes out of the science of general relativity. General relativity collects space and time together into a single "structure" called spacetime. This structure is curved, and its curvature is determined by the material that is in it. Absent forces acting between matter, particles of matter move through spacetime on the shortest possible paths in this curved environment.
<blockquote>
"Spacetime tells matter how to move; matter tells spacetime how to curve." - John Archibald Wheeler
</blockquote>
All of this is encompassed in the Einstein field equation:
<div align=center>
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 8πG
<td rowspan="2" nowrap="nowrap" align="right"> T = R -  
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right"> R g
<tr>
<td style="border-top:solid 1px black;"> c4
<td style="border-top:solid 1px black;"> 2
</table>
</div>
Unfortunately, by convention, both the symbols G and R get used for a lot of different things. Here, symbols in bold are second rank tensors – constructs of arrays of 4×4 numbers where 4 is necessary because that is the number of dimensions of spacetime. Conventionally, these numbers are indexed by a value going from 0 to 3, with 0 being the part corresponding to the coordinate of time and 1 to 3 being the Cartesian coordinates that you might know as x, y, and z.

T is the stress-energy tensor: the (0,0) component represents the energy density at a given point; the (0,i) and (i, 0) components (with i going from 1 to 3) represent the momentum density; and the (i,j) components represent the stresses present, such as pressure or tension or shear.

The other term, R - R g, involves a partial description of the curvature. A full description of the curvature requires a construct of 4×4×4×4 components; too much for the stress energy tensor to fully constrain. Which is okay, because this allows curvature to exist away from where matter is, allowing gravity to have long range effects across empty space as well as for phenomena like gravitational waves. But it is enough that a given distribution of matter T gives enough constraints (along with specifying how the geometry behaves far away) that you can generally figure out what the actual gravity is.

Describing spacetime - lapse function & shift vectors (foliations, etc.)
Diagram of what the shift vector does.

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Lapse_shift.png]]
<tr><td>A folliation of space-time into slices of constant time coordinates (grey). At an event (black circle), the lapse function (yellow) shows how far into the future the point at that event will be projected in a unit amount of proper time experienced by an observer initially at the event. The shift vector (red) shows how that point changes coordinates after the unit amount of proper time. The cyan vector shows the new coordinates the point is projected onto. The lower black surface indicates a foliation of spacetime at the same time coordinate; after unit proper time, the events on the lower foliation get projected to the curved upper black foliation according to the shift vectors and lapse functions at each of their respective points.
</table>
</table>

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Warp_shift.png]]
<tr><td>The shift vector of a warp geometry, in red. At the edges there is no shift. At the center (inside the warp bubble) there is a region of uniform shift so that at a later time the inside of the warp bubble is rigidly translated to different spatial coordinates (in blue). The two foliations remain the same distance apart, because warp drives maintain a unit lapse function.
</table>
</table>

Solve backwards - assume a structure of space-time, figure out what distribution of matter is required for this from the EFE.

What sort of spacetime geometry produces a warp?

Unit lapse function <-> no time dilation

Region of near constant shift vector - that region is "moving" spacetime, the warp bubble.

Shift vector falls off to zero away from warp bubble -> normal flat spacetime.

=== Alcubierre warp interactions with light and matter ===

Space is not entirely empty. It is filled with a diffuse plasma in the form of the interstellar medium, as well as cosmic radiation, light from stars, and cosmic microwave background radiation. A warp drive propagating through space will encounter this stuff. What happens when this matter and radiation have a warp bubble pass across them?

The first analysis of matter encountering a warp bubble was performed by Pfenning and Ford<ref name="PfenningFord_2001"></ref>. They looked at the warp bubble interaction with a massive object at rest with the frame of reference of the warp drive (keep in mind that the rest frame is the same inside and outside the warp bubble; in this rest frame objects inside the bubble have no momentum even though they are, in some sense, changing location rapidly with time). When the warp bubble passes, the object experiences an acceleration in the direction of the warp bubble motion. When the warp shell passes and the object is inside the bubble, it will be moving with approximately the speed of the bubble. Pfenning and Ford analyzed this problem with a continuous distribution of shell energy that, strictly speaking, never falls to zero except at the bubble center and at spatial infinity, so unless the object passes through the center in Pfenning and Ford's description it will never quite get up to the bubble's speed. In this case, the object will move almost as fast as the object but will pass through the bubble in a finite time, after which it will again be at rest with respect to the reference frame but displaced along the direction of the warp bubble motion by some distance. With this description of the warp bubble, a spacecraft of finite size will always be moving a little slower than the warp bubble and would have to use rockets to keep up with it. in addition, the spacecraft would experience tidal forces that would cause stress on the spacecraft's structure.

Pfenning and Ford also examined cases where the shell is of a finite (possibly infinitesimal) width and falls to zero both inside and outside the bubble. In this case any matter encountering the bubble would be collected at the bow of the bubble and thereafter move along with it.

McMonigal et al. analyzed the situation for both massive particles and light moving along the axis of travel of the bubble<ref name="McMonigalEtAl_2012">B. McMonigal, G. F. Lewis, and P. O'Byrne, "Alcubierre warp drive: On the matter of matter". Physical Review D. 85 (6) 064024 (20 March 2012). arXiv:[https://arxiv.org/abs/1202.5708 1202.5708]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2012PhRvD..85f4024M 2012PhRvD..85f4024M]. doi:[https://doi.org/10.1103%2FPhysRevD.85.064024 10.1103/PhysRevD.85.064024]. S2CID [https://api.semanticscholar.org/CorpusID:3993148 3993148].</ref>. They found that light moving opposite the warp direction passed through the bubble without incident, being only somewhat delayed by passing through the bubble. A warp bubble that is warping at sub-luminal speeds can have light catch up from behind it. This light is able to pass through the bubble, and is somewhat advanced in its path by the speed of the bubble. For super-luminal warp bubbles, however, the situation is different. The bubble will catch up to light moving it its own direction that is originally in front of it. This light cannot escape forward, the bubble being too fast. Nor can it escape backward, as the light is propagating forward and the interior of the bubble is at rest. Thus, the light gets caught at the bow of the warp bubble, unable to escape for so long as the warp bubble is active. This light is strongly blue shifted to extremely energetic x-rays and gamma rays. The space behind the bubble is swept clear of forward-moving light.

The situation for matter over-run by a super-luminal warp bubble is similar. Matter moving backward passes through the bubble, being only swept a ways along the bubble's path as sign of its passing but otherwise continuing on their way. Matter at rest behaves the same way as Pfenning and Ford discovered. Meanwhile, matter moving in the same direction of the warp is overtaken and collects at the bow of the bubble, unable to leave for so long as the bubble is warping. This matter "surfing" on the bubble bow is highly accelerated to relativistic speeds, experiencing extreme time dilation.

For a sub-luminal bubble, there is a speed faster than the warp speed where particles coming up from behind overtake the bubble and pass through, out the front. Particles below this speed but still faster than the warp speed still overtake the bubble, but then bounce backwards out of the bubble from behind. They are still moving forward, but are now moving more slowly than the warp speed. Particles moving forward but slower than the bubble will be overtaken and then bounced out the front with a speed higher than the warp speed. Those particles moving backward will pass through the bubble and exit with their initial speed.

The spacecraft inside the bubble will observe that light moving backward compared to the warp direction is blue-shifted, and matter moving backward passes through with increased energy. Meanwhile, a sub-luminal bubble will see light that was moving forward as red-shifted and matter catching up to the bubble from behind will pass by with reduced energy (superluminal bubbles, of course, do not have any matter or radiation catching up to them from behind).

When a superluminal warp bubble that has been collecting matter and radiation for the duration of its journey and blue-shifting it to much higher energies is turned off, all that matter and energy is released as a blast of radiation in the direction the bubble was warping.

McMonigal et al. conclude by noting that any spacecraft in the warp bubble would need shielding to protect against the blue-shifted radiation and matter of increased energy. In addition the destination would be "blasted into oblivion" by the release of matter and radiation that had been caught in the bubble during the trip.

Clark et al.<ref>Chad Clark, William A. Hiscock, and Shane L. Larson, "Null geodesics in the Alcubierre warp drive spacetime: the view from the bridge", arXiv:gr-qc/9907019 (2018) https://doi.org/10.48550/arXiv.gr-qc/9907019</ref> examine what it would look like from within an operating warp drive. They find that light moving at 90 degrees to the direction of motion will look like it is still coming from 90 degrees. However, light from the front hemisphere is squished progressively toward the forward point the faster you go. For a warp bubble moving faster than the speed of light, there is a cone behind the warp bubble where no light can reach it. However, you don't see a black spot behind you – light from the regions in the backward hemisphere from outside that cone that can reach you is distorted toward the direction behind you to fill in that spot. Light from the forward direction is blue-shifted and light from behind is red-shifted. At speeds of around 200 times light speed, the cosmic microwave background would be blue shifted to energies similar to that of the solar photosphere, heating anything in the warp bubble as if they were close to the sun.

== Van Den Broeck warp drive ==

If an Alcubierre warp bubble a hundred meters across requires a magnitude of energy greater than the entire energy in the observable universe, one option to reduce the magnitude of energy used is to make the warp bubble smaller. Much smaller. Pfenning and Ford<ref name="PfenningFord_2001"></ref> suggested making the warp bubble smaller than an atom. Of course, that brings up the problem of how to stuff a spacecraft in there. Van Den Broeck proposed a solution<ref name="VanDenBroeck_1999"></ref>: make the warp bubble only about the size of an atomic nucleus but expand the space inside the bubble enormously so that a spacecraft could fit in. This is somewhat like a nuclear sized wormhole that leads to a pocket universe that holds the spacecraft. Unfortunately, the metric doesn't fit neatly into any [[Wormholes#Three_dimensions|embedding diagram]] that I can figure out, but the math works out so that spatial coordinates inside the bubble are expanded by a factor of 1017 and a spacecraft would have a few hundred meters of bubble interior to putz around in. This trick manages to reduce the magnitude of energy to only about the mass energy of a few stars similar to our own. Other than that, it is otherwise a normal Alcubierre drive.

Further work<ref>S. Krasnikov, "The quantum inequalities do not forbid spacetime shortcuts", Physical Review D. 67 (10) 104013. https://doi.org/10.1103%2FPhysRevD.67.104013, arXiv:gr-qc/0207057 (2003) https://doi.org/10.48550/arXiv.gr-qc/0207057 </ref> has managed to use Van Den Broeck's trick to reduce the magnitude of the negative energy to only about a milligram.

== Natário warp drive ==

The Alcubierre drive is not the only way to construct a warp drive. José Natário<ref name="Natario"></ref> showed that it was just one example of an entire class of warp drives. These are space-time geometries with a smooth and localized shift vector field a shift vector is a vector field that describes how one "slice" of space out of space-time connects to the next) and a unit lapse function (the lapse function describes how the time you experience changes when going from a "slice" of space-time to the next).

Natário then went on to find in that class a set of warp drives that do not have any expansion or contraction of space at all. Instead, space encountering the front boundary of the warp bubble instead "slides" around the outside of the bubble until it gets to the corresponding place on the back and is left behind there. Another way to think of it is that space entering the bubble shell is compressed in the radial direction of the shell but is simultaneously expanded in the tangential direction so that there is no net change in volume. When it gets to the back, the opposite occurs. In the language of mathematics, the shift vector field is divergenceless.

<table align=center>
<tr><td width=360>[[File:Natario_drive_flow_of_space.png]]
<td width=445>[[File:Natario_warp_drive_horizons.png]]
<tr><td style="vertical-align: top;">A representation of the motion of space in a zero-expansion Natário drive with a vanishingly thin warp shell around the bubble.
<td>The horizons of the generalized Natário class of warp drives (including the Alcubierre warp drive and all fell-Heisenberg drives) at superluminal speed in the direction of the green arrow. No event inside the bubble can affect events outside the bubble in front of the blue line. No events outside the bubble behind the red line can affect events inside the bubble. The lines have an angle α with respect to the direction of motion with sin(α) = 1/vs for vs the speed of the bubble relative to light speed. This is analogous to the Mach cone of supersonic objects.
</table>

NOTE: There are two different things that can be meant by the term "Natário warp drive". The generalized Natário warp drive is the general theory of all warp drives that includes within it the Alcubierre, Fell-Heisenberg, and Lentz warp drives, among infinitely many others. The zero-expansion Natário warp drive is the specific implementation of the generalized Natário warp drive that has a divergenceless shift vector field.

The generalized class of warp drives developed by Natário (including both the Alcubierre and this zero-expansion warp drive) are shown to always have regions in the warp shell where the energy density is negative to at least some observers. You can't get away from it – to warp, you need negative energy density.

The Natário class of warp drives blueshift light coming in from the front and redshift light catching up from the back. For the case of a super-luminal warp bubble, there will of course be a horizon at the back of the bubble and light will not be able to get through from behind. For a bubble speed vs relative to the speed of light, light coming from straight ahead will be blueshifted up in frequency and photon energy by a factor of 1 + vs. For sub-luminal travel, light coming from behind will be redshifted by a factor 1 - vs. In general for n as the unit direction along which the light is propagating, the blueshift factor will be 1 + vs ⋅ n. If the warp shell is infinitesimally thin, light from outside that reaches the center of the bubble will not be distorted in direction although it will be frequency shifted, but observers still inside the bubble but displaced from the center will see the field of view distorted as well as frequency shifted.

Most of the difficulties that apply to the Alcubierre warp drive also apply to the generalized Natário warp drive (and thus to the zero-expansion Natário drive and Fell-Heisenberg drive as well). In particular, quantum energy inequalities, high magnitudes of the energy required, formation of time machines, instabilities related to the stress-energy making up the warp shell being unable to keep up with the bubble, the chicken-and-egg problem of needing FTL to achieve FTL, the formation of horizons making warp bubbles uncontrollable, and Hawking radiation at the horizons, are all still concerns for generic warp drives.

== Fell-Heisenberg warp drives ==

S. D. B. Fell and L. Heisenberg investigated what sort of warp structure is necessary to produce a warp drive with positive energy density everywhere<ref name="Fell-Heisenberg">S. D. B. Fell and L> Heisenberg, "Positive energy warp drive from hidden geometric structures", Classical and Quantum Gravity 38 155020 (2021) https://doi.org/10.1088/1361-6382/ac0e47 https://arxiv.org/abs/2104.06488</ref>. They found that warp drives of the generic Natário form with irrotational shift vectors had a simple expression for the energy density and vanishing momentum (irrotational means that the shift vector can be described as the gradient of a scalar potential field or, equivalently, that the curl of the shift vector field vanishes everywhere). Fell and Heisenberg proposed several potential fields that either had positive energy density everywhere except at discontinuities, or that were continuous but had regions that violated the weak energy condition (i. e. negative energy densities in some frames of reference). One specific example they provided of a warp field had an energy that is about 10,000 times less than the mass-energy of our sun. Or only about half the mass-energy of Jupiter. This is a significant improvement over many other proposed drives.

Despite the introduction discussing warp drive configurations that satisfy the various energy conditions, the configurations described in the paper are shown to locally violate the weak and strong energy conditions. Nonetheless, the energy density is still mostly positive. Santiago, Schuster, and Visser's<ref name="Santiago Schuster Visser 2022">J. Santiago, S. Schuster, and M. Visser, "Generic warp drives violate the null energy condition", Physical Review D 105, 064038 (2022) https://doi.org/10.1103/PhysRevD.105.064038</ref> work shows that the various energy conditions must still be violated by this warp drive; to not violate these, energy density must be positive in all reference frames not just those of the co-moving observer.

Any vector field can be described by its divergence, its curl, and a constant. The constant part of a warp drive's shift vector can be discarded because then the warp drive would not be localized. It is interesting, therefore, that the Natário zero-expansion drive has zero divergence and can be described only by its curl, while the Fell-Heisenberg drive has zero curl and can be described only by its divergence. These two classes of drives thus rely on opposite choices of setting one of the two different parts describing a vector field to zero. As might be expected, the Fell-Heisenberg drive is not zero expansion – space-time will change in volume as it passes through the warp bubble (although after going all the way through it will return to its original volume). However, objects in space-time will retain their original orientation as they pass through the warp bubble.

Previous examples of warp drives, such as the Alcubierre and zero-expansion Natário drives, tend toward a flat space-time as you go far enough away faster than would occur if there were any mass present in the warp drive. That is, the ADM mass of such drives is zero. However, one of the Fell-Heisenberg examples described a spacetime where the geometry approaches that of what you would get for a massive object at the warp location, giving one of the first examples of a warp drive with mass.

In the Fell-Heisenberg drive the momentum vanishes everywhere. Yet the energy occupies regions where the shift vector is varying rapidly. The lack of momentum means that the energy will not move to keep up with the differential expansion and movement of the spacetime. As a consequence, at later times the energy will have a different distribution than what is necessary to maintain the given warp configuration. The exact time evolution is not solved but this likely leads to collapse of warp bubble. Note that this is not a problem unique to the Fell-Heisenberg drive. Assuming a geometry that describes a warp produces a specific distribution of the stress-energy tensor that satisfies Einstein's equation. But there is no guarantee that this stress energy tensor satisfies the continuity equation, which describes how the distribution of matter and momentum change with time. The issue of stability of any warp drive is not well understood, but most warp drive geometries will be unstable in the sense that their matter that generates the warp fields will not be able to keep up with the motion of the warp. The Fell-Heisenberg drive is unique only in that, with vanishing momentum, there is no configuration that allows stable propagation.

(n. b. The Heisenberg here is Lavinia Heisenberg, not the Werner Heisenberg of quantum physics and uncertainty principle fame.)

== Lentz warp drive ==

E. Lentz<ref name="Lentz">E. W. Lentz, "Breaking the warp barrier: hyper-fast solitons in Einstein–Maxwell-plasma theory", Classical and Quantum Gravity. 38 075015 (2021). arXiv:[https://arxiv.org/abs/2006.07125 2006.07125]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2021CQGra..38g5015L 2021CQGra..38g5015L]. doi:[https://doi.org/10.1088%2F1361-6382%2Fabe692 10.1088/1361-6382/abe692]. ISSN [https://search.worldcat.org/issn/0264-9381 0264-9381]. S2CID [https://api.semanticscholar.org/CorpusID:219635854 219635854].</ref> investigated a particular configuration of the Fell-Heisenberg class of warp drives, where the scalar field has a hyperbolic shape. The necessary conditions were found to be satisfied by a selection of rhomboid regions of the scalar field, resulting in an energy density that is everywhere positive and a shift vector with vanishing tidal force at the center for a comfortable ride.

However, Santiago, Schuster, and Visser<ref name="Santiago Schuster Visser 2022"></ref> dispute claims that the Lentz drive satisfies the energy conditions, noting that everywhere positive energy density in one frame of reference is insufficient to establish that the energy density is positive in all reference frames; knowledge of the Cauchy stress tensor is also needed. They show that any generic warp drive will violate the strong energy condition, null energy condition, and weak energy condition.

== Conservation laws ==

(Discussion of asymptotic flatness: what is it? Where does it apply?)

(The "big four" conservation laws and how they relate to asymptotic flatness.)

(ADM mass -> 0; conservation of energy/mass issues.)<ref>Sebastian Schuster, Jessica Santiago,and Matt Visser, "ADM mass in warp drive spacetimes" General Relativity and Gravitation (2023) 55:14 https://doi.org/10.1007/s10714-022-03061-9</ref>

(Discuss issues of conservation of angular momentum.)

(If angular momentum conservation is ignored, discuss how momentum & energy are affected by outside forces. In a gravitational field equivalent to inertial frame ... like being in an accelerated elevator; will acquire momentum buildup while staying "at rest".)

== Warp Railroads ==

Van Den Broeck<ref name="VanDenBroeck_2000"></ref> suggested a warp railroad,

You can set up devices ahead of time along the path of the bubble that produce the negative energy regions for the warp bubble at the appropriate time without needing the negative energy to ever move with the bubble at all. At least for bubbles on a predictable schedule and route.

helps with:

-stability (elaborate)

-conservation laws (elaborate)

Mention Krasnikov<ref name="Krasnikov 1998">S. V. Krasnikov, "Hyperfast travel in general relativity" Phys. Rev. D 57 4760</ref>

Hyperwave<ref name="Pieri 2024">Lorenzo Pieri, "Hyperwave: hyper-Fast Communication within General Relativity", arXiv:2311.12069 [gr-qc] https://doi.org/10.48550/arXiv.2311.12069</ref> makes use of this warp railroad idea.

== Quantum effects ==

(quantum stuff here<ref name="Hiscock1997">W. A. Hiscock, "Quantum effects in the Alcubierre warp drive spacetime", Classical and Quantum Gravity 14 L183 https://doi.org/10.1088/0264-9381/14/11/002 https://arxiv.org/abs/gr-qc/9707024</ref><ref name="Finazzi et al 2009">S. Finazzi, S. Liberati, C. Barceló, "Semiclassical instability of dynamical warp drives", Physical Review D 79, 124017 (2009)https://doi.org/10.1103/PhysRevD.79.124017 https://arxiv.org/abs/0904.0141</ref>)

From<ref name="Pieri 2024"></ref>
<blockquote>
It should be emphasised that Quantum Inequalities have not been proven for realistic interacting quantum fields, where the they are expected to be weaker if they exist [Cadamuro, 2019], and they often require Minkowskian conditions at infinity, a condition which is not appropriate for fields inside bounded regions, as in a Casimir devices.
</blockquote>

== Warp drives and black holes ==

What happens if you drive your warp drive into a black hole?

The pop-sci reason that nothing can escape a black hole is that at the event horizon the escape velocity is faster than light and nothing can go faster than light. A warp drive can go faster than light.

The more detailed reason from general relativity is that at the event horizon space and time are rotated sufficiently that "inward" becomes "forward in time." You can no more go farther away from the singularity in a black hole once you are inside the event horizon than you can go backward in time. Any super-luminal travel can go backward in time, and warp drives can engage in super-luminal travel.

The Hawking area theorem that says that black holes can only grow, never shrink relies on the condition that the energy is everywhere positive. Warp drives have negative energy density.

So in principle, there's nothing that prevents a warp drive from taking a dip into a black hole and coming back out again.

Garattini and Zatrimaylov looked into the problem of a warp drive near (and in) a black hole<ref name="Garattini and Zatrimaylov">R. Garattini and K. Zatrimaylov, "Black holes, warp drives, and energy conditions", Physics Letters B 856 138910 (2024) https://doi.org/10.1016/j.physletb.2024.138910</ref>. The found that the black hole reduces the negative energy requirement for the warp bubble to move inward toward the black hole singularity, but increases the negative energy needed to move away from the center of the black hole. In addition, a warp drive parked at the event horizon would allow light from inside the horizon to pass through the bubble and escape back outside.

== Credit ==
Author: Luke Campbell

== References ==

[[Category:Physics]][[Category:Infrastructure]][[Category:Physics & Engineering‏‎]][[Category:Physics & Math & Engineering]][[Category:Transportation & Infrastructure‏‎]][[Category:Metric Engineering]]

Warp Drives

2026-07-26T23:07:54Z

Lwcamp: /* So what is going on? */

{{PageConstructionNotice}}

Science fiction often features spacecraft that can seemingly move across space and get between the place of departure and the destination much faster than light could have done. This appears to contradict the theory of relativity, which predicts unequivocally that nothing can move through space faster than light. Because relativity has been incredibly successful at describing nature, with its many other predictions regularly being confirmed to extraordinary accuracy and within the bounds of uncertainty of all the experiments that tested them, it gives confidence that relativity is a correct description of reality. Which seems to rather throw a wet towel on our hopes for rapid travel between stars.

However, while relativity does not allow things to move through space faster than light, it places no such restrictions on how fast space-time itself can expand, contract, or move around. This leads to the idea of a warp drive – the spacecraft remains stationary within a region of highly curved space-time, and that region moves at super-luminal speeds rather than the spacecraft.

== The Alcubierre warp drive ==

The first warp drive geometry that satisfied the Einstein field equations of relativity was proposed by Miguel Alcubierre<ref>M. Alcubierre, "The warp drive: hyper-fast travel within general relativity." Classical and Quantum Gravity. 11 (5): L73–L77 (1994). [https://arxiv.org/abs/gr-qc/0009013 arXiv:gr-qc/0009013]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1994CQGra..11L..73A 1994CQGra..11L..73A]. doi:[https://doi.org/10.1088%2F0264-9381%2F11%2F5%2F001 10.1088/0264-9381/11/5/001]. S2CID [https://api.semanticscholar.org/CorpusID:4797900 4797900].</ref>. In this geometry, a sphere of space-time moves at an arbitrary speed (potentially but not necessarily a speed much faster than light). Objects within the sphere are moved along with the sphere; an object at rest within the sphere would be moved along with the sphere indefinitely. Space is expanding at the rear boundary and contracting at the front boundary in order to keep the sphere moving. In order to satisfy the Einstein field equations, the boundary of the sphere must have a negative energy density. The challenges of space-time geometries with negative energy densities are described in our page on [[Wormholes#Exotic_energy_conditions|wormholes]], for our purposes it is enough to note that negative energy densities can pose problems if not handled carefully, there may be limits on how much negative energy you can have without nearby positive energy density, and it may not be possible to get enough negative energy to support a warp drive; although none of this is ruled out by physics – yet!

<table align=center>
<tr><td width=650>[[File:Energy_magnitude_of_Alcubierre_drive.png]]
<td width=650>[[File:Expansion_of_Alcubierre_drive.png]]
<tr><td>The magnitude of the energy of the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. (The energy density is negative everywhere in the shell, here only the magnitude is shown to aid visual comprehension.)
<td>The expansion of space in the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. Negative expansion means that space is contracting in that direction, positive expansion that it is expanding.
</table>

NOTE: What the? The energy distribution is symmetric! How does the drive know which way to go? There must be significant contributions of other components of the stress-energy tensor, and those have got to be asymmetric along the forward/backward axis. Check this when I get time.

Almost as soon as Alcubierre proposed his warp drive, others began picking it apart.
<ol>
<li>Pfenning and Ford<ref name="PfenningFord_2001">M. J. Pfenning and L. H. Ford, "The unphysical nature of 'Warp Drive'", Classical and Quantum Gravity. 14 (7): 1743–1751 (1997). arXiv:[https://arxiv.org/abs/gr-qc/9702026 gr-qc/9702026]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1997CQGra..14.1743P 1997CQGra..14.1743P]. doi:[https://doi.org/10.1088%2F0264-9381%2F14%2F7%2F011 10.1088/0264-9381/14/7/011]. S2CID [https://api.semanticscholar.org/CorpusID:15279207 15279207].</ref> found that if the negative energy density wall around the bubble obeys [[Wormholes#Quantum_energy_inequalities|quantum energy inequalities]], then for warp speeds of around light speed the shell thickness would be on the order of a hundred Planck lengths; a distance far smaller than any other known physical phenomenon or object.
<li>In the same work, Pfenning and Ford also found that for a bubble a hundred meters in radius, a warp speed of around the speed of light, and the required thickness of about 100 Planck lengths the energy in the bubble shell would be E ≈ -1063 kg c2. The magnitude of this latter value is ten orders of magnitude larger than the energy of the entire visible universe. They do note, however, that if quantum energy inequalities can be ignored then a warp drive with a hundred meters radius but a shell thickness of one meter would "only" have an energy magnitude of about a quarter solar mass.
<li>As with any method of faster than light travel, the warp drive could be used to make a time machine.
<li>Perhaps most seriously, if the warp drive is going faster than light speed, the negative energy regions on the outside of the shell won't be in the warp parts of the bubble. They will have to be moving through space at faster than light speed if the bubble is to maintain its integrity, which is the very problem that the warp drive was designed to avoid. The outside of the warp bubble shell that maintains the warp bubble would fall away and the warp bubble would quickly erode to sub-luminal speeds.<ref name="Van Den Broeck 2000">Chris Van Den Broeck, "On the (im)possibility of warp bubbles", arXiv:gr-qc/9906050 https://doi.org/10.48550/arXiv.gr-qc/9906050</ref>
<li>A more general problem that includes the previous one is that the geometry of the warp drive determines the distribution of energy, momentum, and stress (pressure, shear, and tension) by Einstein's field equation, which states that the stress energy tensor is directly proportional to a reduced order curvature tensor of space-time. However, the stress-energy tensor must also obey the continuity equation – basically, for the energy to change in a given volume of space that energy must come in or go out of the boundary of that volume and similarly for the momentum. So a warp drive can have the warp condition for one moment in time. But after that, the stress-energy that maintains the warp will start to move according to the forces on it and according to the natural paths that it takes through the warped space-time. There is no guarantee that the configuration of the stress-energy at later times will be sufficient to maintain the warp.
<blockquote>
From the perspective of simulating the warp drive dynamically, the key challenge is stability. One can straightforwardly generate the initial metric that describes the Alcubierre metric and infer the instantaneous matter configuration that must support it using the Einstein Equation, but a time-domain evolution of the coupled matter-gravity system necessitates that we specify the equation of state of the matter, which in turn determines how it evolves. In a perfect fluid, this involves specifying the relation between the fluid pressure and density, but in a more general case, the full relation between the
stress-energy components must be given. There is (to our knowledge) no known equation of state that would maintain the warp drive metric in a stable configuration over time - therefore, whilst one can require that initially, the warp bubble is constant, it will quickly evolve away from that state and, in most cases, the warp fluid and spacetime deformations will disperse or collapse into a central point.
 
– Clough, Dietrich, and Khan<ref>Katy Clough, Tim Dietrich, and Sebastian Khan, "What no one has seen before: gravitational waveforms from warp drive collapse", arXiv:2406.02466 [gr-qc] https://doi.org/10.48550/arXiv.2406.02466</ref>
</blockquote>
<li>In order to get the matter (or more generally, stress-energy) that forms the warp shell up to speeds that could allow the shell to exist, you need to be able to, well, accelerate that matter to warp speeds. Which gives a chicken-and-egg problem - in order to achieve superluminal travel you need to be able to get matter or energy moving superluminally in the first place<ref name="Bobrick and Martire">Alexey Bobrick and Gianni Martire, "Introducing Physical Warp Drives", arXiv:2102.06824 [gr-qc] https://doi.org/10.48550/arXiv.2102.06824</ref><ref>D. H. Coule, "No warp drive", Class. Quantum Grav. 15 (1998) 2523–2527. https://doi.org/10.1088/0264-9381/15/8/026</ref>, which rather makes warp drives unnecessary from the get-go.
<li>A warp bubble moving faster than light will not be able to get any signals in front of it as it will overtake its own signal. In fact, anyone in the interior of the bubble would not be able to send signals to the front surface of the bubble. In science-speak, a horizon forms at the front of the bubble<ref name="Natario">José Natário, "Warp drive with zero expansion", Classical and Quantum Gravity. 19 (6): 1157–1166 (2002). arXiv:[https://arxiv.org/abs/gr-qc/0110086 gr-qc/0110086]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2002CQGra..19.1157N 2002CQGra..19.1157N]. doi:[https://doi.org/10.1088%2F0264-9381%2F19%2F6%2F308 10.1088/0264-9381/19/6/308]. S2CID [https://api.semanticscholar.org/CorpusID:15859984 15859984].</ref>. As a consequence, the bubble cannot be controlled, steered, nor stopped from within the bubble itself.
<li>A Horizon will naturally emit Hawking radiation. It is estimated that the radiation will raise the temperature of the interior of the bubble to approximately 1032 K<ref name="Finazzi 2009">Stefano Finazzi, Stefano Liberati, and Carlos Barceló, "Semiclassical instability of dynamical warp drives", Phys. Rev. D 79, 124017 (12 June, 2009) https://doi.org/10.1103/PhysRevD.79.124017 , arXiv:0904.0141 [gr-qc] https://doi.org/10.48550/arXiv.0904.0141</ref> (at this level of accuracy, the result is the same if you replace temperatures in K by the same number in °C or °F). This temperature, however, may be lower if quantum inequalities do not have to be fulfilled.
<li>Semi-classical instabilities at the horizon are expected to lead to a divergence of the stress-energy tensor at the boundary in a short period of time, thus leading to the destruction of the warp bubble<ref name="Finazzi 2009"></ref>.
</ol>
Helpfully, Van Den Broeck<ref name="VanDenBroeck_2000">C. Van Den Broeck, "Alcubierre’s Warp Drive: Problems and Prospects" AIP Conference Proceedings. 504: 1105–1110 (2000). Bibcode:[https://ui.adsabs.harvard.edu/abs/2000AIPC..504.1105V 2000AIPC..504.1105V]. doi:[https://doi.org/10.1063%2F1.1290913 10.1063/1.1290913].</ref> suggested several solutions – or at least mitigations – for some of these problems
<ol>
<li>Quantum inequalities had not been shown to be true in general for highly curved space-times.
<li>A proposed warp drive geometry<ref name="VanDenBroeck_1999">C. Van Den Broeck, "A 'warp drive' with more reasonable total energy requirements". Classical and Quantum Gravity. 16 (12): 3973–3979 (1999). arXiv:[https://arxiv.org/abs/gr-qc/9905084 gr-qc/9905084]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1999CQGra..16.3973V 1999CQGra..16.3973V]. doi:[https://doi.org/10.1088%2F0264-9381%2F16%2F12%2F314 10.1088/0264-9381/16/12/314]. S2CID [https://api.semanticscholar.org/CorpusID:15466313 15466313]. </ref> (see the van Den Broeck drive, below) would be able to minimize the magnitude of the energy required to about that of the mass of our sun. While still large, it is not unphysically large.
<li>Time travel is always going to be a worry with faster than light travel. There's no neat solution to this one.
<li>It may be possible for the negative energy outside of the superluminal region to compress into the superluminal region to form a shock in space-time. The front surface of the bubble would then be a singularity. It is not clear if such a sharp jump in physical properties is possible, but neither is sure that it is impossible, either. However, while this might help with the negative energy in front of the bubble getting swept up because it cannot move faster than light, it does little to help with the material at the back of the bubble on the outside getting left behind because it cannot keep up.
</ol>
Other mitigating research has also shown
<ol start=9>
<li>Semi-classical divergence only occurs in warp drives with only one spatial dimensions. Those with two or more spatial dimensions (as in our universe) only contain points where divergences occur, and these divergences can be managed. In fact, their effects can be reduced by making the warp shape more elongated<ref>Carlos Barceló, Valentin Boyanov, Luis J. Garay, Eduardo Martín-Martínez,
and Jose M. Sánchez Velázquez, "Warp drive aerodynamics", JHEP08 (2022) 288 https://doi.org/10.1007/JHEP08(2022)288 , arXiv:2207.06458 [gr-qc] https://doi.org/10.48550/arXiv.2207.06458</ref>.
</ol>

(And what's with the ring in all the artwork, anyway?)

=== So what is going on? ===

What is it that makes a warp drive a warp drive, and how does it work?

A warp drive is a concept that comes out of the science of general relativity. General relativity collects space and time together into a single "structure" called spacetime. This structure is curved, and its curvature is determined by the material that is in it. Absent forces acting between matter, particles of matter move through spacetime on the shortest possible paths in this curved environment.

"Spacetime tells matter how to move; matter tells spacetime how to curve." - John Archibald Wheeler

All of this is encompassed in the Einstein field equation:
<div align=center>
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 8πG
<td rowspan="2" nowrap="nowrap" align="right"> T = R -  
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right"> R g
<tr>
<td style="border-top:solid 1px black;"> c4
<td style="border-top:solid 1px black;"> 2
</table>
</div>
Unfortunately, by convention, both the symbols G and R get used for a lot of different things. Here, symbols in bold are second rank tensors – constructs of arrays of 4×4 numbers where 4 is necessary because that is the number of dimensions of spacetime. Conventionally, these numbers are indexed by a value going from 0 to 3, with 0 being the part corresponding to the coordinate of time and 1 to 3 being the Cartesian coordinates that you might know as x, y, and z.

T is the stress-energy tensor: the (0,0) component represents the energy density at a given point; the (0,i) and (i, 0) components (with i going from 1 to 3) represent the momentum density; and the (i,j) components represent the stresses present, such as pressure or tension or shear.

The other term, R - R g, involves a partial description of the curvature. A full description of the curvature requires a construct of 4×4×4×4 components; too much for the stress energy tensor to fully constrain. Which is okay, because this allows curvature to exist away from where matter is, allowing gravity to have long range effects across empty space as well as for phenomena like gravitational waves. But it is enough that a given distribution of matter T gives enough constraints (along with specifying how the geometry behaves far away) that you can generally figure out what the actual gravity is.

Describing spacetime - lapse function & shift vectors (foliations, etc.)
Diagram of what the shift vector does.

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Lapse_shift.png]]
<tr><td>A folliation of space-time into slices of constant time coordinates (grey). At an event (black circle), the lapse function (yellow) shows how far into the future the point at that event will be projected in a unit amount of proper time experienced by an observer initially at the event. The shift vector (red) shows how that point changes coordinates after the unit amount of proper time. The cyan vector shows the new coordinates the point is projected onto. The lower black surface indicates a foliation of spacetime at the same time coordinate; after unit proper time, the events on the lower foliation get projected to the curved upper black foliation according to the shift vectors and lapse functions at each of their respective points.
</table>
</table>

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Warp_shift.png]]
<tr><td>The shift vector of a warp geometry, in red. At the edges there is no shift. At the center (inside the warp bubble) there is a region of uniform shift so that at a later time the inside of the warp bubble is rigidly translated to different spatial coordinates (in blue). The two foliations remain the same distance apart, because warp drives maintain a unit lapse function.
</table>
</table>

Solve backwards - assume a structure of space-time, figure out what distribution of matter is required for this from the EFE.

What sort of spacetime geometry produces a warp?

Unit lapse function <-> no time dilation

Region of near constant shift vector - that region is "moving" spacetime, the warp bubble.

Shift vector falls off to zero away from warp bubble -> normal flat spacetime.

=== Alcubierre warp interactions with light and matter ===

Space is not entirely empty. It is filled with a diffuse plasma in the form of the interstellar medium, as well as cosmic radiation, light from stars, and cosmic microwave background radiation. A warp drive propagating through space will encounter this stuff. What happens when this matter and radiation have a warp bubble pass across them?

The first analysis of matter encountering a warp bubble was performed by Pfenning and Ford<ref name="PfenningFord_2001"></ref>. They looked at the warp bubble interaction with a massive object at rest with the frame of reference of the warp drive (keep in mind that the rest frame is the same inside and outside the warp bubble; in this rest frame objects inside the bubble have no momentum even though they are, in some sense, changing location rapidly with time). When the warp bubble passes, the object experiences an acceleration in the direction of the warp bubble motion. When the warp shell passes and the object is inside the bubble, it will be moving with approximately the speed of the bubble. Pfenning and Ford analyzed this problem with a continuous distribution of shell energy that, strictly speaking, never falls to zero except at the bubble center and at spatial infinity, so unless the object passes through the center in Pfenning and Ford's description it will never quite get up to the bubble's speed. In this case, the object will move almost as fast as the object but will pass through the bubble in a finite time, after which it will again be at rest with respect to the reference frame but displaced along the direction of the warp bubble motion by some distance. With this description of the warp bubble, a spacecraft of finite size will always be moving a little slower than the warp bubble and would have to use rockets to keep up with it. in addition, the spacecraft would experience tidal forces that would cause stress on the spacecraft's structure.

Pfenning and Ford also examined cases where the shell is of a finite (possibly infinitesimal) width and falls to zero both inside and outside the bubble. In this case any matter encountering the bubble would be collected at the bow of the bubble and thereafter move along with it.

McMonigal et al. analyzed the situation for both massive particles and light moving along the axis of travel of the bubble<ref name="McMonigalEtAl_2012">B. McMonigal, G. F. Lewis, and P. O'Byrne, "Alcubierre warp drive: On the matter of matter". Physical Review D. 85 (6) 064024 (20 March 2012). arXiv:[https://arxiv.org/abs/1202.5708 1202.5708]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2012PhRvD..85f4024M 2012PhRvD..85f4024M]. doi:[https://doi.org/10.1103%2FPhysRevD.85.064024 10.1103/PhysRevD.85.064024]. S2CID [https://api.semanticscholar.org/CorpusID:3993148 3993148].</ref>. They found that light moving opposite the warp direction passed through the bubble without incident, being only somewhat delayed by passing through the bubble. A warp bubble that is warping at sub-luminal speeds can have light catch up from behind it. This light is able to pass through the bubble, and is somewhat advanced in its path by the speed of the bubble. For super-luminal warp bubbles, however, the situation is different. The bubble will catch up to light moving it its own direction that is originally in front of it. This light cannot escape forward, the bubble being too fast. Nor can it escape backward, as the light is propagating forward and the interior of the bubble is at rest. Thus, the light gets caught at the bow of the warp bubble, unable to escape for so long as the warp bubble is active. This light is strongly blue shifted to extremely energetic x-rays and gamma rays. The space behind the bubble is swept clear of forward-moving light.

The situation for matter over-run by a super-luminal warp bubble is similar. Matter moving backward passes through the bubble, being only swept a ways along the bubble's path as sign of its passing but otherwise continuing on their way. Matter at rest behaves the same way as Pfenning and Ford discovered. Meanwhile, matter moving in the same direction of the warp is overtaken and collects at the bow of the bubble, unable to leave for so long as the bubble is warping. This matter "surfing" on the bubble bow is highly accelerated to relativistic speeds, experiencing extreme time dilation.

For a sub-luminal bubble, there is a speed faster than the warp speed where particles coming up from behind overtake the bubble and pass through, out the front. Particles below this speed but still faster than the warp speed still overtake the bubble, but then bounce backwards out of the bubble from behind. They are still moving forward, but are now moving more slowly than the warp speed. Particles moving forward but slower than the bubble will be overtaken and then bounced out the front with a speed higher than the warp speed. Those particles moving backward will pass through the bubble and exit with their initial speed.

The spacecraft inside the bubble will observe that light moving backward compared to the warp direction is blue-shifted, and matter moving backward passes through with increased energy. Meanwhile, a sub-luminal bubble will see light that was moving forward as red-shifted and matter catching up to the bubble from behind will pass by with reduced energy (superluminal bubbles, of course, do not have any matter or radiation catching up to them from behind).

When a superluminal warp bubble that has been collecting matter and radiation for the duration of its journey and blue-shifting it to much higher energies is turned off, all that matter and energy is released as a blast of radiation in the direction the bubble was warping.

McMonigal et al. conclude by noting that any spacecraft in the warp bubble would need shielding to protect against the blue-shifted radiation and matter of increased energy. In addition the destination would be "blasted into oblivion" by the release of matter and radiation that had been caught in the bubble during the trip.

Clark et al.<ref>Chad Clark, William A. Hiscock, and Shane L. Larson, "Null geodesics in the Alcubierre warp drive spacetime: the view from the bridge", arXiv:gr-qc/9907019 (2018) https://doi.org/10.48550/arXiv.gr-qc/9907019</ref> examine what it would look like from within an operating warp drive. They find that light moving at 90 degrees to the direction of motion will look like it is still coming from 90 degrees. However, light from the front hemisphere is squished progressively toward the forward point the faster you go. For a warp bubble moving faster than the speed of light, there is a cone behind the warp bubble where no light can reach it. However, you don't see a black spot behind you – light from the regions in the backward hemisphere from outside that cone that can reach you is distorted toward the direction behind you to fill in that spot. Light from the forward direction is blue-shifted and light from behind is red-shifted. At speeds of around 200 times light speed, the cosmic microwave background would be blue shifted to energies similar to that of the solar photosphere, heating anything in the warp bubble as if they were close to the sun.

== Van Den Broeck warp drive ==

If an Alcubierre warp bubble a hundred meters across requires a magnitude of energy greater than the entire energy in the observable universe, one option to reduce the magnitude of energy used is to make the warp bubble smaller. Much smaller. Pfenning and Ford<ref name="PfenningFord_2001"></ref> suggested making the warp bubble smaller than an atom. Of course, that brings up the problem of how to stuff a spacecraft in there. Van Den Broeck proposed a solution<ref name="VanDenBroeck_1999"></ref>: make the warp bubble only about the size of an atomic nucleus but expand the space inside the bubble enormously so that a spacecraft could fit in. This is somewhat like a nuclear sized wormhole that leads to a pocket universe that holds the spacecraft. Unfortunately, the metric doesn't fit neatly into any [[Wormholes#Three_dimensions|embedding diagram]] that I can figure out, but the math works out so that spatial coordinates inside the bubble are expanded by a factor of 1017 and a spacecraft would have a few hundred meters of bubble interior to putz around in. This trick manages to reduce the magnitude of energy to only about the mass energy of a few stars similar to our own. Other than that, it is otherwise a normal Alcubierre drive.

Further work<ref>S. Krasnikov, "The quantum inequalities do not forbid spacetime shortcuts", Physical Review D. 67 (10) 104013. https://doi.org/10.1103%2FPhysRevD.67.104013, arXiv:gr-qc/0207057 (2003) https://doi.org/10.48550/arXiv.gr-qc/0207057 </ref> has managed to use Van Den Broeck's trick to reduce the magnitude of the negative energy to only about a milligram.

== Natário warp drive ==

The Alcubierre drive is not the only way to construct a warp drive. José Natário<ref name="Natario"></ref> showed that it was just one example of an entire class of warp drives. These are space-time geometries with a smooth and localized shift vector field a shift vector is a vector field that describes how one "slice" of space out of space-time connects to the next) and a unit lapse function (the lapse function describes how the time you experience changes when going from a "slice" of space-time to the next).

Natário then went on to find in that class a set of warp drives that do not have any expansion or contraction of space at all. Instead, space encountering the front boundary of the warp bubble instead "slides" around the outside of the bubble until it gets to the corresponding place on the back and is left behind there. Another way to think of it is that space entering the bubble shell is compressed in the radial direction of the shell but is simultaneously expanded in the tangential direction so that there is no net change in volume. When it gets to the back, the opposite occurs. In the language of mathematics, the shift vector field is divergenceless.

<table align=center>
<tr><td width=360>[[File:Natario_drive_flow_of_space.png]]
<td width=445>[[File:Natario_warp_drive_horizons.png]]
<tr><td style="vertical-align: top;">A representation of the motion of space in a zero-expansion Natário drive with a vanishingly thin warp shell around the bubble.
<td>The horizons of the generalized Natário class of warp drives (including the Alcubierre warp drive and all fell-Heisenberg drives) at superluminal speed in the direction of the green arrow. No event inside the bubble can affect events outside the bubble in front of the blue line. No events outside the bubble behind the red line can affect events inside the bubble. The lines have an angle α with respect to the direction of motion with sin(α) = 1/vs for vs the speed of the bubble relative to light speed. This is analogous to the Mach cone of supersonic objects.
</table>

NOTE: There are two different things that can be meant by the term "Natário warp drive". The generalized Natário warp drive is the general theory of all warp drives that includes within it the Alcubierre, Fell-Heisenberg, and Lentz warp drives, among infinitely many others. The zero-expansion Natário warp drive is the specific implementation of the generalized Natário warp drive that has a divergenceless shift vector field.

The generalized class of warp drives developed by Natário (including both the Alcubierre and this zero-expansion warp drive) are shown to always have regions in the warp shell where the energy density is negative to at least some observers. You can't get away from it – to warp, you need negative energy density.

The Natário class of warp drives blueshift light coming in from the front and redshift light catching up from the back. For the case of a super-luminal warp bubble, there will of course be a horizon at the back of the bubble and light will not be able to get through from behind. For a bubble speed vs relative to the speed of light, light coming from straight ahead will be blueshifted up in frequency and photon energy by a factor of 1 + vs. For sub-luminal travel, light coming from behind will be redshifted by a factor 1 - vs. In general for n as the unit direction along which the light is propagating, the blueshift factor will be 1 + vs ⋅ n. If the warp shell is infinitesimally thin, light from outside that reaches the center of the bubble will not be distorted in direction although it will be frequency shifted, but observers still inside the bubble but displaced from the center will see the field of view distorted as well as frequency shifted.

Most of the difficulties that apply to the Alcubierre warp drive also apply to the generalized Natário warp drive (and thus to the zero-expansion Natário drive and Fell-Heisenberg drive as well). In particular, quantum energy inequalities, high magnitudes of the energy required, formation of time machines, instabilities related to the stress-energy making up the warp shell being unable to keep up with the bubble, the chicken-and-egg problem of needing FTL to achieve FTL, the formation of horizons making warp bubbles uncontrollable, and Hawking radiation at the horizons, are all still concerns for generic warp drives.

== Fell-Heisenberg warp drives ==

S. D. B. Fell and L. Heisenberg investigated what sort of warp structure is necessary to produce a warp drive with positive energy density everywhere<ref name="Fell-Heisenberg">S. D. B. Fell and L> Heisenberg, "Positive energy warp drive from hidden geometric structures", Classical and Quantum Gravity 38 155020 (2021) https://doi.org/10.1088/1361-6382/ac0e47 https://arxiv.org/abs/2104.06488</ref>. They found that warp drives of the generic Natário form with irrotational shift vectors had a simple expression for the energy density and vanishing momentum (irrotational means that the shift vector can be described as the gradient of a scalar potential field or, equivalently, that the curl of the shift vector field vanishes everywhere). Fell and Heisenberg proposed several potential fields that either had positive energy density everywhere except at discontinuities, or that were continuous but had regions that violated the weak energy condition (i. e. negative energy densities in some frames of reference). One specific example they provided of a warp field had an energy that is about 10,000 times less than the mass-energy of our sun. Or only about half the mass-energy of Jupiter. This is a significant improvement over many other proposed drives.

Despite the introduction discussing warp drive configurations that satisfy the various energy conditions, the configurations described in the paper are shown to locally violate the weak and strong energy conditions. Nonetheless, the energy density is still mostly positive. Santiago, Schuster, and Visser's<ref name="Santiago Schuster Visser 2022">J. Santiago, S. Schuster, and M. Visser, "Generic warp drives violate the null energy condition", Physical Review D 105, 064038 (2022) https://doi.org/10.1103/PhysRevD.105.064038</ref> work shows that the various energy conditions must still be violated by this warp drive; to not violate these, energy density must be positive in all reference frames not just those of the co-moving observer.

Any vector field can be described by its divergence, its curl, and a constant. The constant part of a warp drive's shift vector can be discarded because then the warp drive would not be localized. It is interesting, therefore, that the Natário zero-expansion drive has zero divergence and can be described only by its curl, while the Fell-Heisenberg drive has zero curl and can be described only by its divergence. These two classes of drives thus rely on opposite choices of setting one of the two different parts describing a vector field to zero. As might be expected, the Fell-Heisenberg drive is not zero expansion – space-time will change in volume as it passes through the warp bubble (although after going all the way through it will return to its original volume). However, objects in space-time will retain their original orientation as they pass through the warp bubble.

Previous examples of warp drives, such as the Alcubierre and zero-expansion Natário drives, tend toward a flat space-time as you go far enough away faster than would occur if there were any mass present in the warp drive. That is, the ADM mass of such drives is zero. However, one of the Fell-Heisenberg examples described a spacetime where the geometry approaches that of what you would get for a massive object at the warp location, giving one of the first examples of a warp drive with mass.

In the Fell-Heisenberg drive the momentum vanishes everywhere. Yet the energy occupies regions where the shift vector is varying rapidly. The lack of momentum means that the energy will not move to keep up with the differential expansion and movement of the spacetime. As a consequence, at later times the energy will have a different distribution than what is necessary to maintain the given warp configuration. The exact time evolution is not solved but this likely leads to collapse of warp bubble. Note that this is not a problem unique to the Fell-Heisenberg drive. Assuming a geometry that describes a warp produces a specific distribution of the stress-energy tensor that satisfies Einstein's equation. But there is no guarantee that this stress energy tensor satisfies the continuity equation, which describes how the distribution of matter and momentum change with time. The issue of stability of any warp drive is not well understood, but most warp drive geometries will be unstable in the sense that their matter that generates the warp fields will not be able to keep up with the motion of the warp. The Fell-Heisenberg drive is unique only in that, with vanishing momentum, there is no configuration that allows stable propagation.

(n. b. The Heisenberg here is Lavinia Heisenberg, not the Werner Heisenberg of quantum physics and uncertainty principle fame.)

== Lentz warp drive ==

E. Lentz<ref name="Lentz">E. W. Lentz, "Breaking the warp barrier: hyper-fast solitons in Einstein–Maxwell-plasma theory", Classical and Quantum Gravity. 38 075015 (2021). arXiv:[https://arxiv.org/abs/2006.07125 2006.07125]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2021CQGra..38g5015L 2021CQGra..38g5015L]. doi:[https://doi.org/10.1088%2F1361-6382%2Fabe692 10.1088/1361-6382/abe692]. ISSN [https://search.worldcat.org/issn/0264-9381 0264-9381]. S2CID [https://api.semanticscholar.org/CorpusID:219635854 219635854].</ref> investigated a particular configuration of the Fell-Heisenberg class of warp drives, where the scalar field has a hyperbolic shape. The necessary conditions were found to be satisfied by a selection of rhomboid regions of the scalar field, resulting in an energy density that is everywhere positive and a shift vector with vanishing tidal force at the center for a comfortable ride.

However, Santiago, Schuster, and Visser<ref name="Santiago Schuster Visser 2022"></ref> dispute claims that the Lentz drive satisfies the energy conditions, noting that everywhere positive energy density in one frame of reference is insufficient to establish that the energy density is positive in all reference frames; knowledge of the Cauchy stress tensor is also needed. They show that any generic warp drive will violate the strong energy condition, null energy condition, and weak energy condition.

== Conservation laws ==

(Discussion of asymptotic flatness: what is it? Where does it apply?)

(The "big four" conservation laws and how they relate to asymptotic flatness.)

(ADM mass -> 0; conservation of energy/mass issues.)<ref>Sebastian Schuster, Jessica Santiago,and Matt Visser, "ADM mass in warp drive spacetimes" General Relativity and Gravitation (2023) 55:14 https://doi.org/10.1007/s10714-022-03061-9</ref>

(Discuss issues of conservation of angular momentum.)

(If angular momentum conservation is ignored, discuss how momentum & energy are affected by outside forces. In a gravitational field equivalent to inertial frame ... like being in an accelerated elevator; will acquire momentum buildup while staying "at rest".)

== Warp Railroads ==

Van Den Broeck<ref name="VanDenBroeck_2000"></ref> suggested a warp railroad,

You can set up devices ahead of time along the path of the bubble that produce the negative energy regions for the warp bubble at the appropriate time without needing the negative energy to ever move with the bubble at all. At least for bubbles on a predictable schedule and route.

helps with:

-stability (elaborate)

-conservation laws (elaborate)

Mention Krasnikov<ref name="Krasnikov 1998">S. V. Krasnikov, "Hyperfast travel in general relativity" Phys. Rev. D 57 4760</ref>

Hyperwave<ref name="Pieri 2024">Lorenzo Pieri, "Hyperwave: hyper-Fast Communication within General Relativity", arXiv:2311.12069 [gr-qc] https://doi.org/10.48550/arXiv.2311.12069</ref> makes use of this warp railroad idea.

== Quantum effects ==

(quantum stuff here<ref name="Hiscock1997">W. A. Hiscock, "Quantum effects in the Alcubierre warp drive spacetime", Classical and Quantum Gravity 14 L183 https://doi.org/10.1088/0264-9381/14/11/002 https://arxiv.org/abs/gr-qc/9707024</ref><ref name="Finazzi et al 2009">S. Finazzi, S. Liberati, C. Barceló, "Semiclassical instability of dynamical warp drives", Physical Review D 79, 124017 (2009)https://doi.org/10.1103/PhysRevD.79.124017 https://arxiv.org/abs/0904.0141</ref>)

From<ref name="Pieri 2024"></ref>
<blockquote>
It should be emphasised that Quantum Inequalities have not been proven for realistic interacting quantum fields, where the they are expected to be weaker if they exist [Cadamuro, 2019], and they often require Minkowskian conditions at infinity, a condition which is not appropriate for fields inside bounded regions, as in a Casimir devices.
</blockquote>

== Warp drives and black holes ==

What happens if you drive your warp drive into a black hole?

The pop-sci reason that nothing can escape a black hole is that at the event horizon the escape velocity is faster than light and nothing can go faster than light. A warp drive can go faster than light.

The more detailed reason from general relativity is that at the event horizon space and time are rotated sufficiently that "inward" becomes "forward in time." You can no more go farther away from the singularity in a black hole once you are inside the event horizon than you can go backward in time. Any super-luminal travel can go backward in time, and warp drives can engage in super-luminal travel.

The Hawking area theorem that says that black holes can only grow, never shrink relies on the condition that the energy is everywhere positive. Warp drives have negative energy density.

So in principle, there's nothing that prevents a warp drive from taking a dip into a black hole and coming back out again.

Garattini and Zatrimaylov looked into the problem of a warp drive near (and in) a black hole<ref name="Garattini and Zatrimaylov">R. Garattini and K. Zatrimaylov, "Black holes, warp drives, and energy conditions", Physics Letters B 856 138910 (2024) https://doi.org/10.1016/j.physletb.2024.138910</ref>. The found that the black hole reduces the negative energy requirement for the warp bubble to move inward toward the black hole singularity, but increases the negative energy needed to move away from the center of the black hole. In addition, a warp drive parked at the event horizon would allow light from inside the horizon to pass through the bubble and escape back outside.

== Credit ==
Author: Luke Campbell

== References ==

[[Category:Physics]][[Category:Infrastructure]][[Category:Physics & Engineering‏‎]][[Category:Physics & Math & Engineering]][[Category:Transportation & Infrastructure‏‎]][[Category:Metric Engineering]]

Warp Drives

2026-07-26T23:07:20Z

Lwcamp: /* So what is going on? */

{{PageConstructionNotice}}

Science fiction often features spacecraft that can seemingly move across space and get between the place of departure and the destination much faster than light could have done. This appears to contradict the theory of relativity, which predicts unequivocally that nothing can move through space faster than light. Because relativity has been incredibly successful at describing nature, with its many other predictions regularly being confirmed to extraordinary accuracy and within the bounds of uncertainty of all the experiments that tested them, it gives confidence that relativity is a correct description of reality. Which seems to rather throw a wet towel on our hopes for rapid travel between stars.

However, while relativity does not allow things to move through space faster than light, it places no such restrictions on how fast space-time itself can expand, contract, or move around. This leads to the idea of a warp drive – the spacecraft remains stationary within a region of highly curved space-time, and that region moves at super-luminal speeds rather than the spacecraft.

== The Alcubierre warp drive ==

The first warp drive geometry that satisfied the Einstein field equations of relativity was proposed by Miguel Alcubierre<ref>M. Alcubierre, "The warp drive: hyper-fast travel within general relativity." Classical and Quantum Gravity. 11 (5): L73–L77 (1994). [https://arxiv.org/abs/gr-qc/0009013 arXiv:gr-qc/0009013]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1994CQGra..11L..73A 1994CQGra..11L..73A]. doi:[https://doi.org/10.1088%2F0264-9381%2F11%2F5%2F001 10.1088/0264-9381/11/5/001]. S2CID [https://api.semanticscholar.org/CorpusID:4797900 4797900].</ref>. In this geometry, a sphere of space-time moves at an arbitrary speed (potentially but not necessarily a speed much faster than light). Objects within the sphere are moved along with the sphere; an object at rest within the sphere would be moved along with the sphere indefinitely. Space is expanding at the rear boundary and contracting at the front boundary in order to keep the sphere moving. In order to satisfy the Einstein field equations, the boundary of the sphere must have a negative energy density. The challenges of space-time geometries with negative energy densities are described in our page on [[Wormholes#Exotic_energy_conditions|wormholes]], for our purposes it is enough to note that negative energy densities can pose problems if not handled carefully, there may be limits on how much negative energy you can have without nearby positive energy density, and it may not be possible to get enough negative energy to support a warp drive; although none of this is ruled out by physics – yet!

<table align=center>
<tr><td width=650>[[File:Energy_magnitude_of_Alcubierre_drive.png]]
<td width=650>[[File:Expansion_of_Alcubierre_drive.png]]
<tr><td>The magnitude of the energy of the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. (The energy density is negative everywhere in the shell, here only the magnitude is shown to aid visual comprehension.)
<td>The expansion of space in the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. Negative expansion means that space is contracting in that direction, positive expansion that it is expanding.
</table>

NOTE: What the? The energy distribution is symmetric! How does the drive know which way to go? There must be significant contributions of other components of the stress-energy tensor, and those have got to be asymmetric along the forward/backward axis. Check this when I get time.

Almost as soon as Alcubierre proposed his warp drive, others began picking it apart.
<ol>
<li>Pfenning and Ford<ref name="PfenningFord_2001">M. J. Pfenning and L. H. Ford, "The unphysical nature of 'Warp Drive'", Classical and Quantum Gravity. 14 (7): 1743–1751 (1997). arXiv:[https://arxiv.org/abs/gr-qc/9702026 gr-qc/9702026]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1997CQGra..14.1743P 1997CQGra..14.1743P]. doi:[https://doi.org/10.1088%2F0264-9381%2F14%2F7%2F011 10.1088/0264-9381/14/7/011]. S2CID [https://api.semanticscholar.org/CorpusID:15279207 15279207].</ref> found that if the negative energy density wall around the bubble obeys [[Wormholes#Quantum_energy_inequalities|quantum energy inequalities]], then for warp speeds of around light speed the shell thickness would be on the order of a hundred Planck lengths; a distance far smaller than any other known physical phenomenon or object.
<li>In the same work, Pfenning and Ford also found that for a bubble a hundred meters in radius, a warp speed of around the speed of light, and the required thickness of about 100 Planck lengths the energy in the bubble shell would be E ≈ -1063 kg c2. The magnitude of this latter value is ten orders of magnitude larger than the energy of the entire visible universe. They do note, however, that if quantum energy inequalities can be ignored then a warp drive with a hundred meters radius but a shell thickness of one meter would "only" have an energy magnitude of about a quarter solar mass.
<li>As with any method of faster than light travel, the warp drive could be used to make a time machine.
<li>Perhaps most seriously, if the warp drive is going faster than light speed, the negative energy regions on the outside of the shell won't be in the warp parts of the bubble. They will have to be moving through space at faster than light speed if the bubble is to maintain its integrity, which is the very problem that the warp drive was designed to avoid. The outside of the warp bubble shell that maintains the warp bubble would fall away and the warp bubble would quickly erode to sub-luminal speeds.<ref name="Van Den Broeck 2000">Chris Van Den Broeck, "On the (im)possibility of warp bubbles", arXiv:gr-qc/9906050 https://doi.org/10.48550/arXiv.gr-qc/9906050</ref>
<li>A more general problem that includes the previous one is that the geometry of the warp drive determines the distribution of energy, momentum, and stress (pressure, shear, and tension) by Einstein's field equation, which states that the stress energy tensor is directly proportional to a reduced order curvature tensor of space-time. However, the stress-energy tensor must also obey the continuity equation – basically, for the energy to change in a given volume of space that energy must come in or go out of the boundary of that volume and similarly for the momentum. So a warp drive can have the warp condition for one moment in time. But after that, the stress-energy that maintains the warp will start to move according to the forces on it and according to the natural paths that it takes through the warped space-time. There is no guarantee that the configuration of the stress-energy at later times will be sufficient to maintain the warp.
<blockquote>
From the perspective of simulating the warp drive dynamically, the key challenge is stability. One can straightforwardly generate the initial metric that describes the Alcubierre metric and infer the instantaneous matter configuration that must support it using the Einstein Equation, but a time-domain evolution of the coupled matter-gravity system necessitates that we specify the equation of state of the matter, which in turn determines how it evolves. In a perfect fluid, this involves specifying the relation between the fluid pressure and density, but in a more general case, the full relation between the
stress-energy components must be given. There is (to our knowledge) no known equation of state that would maintain the warp drive metric in a stable configuration over time - therefore, whilst one can require that initially, the warp bubble is constant, it will quickly evolve away from that state and, in most cases, the warp fluid and spacetime deformations will disperse or collapse into a central point.
 
– Clough, Dietrich, and Khan<ref>Katy Clough, Tim Dietrich, and Sebastian Khan, "What no one has seen before: gravitational waveforms from warp drive collapse", arXiv:2406.02466 [gr-qc] https://doi.org/10.48550/arXiv.2406.02466</ref>
</blockquote>
<li>In order to get the matter (or more generally, stress-energy) that forms the warp shell up to speeds that could allow the shell to exist, you need to be able to, well, accelerate that matter to warp speeds. Which gives a chicken-and-egg problem - in order to achieve superluminal travel you need to be able to get matter or energy moving superluminally in the first place<ref name="Bobrick and Martire">Alexey Bobrick and Gianni Martire, "Introducing Physical Warp Drives", arXiv:2102.06824 [gr-qc] https://doi.org/10.48550/arXiv.2102.06824</ref><ref>D. H. Coule, "No warp drive", Class. Quantum Grav. 15 (1998) 2523–2527. https://doi.org/10.1088/0264-9381/15/8/026</ref>, which rather makes warp drives unnecessary from the get-go.
<li>A warp bubble moving faster than light will not be able to get any signals in front of it as it will overtake its own signal. In fact, anyone in the interior of the bubble would not be able to send signals to the front surface of the bubble. In science-speak, a horizon forms at the front of the bubble<ref name="Natario">José Natário, "Warp drive with zero expansion", Classical and Quantum Gravity. 19 (6): 1157–1166 (2002). arXiv:[https://arxiv.org/abs/gr-qc/0110086 gr-qc/0110086]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2002CQGra..19.1157N 2002CQGra..19.1157N]. doi:[https://doi.org/10.1088%2F0264-9381%2F19%2F6%2F308 10.1088/0264-9381/19/6/308]. S2CID [https://api.semanticscholar.org/CorpusID:15859984 15859984].</ref>. As a consequence, the bubble cannot be controlled, steered, nor stopped from within the bubble itself.
<li>A Horizon will naturally emit Hawking radiation. It is estimated that the radiation will raise the temperature of the interior of the bubble to approximately 1032 K<ref name="Finazzi 2009">Stefano Finazzi, Stefano Liberati, and Carlos Barceló, "Semiclassical instability of dynamical warp drives", Phys. Rev. D 79, 124017 (12 June, 2009) https://doi.org/10.1103/PhysRevD.79.124017 , arXiv:0904.0141 [gr-qc] https://doi.org/10.48550/arXiv.0904.0141</ref> (at this level of accuracy, the result is the same if you replace temperatures in K by the same number in °C or °F). This temperature, however, may be lower if quantum inequalities do not have to be fulfilled.
<li>Semi-classical instabilities at the horizon are expected to lead to a divergence of the stress-energy tensor at the boundary in a short period of time, thus leading to the destruction of the warp bubble<ref name="Finazzi 2009"></ref>.
</ol>
Helpfully, Van Den Broeck<ref name="VanDenBroeck_2000">C. Van Den Broeck, "Alcubierre’s Warp Drive: Problems and Prospects" AIP Conference Proceedings. 504: 1105–1110 (2000). Bibcode:[https://ui.adsabs.harvard.edu/abs/2000AIPC..504.1105V 2000AIPC..504.1105V]. doi:[https://doi.org/10.1063%2F1.1290913 10.1063/1.1290913].</ref> suggested several solutions – or at least mitigations – for some of these problems
<ol>
<li>Quantum inequalities had not been shown to be true in general for highly curved space-times.
<li>A proposed warp drive geometry<ref name="VanDenBroeck_1999">C. Van Den Broeck, "A 'warp drive' with more reasonable total energy requirements". Classical and Quantum Gravity. 16 (12): 3973–3979 (1999). arXiv:[https://arxiv.org/abs/gr-qc/9905084 gr-qc/9905084]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1999CQGra..16.3973V 1999CQGra..16.3973V]. doi:[https://doi.org/10.1088%2F0264-9381%2F16%2F12%2F314 10.1088/0264-9381/16/12/314]. S2CID [https://api.semanticscholar.org/CorpusID:15466313 15466313]. </ref> (see the van Den Broeck drive, below) would be able to minimize the magnitude of the energy required to about that of the mass of our sun. While still large, it is not unphysically large.
<li>Time travel is always going to be a worry with faster than light travel. There's no neat solution to this one.
<li>It may be possible for the negative energy outside of the superluminal region to compress into the superluminal region to form a shock in space-time. The front surface of the bubble would then be a singularity. It is not clear if such a sharp jump in physical properties is possible, but neither is sure that it is impossible, either. However, while this might help with the negative energy in front of the bubble getting swept up because it cannot move faster than light, it does little to help with the material at the back of the bubble on the outside getting left behind because it cannot keep up.
</ol>
Other mitigating research has also shown
<ol start=9>
<li>Semi-classical divergence only occurs in warp drives with only one spatial dimensions. Those with two or more spatial dimensions (as in our universe) only contain points where divergences occur, and these divergences can be managed. In fact, their effects can be reduced by making the warp shape more elongated<ref>Carlos Barceló, Valentin Boyanov, Luis J. Garay, Eduardo Martín-Martínez,
and Jose M. Sánchez Velázquez, "Warp drive aerodynamics", JHEP08 (2022) 288 https://doi.org/10.1007/JHEP08(2022)288 , arXiv:2207.06458 [gr-qc] https://doi.org/10.48550/arXiv.2207.06458</ref>.
</ol>

(And what's with the ring in all the artwork, anyway?)

=== So what is going on? ===

What is it that makes a warp drive a warp drive, and how does it work?

A warp drive is a concept that comes out of the science of general relativity. General relativity collects space and time together into a single "structure" called spacetime. This structure is curved, and its curvature is determined by the material that is in it. Absent forces acting between matter, particles of matter move through spacetime on the shortest possible paths in this curved environment.

"Spacetime tells matter how to move; matter tells spacetime how to curve." - John Archibald Wheeler

All of this is encompassed in the Einstein field equation:
<div align=center>
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 8πG
<td rowspan="2" nowrap="nowrap" align="right"> T = R -  
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right"> R g
<tr>
<td style="border-top:solid 1px black;"> c4
<td style="border-top:solid 1px black;"> 2
</table>
</div>
Unfortunately, by convention, both the symbols G and R get used for a lot of different things. Here, symbols in bold are second rank tensors – constructs of arrays of 4×4 numbers where 4 is necessary because that is the number of dimensions of spacetime. Conventionally, these numbers are indexed by a value going from 0 to 3, with 0 being the part corresponding to the coordinate of time and 1 to 3 being the Cartesian coordinates that you might know as x, y, and z.

T is the stress-energy tensor: the (0,0) component represents the energy density at a given point; the (0,i) components (with i going from 1 to 3) represent the momentum density; and the (i,j) components represent the stresses present, such as pressure or tension or shear.

The other term, R - R g, involves a partial description of the curvature. A full description of the curvature requires a construct of 4×4×4×4 components; too much for the stress energy tensor to fully constrain. Which is okay, because this allows curvature to exist away from where matter is, allowing gravity to have long range effects across empty space as well as for phenomena like gravitational waves. But it is enough that a given distribution of matter T gives enough constraints (along with specifying how the geometry behaves far away) that you can generally figure out what the actual gravity is.

Describing spacetime - lapse function & shift vectors (foliations, etc.)
Diagram of what the shift vector does.

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Lapse_shift.png]]
<tr><td>A folliation of space-time into slices of constant time coordinates (grey). At an event (black circle), the lapse function (yellow) shows how far into the future the point at that event will be projected in a unit amount of proper time experienced by an observer initially at the event. The shift vector (red) shows how that point changes coordinates after the unit amount of proper time. The cyan vector shows the new coordinates the point is projected onto. The lower black surface indicates a foliation of spacetime at the same time coordinate; after unit proper time, the events on the lower foliation get projected to the curved upper black foliation according to the shift vectors and lapse functions at each of their respective points.
</table>
</table>

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Warp_shift.png]]
<tr><td>The shift vector of a warp geometry, in red. At the edges there is no shift. At the center (inside the warp bubble) there is a region of uniform shift so that at a later time the inside of the warp bubble is rigidly translated to different spatial coordinates (in blue). The two foliations remain the same distance apart, because warp drives maintain a unit lapse function.
</table>
</table>

Solve backwards - assume a structure of space-time, figure out what distribution of matter is required for this from the EFE.

What sort of spacetime geometry produces a warp?

Unit lapse function <-> no time dilation

Region of near constant shift vector - that region is "moving" spacetime, the warp bubble.

Shift vector falls off to zero away from warp bubble -> normal flat spacetime.

=== Alcubierre warp interactions with light and matter ===

Space is not entirely empty. It is filled with a diffuse plasma in the form of the interstellar medium, as well as cosmic radiation, light from stars, and cosmic microwave background radiation. A warp drive propagating through space will encounter this stuff. What happens when this matter and radiation have a warp bubble pass across them?

The first analysis of matter encountering a warp bubble was performed by Pfenning and Ford<ref name="PfenningFord_2001"></ref>. They looked at the warp bubble interaction with a massive object at rest with the frame of reference of the warp drive (keep in mind that the rest frame is the same inside and outside the warp bubble; in this rest frame objects inside the bubble have no momentum even though they are, in some sense, changing location rapidly with time). When the warp bubble passes, the object experiences an acceleration in the direction of the warp bubble motion. When the warp shell passes and the object is inside the bubble, it will be moving with approximately the speed of the bubble. Pfenning and Ford analyzed this problem with a continuous distribution of shell energy that, strictly speaking, never falls to zero except at the bubble center and at spatial infinity, so unless the object passes through the center in Pfenning and Ford's description it will never quite get up to the bubble's speed. In this case, the object will move almost as fast as the object but will pass through the bubble in a finite time, after which it will again be at rest with respect to the reference frame but displaced along the direction of the warp bubble motion by some distance. With this description of the warp bubble, a spacecraft of finite size will always be moving a little slower than the warp bubble and would have to use rockets to keep up with it. in addition, the spacecraft would experience tidal forces that would cause stress on the spacecraft's structure.

Pfenning and Ford also examined cases where the shell is of a finite (possibly infinitesimal) width and falls to zero both inside and outside the bubble. In this case any matter encountering the bubble would be collected at the bow of the bubble and thereafter move along with it.

McMonigal et al. analyzed the situation for both massive particles and light moving along the axis of travel of the bubble<ref name="McMonigalEtAl_2012">B. McMonigal, G. F. Lewis, and P. O'Byrne, "Alcubierre warp drive: On the matter of matter". Physical Review D. 85 (6) 064024 (20 March 2012). arXiv:[https://arxiv.org/abs/1202.5708 1202.5708]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2012PhRvD..85f4024M 2012PhRvD..85f4024M]. doi:[https://doi.org/10.1103%2FPhysRevD.85.064024 10.1103/PhysRevD.85.064024]. S2CID [https://api.semanticscholar.org/CorpusID:3993148 3993148].</ref>. They found that light moving opposite the warp direction passed through the bubble without incident, being only somewhat delayed by passing through the bubble. A warp bubble that is warping at sub-luminal speeds can have light catch up from behind it. This light is able to pass through the bubble, and is somewhat advanced in its path by the speed of the bubble. For super-luminal warp bubbles, however, the situation is different. The bubble will catch up to light moving it its own direction that is originally in front of it. This light cannot escape forward, the bubble being too fast. Nor can it escape backward, as the light is propagating forward and the interior of the bubble is at rest. Thus, the light gets caught at the bow of the warp bubble, unable to escape for so long as the warp bubble is active. This light is strongly blue shifted to extremely energetic x-rays and gamma rays. The space behind the bubble is swept clear of forward-moving light.

The situation for matter over-run by a super-luminal warp bubble is similar. Matter moving backward passes through the bubble, being only swept a ways along the bubble's path as sign of its passing but otherwise continuing on their way. Matter at rest behaves the same way as Pfenning and Ford discovered. Meanwhile, matter moving in the same direction of the warp is overtaken and collects at the bow of the bubble, unable to leave for so long as the bubble is warping. This matter "surfing" on the bubble bow is highly accelerated to relativistic speeds, experiencing extreme time dilation.

For a sub-luminal bubble, there is a speed faster than the warp speed where particles coming up from behind overtake the bubble and pass through, out the front. Particles below this speed but still faster than the warp speed still overtake the bubble, but then bounce backwards out of the bubble from behind. They are still moving forward, but are now moving more slowly than the warp speed. Particles moving forward but slower than the bubble will be overtaken and then bounced out the front with a speed higher than the warp speed. Those particles moving backward will pass through the bubble and exit with their initial speed.

The spacecraft inside the bubble will observe that light moving backward compared to the warp direction is blue-shifted, and matter moving backward passes through with increased energy. Meanwhile, a sub-luminal bubble will see light that was moving forward as red-shifted and matter catching up to the bubble from behind will pass by with reduced energy (superluminal bubbles, of course, do not have any matter or radiation catching up to them from behind).

When a superluminal warp bubble that has been collecting matter and radiation for the duration of its journey and blue-shifting it to much higher energies is turned off, all that matter and energy is released as a blast of radiation in the direction the bubble was warping.

McMonigal et al. conclude by noting that any spacecraft in the warp bubble would need shielding to protect against the blue-shifted radiation and matter of increased energy. In addition the destination would be "blasted into oblivion" by the release of matter and radiation that had been caught in the bubble during the trip.

Clark et al.<ref>Chad Clark, William A. Hiscock, and Shane L. Larson, "Null geodesics in the Alcubierre warp drive spacetime: the view from the bridge", arXiv:gr-qc/9907019 (2018) https://doi.org/10.48550/arXiv.gr-qc/9907019</ref> examine what it would look like from within an operating warp drive. They find that light moving at 90 degrees to the direction of motion will look like it is still coming from 90 degrees. However, light from the front hemisphere is squished progressively toward the forward point the faster you go. For a warp bubble moving faster than the speed of light, there is a cone behind the warp bubble where no light can reach it. However, you don't see a black spot behind you – light from the regions in the backward hemisphere from outside that cone that can reach you is distorted toward the direction behind you to fill in that spot. Light from the forward direction is blue-shifted and light from behind is red-shifted. At speeds of around 200 times light speed, the cosmic microwave background would be blue shifted to energies similar to that of the solar photosphere, heating anything in the warp bubble as if they were close to the sun.

== Van Den Broeck warp drive ==

If an Alcubierre warp bubble a hundred meters across requires a magnitude of energy greater than the entire energy in the observable universe, one option to reduce the magnitude of energy used is to make the warp bubble smaller. Much smaller. Pfenning and Ford<ref name="PfenningFord_2001"></ref> suggested making the warp bubble smaller than an atom. Of course, that brings up the problem of how to stuff a spacecraft in there. Van Den Broeck proposed a solution<ref name="VanDenBroeck_1999"></ref>: make the warp bubble only about the size of an atomic nucleus but expand the space inside the bubble enormously so that a spacecraft could fit in. This is somewhat like a nuclear sized wormhole that leads to a pocket universe that holds the spacecraft. Unfortunately, the metric doesn't fit neatly into any [[Wormholes#Three_dimensions|embedding diagram]] that I can figure out, but the math works out so that spatial coordinates inside the bubble are expanded by a factor of 1017 and a spacecraft would have a few hundred meters of bubble interior to putz around in. This trick manages to reduce the magnitude of energy to only about the mass energy of a few stars similar to our own. Other than that, it is otherwise a normal Alcubierre drive.

Further work<ref>S. Krasnikov, "The quantum inequalities do not forbid spacetime shortcuts", Physical Review D. 67 (10) 104013. https://doi.org/10.1103%2FPhysRevD.67.104013, arXiv:gr-qc/0207057 (2003) https://doi.org/10.48550/arXiv.gr-qc/0207057 </ref> has managed to use Van Den Broeck's trick to reduce the magnitude of the negative energy to only about a milligram.

== Natário warp drive ==

The Alcubierre drive is not the only way to construct a warp drive. José Natário<ref name="Natario"></ref> showed that it was just one example of an entire class of warp drives. These are space-time geometries with a smooth and localized shift vector field a shift vector is a vector field that describes how one "slice" of space out of space-time connects to the next) and a unit lapse function (the lapse function describes how the time you experience changes when going from a "slice" of space-time to the next).

Natário then went on to find in that class a set of warp drives that do not have any expansion or contraction of space at all. Instead, space encountering the front boundary of the warp bubble instead "slides" around the outside of the bubble until it gets to the corresponding place on the back and is left behind there. Another way to think of it is that space entering the bubble shell is compressed in the radial direction of the shell but is simultaneously expanded in the tangential direction so that there is no net change in volume. When it gets to the back, the opposite occurs. In the language of mathematics, the shift vector field is divergenceless.

<table align=center>
<tr><td width=360>[[File:Natario_drive_flow_of_space.png]]
<td width=445>[[File:Natario_warp_drive_horizons.png]]
<tr><td style="vertical-align: top;">A representation of the motion of space in a zero-expansion Natário drive with a vanishingly thin warp shell around the bubble.
<td>The horizons of the generalized Natário class of warp drives (including the Alcubierre warp drive and all fell-Heisenberg drives) at superluminal speed in the direction of the green arrow. No event inside the bubble can affect events outside the bubble in front of the blue line. No events outside the bubble behind the red line can affect events inside the bubble. The lines have an angle α with respect to the direction of motion with sin(α) = 1/vs for vs the speed of the bubble relative to light speed. This is analogous to the Mach cone of supersonic objects.
</table>

NOTE: There are two different things that can be meant by the term "Natário warp drive". The generalized Natário warp drive is the general theory of all warp drives that includes within it the Alcubierre, Fell-Heisenberg, and Lentz warp drives, among infinitely many others. The zero-expansion Natário warp drive is the specific implementation of the generalized Natário warp drive that has a divergenceless shift vector field.

The generalized class of warp drives developed by Natário (including both the Alcubierre and this zero-expansion warp drive) are shown to always have regions in the warp shell where the energy density is negative to at least some observers. You can't get away from it – to warp, you need negative energy density.

The Natário class of warp drives blueshift light coming in from the front and redshift light catching up from the back. For the case of a super-luminal warp bubble, there will of course be a horizon at the back of the bubble and light will not be able to get through from behind. For a bubble speed vs relative to the speed of light, light coming from straight ahead will be blueshifted up in frequency and photon energy by a factor of 1 + vs. For sub-luminal travel, light coming from behind will be redshifted by a factor 1 - vs. In general for n as the unit direction along which the light is propagating, the blueshift factor will be 1 + vs ⋅ n. If the warp shell is infinitesimally thin, light from outside that reaches the center of the bubble will not be distorted in direction although it will be frequency shifted, but observers still inside the bubble but displaced from the center will see the field of view distorted as well as frequency shifted.

Most of the difficulties that apply to the Alcubierre warp drive also apply to the generalized Natário warp drive (and thus to the zero-expansion Natário drive and Fell-Heisenberg drive as well). In particular, quantum energy inequalities, high magnitudes of the energy required, formation of time machines, instabilities related to the stress-energy making up the warp shell being unable to keep up with the bubble, the chicken-and-egg problem of needing FTL to achieve FTL, the formation of horizons making warp bubbles uncontrollable, and Hawking radiation at the horizons, are all still concerns for generic warp drives.

== Fell-Heisenberg warp drives ==

S. D. B. Fell and L. Heisenberg investigated what sort of warp structure is necessary to produce a warp drive with positive energy density everywhere<ref name="Fell-Heisenberg">S. D. B. Fell and L> Heisenberg, "Positive energy warp drive from hidden geometric structures", Classical and Quantum Gravity 38 155020 (2021) https://doi.org/10.1088/1361-6382/ac0e47 https://arxiv.org/abs/2104.06488</ref>. They found that warp drives of the generic Natário form with irrotational shift vectors had a simple expression for the energy density and vanishing momentum (irrotational means that the shift vector can be described as the gradient of a scalar potential field or, equivalently, that the curl of the shift vector field vanishes everywhere). Fell and Heisenberg proposed several potential fields that either had positive energy density everywhere except at discontinuities, or that were continuous but had regions that violated the weak energy condition (i. e. negative energy densities in some frames of reference). One specific example they provided of a warp field had an energy that is about 10,000 times less than the mass-energy of our sun. Or only about half the mass-energy of Jupiter. This is a significant improvement over many other proposed drives.

Despite the introduction discussing warp drive configurations that satisfy the various energy conditions, the configurations described in the paper are shown to locally violate the weak and strong energy conditions. Nonetheless, the energy density is still mostly positive. Santiago, Schuster, and Visser's<ref name="Santiago Schuster Visser 2022">J. Santiago, S. Schuster, and M. Visser, "Generic warp drives violate the null energy condition", Physical Review D 105, 064038 (2022) https://doi.org/10.1103/PhysRevD.105.064038</ref> work shows that the various energy conditions must still be violated by this warp drive; to not violate these, energy density must be positive in all reference frames not just those of the co-moving observer.

Any vector field can be described by its divergence, its curl, and a constant. The constant part of a warp drive's shift vector can be discarded because then the warp drive would not be localized. It is interesting, therefore, that the Natário zero-expansion drive has zero divergence and can be described only by its curl, while the Fell-Heisenberg drive has zero curl and can be described only by its divergence. These two classes of drives thus rely on opposite choices of setting one of the two different parts describing a vector field to zero. As might be expected, the Fell-Heisenberg drive is not zero expansion – space-time will change in volume as it passes through the warp bubble (although after going all the way through it will return to its original volume). However, objects in space-time will retain their original orientation as they pass through the warp bubble.

Previous examples of warp drives, such as the Alcubierre and zero-expansion Natário drives, tend toward a flat space-time as you go far enough away faster than would occur if there were any mass present in the warp drive. That is, the ADM mass of such drives is zero. However, one of the Fell-Heisenberg examples described a spacetime where the geometry approaches that of what you would get for a massive object at the warp location, giving one of the first examples of a warp drive with mass.

In the Fell-Heisenberg drive the momentum vanishes everywhere. Yet the energy occupies regions where the shift vector is varying rapidly. The lack of momentum means that the energy will not move to keep up with the differential expansion and movement of the spacetime. As a consequence, at later times the energy will have a different distribution than what is necessary to maintain the given warp configuration. The exact time evolution is not solved but this likely leads to collapse of warp bubble. Note that this is not a problem unique to the Fell-Heisenberg drive. Assuming a geometry that describes a warp produces a specific distribution of the stress-energy tensor that satisfies Einstein's equation. But there is no guarantee that this stress energy tensor satisfies the continuity equation, which describes how the distribution of matter and momentum change with time. The issue of stability of any warp drive is not well understood, but most warp drive geometries will be unstable in the sense that their matter that generates the warp fields will not be able to keep up with the motion of the warp. The Fell-Heisenberg drive is unique only in that, with vanishing momentum, there is no configuration that allows stable propagation.

(n. b. The Heisenberg here is Lavinia Heisenberg, not the Werner Heisenberg of quantum physics and uncertainty principle fame.)

== Lentz warp drive ==

E. Lentz<ref name="Lentz">E. W. Lentz, "Breaking the warp barrier: hyper-fast solitons in Einstein–Maxwell-plasma theory", Classical and Quantum Gravity. 38 075015 (2021). arXiv:[https://arxiv.org/abs/2006.07125 2006.07125]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2021CQGra..38g5015L 2021CQGra..38g5015L]. doi:[https://doi.org/10.1088%2F1361-6382%2Fabe692 10.1088/1361-6382/abe692]. ISSN [https://search.worldcat.org/issn/0264-9381 0264-9381]. S2CID [https://api.semanticscholar.org/CorpusID:219635854 219635854].</ref> investigated a particular configuration of the Fell-Heisenberg class of warp drives, where the scalar field has a hyperbolic shape. The necessary conditions were found to be satisfied by a selection of rhomboid regions of the scalar field, resulting in an energy density that is everywhere positive and a shift vector with vanishing tidal force at the center for a comfortable ride.

However, Santiago, Schuster, and Visser<ref name="Santiago Schuster Visser 2022"></ref> dispute claims that the Lentz drive satisfies the energy conditions, noting that everywhere positive energy density in one frame of reference is insufficient to establish that the energy density is positive in all reference frames; knowledge of the Cauchy stress tensor is also needed. They show that any generic warp drive will violate the strong energy condition, null energy condition, and weak energy condition.

== Conservation laws ==

(Discussion of asymptotic flatness: what is it? Where does it apply?)

(The "big four" conservation laws and how they relate to asymptotic flatness.)

(ADM mass -> 0; conservation of energy/mass issues.)<ref>Sebastian Schuster, Jessica Santiago,and Matt Visser, "ADM mass in warp drive spacetimes" General Relativity and Gravitation (2023) 55:14 https://doi.org/10.1007/s10714-022-03061-9</ref>

(Discuss issues of conservation of angular momentum.)

(If angular momentum conservation is ignored, discuss how momentum & energy are affected by outside forces. In a gravitational field equivalent to inertial frame ... like being in an accelerated elevator; will acquire momentum buildup while staying "at rest".)

== Warp Railroads ==

Van Den Broeck<ref name="VanDenBroeck_2000"></ref> suggested a warp railroad,

You can set up devices ahead of time along the path of the bubble that produce the negative energy regions for the warp bubble at the appropriate time without needing the negative energy to ever move with the bubble at all. At least for bubbles on a predictable schedule and route.

helps with:

-stability (elaborate)

-conservation laws (elaborate)

Mention Krasnikov<ref name="Krasnikov 1998">S. V. Krasnikov, "Hyperfast travel in general relativity" Phys. Rev. D 57 4760</ref>

Hyperwave<ref name="Pieri 2024">Lorenzo Pieri, "Hyperwave: hyper-Fast Communication within General Relativity", arXiv:2311.12069 [gr-qc] https://doi.org/10.48550/arXiv.2311.12069</ref> makes use of this warp railroad idea.

== Quantum effects ==

(quantum stuff here<ref name="Hiscock1997">W. A. Hiscock, "Quantum effects in the Alcubierre warp drive spacetime", Classical and Quantum Gravity 14 L183 https://doi.org/10.1088/0264-9381/14/11/002 https://arxiv.org/abs/gr-qc/9707024</ref><ref name="Finazzi et al 2009">S. Finazzi, S. Liberati, C. Barceló, "Semiclassical instability of dynamical warp drives", Physical Review D 79, 124017 (2009)https://doi.org/10.1103/PhysRevD.79.124017 https://arxiv.org/abs/0904.0141</ref>)

From<ref name="Pieri 2024"></ref>
<blockquote>
It should be emphasised that Quantum Inequalities have not been proven for realistic interacting quantum fields, where the they are expected to be weaker if they exist [Cadamuro, 2019], and they often require Minkowskian conditions at infinity, a condition which is not appropriate for fields inside bounded regions, as in a Casimir devices.
</blockquote>

== Warp drives and black holes ==

What happens if you drive your warp drive into a black hole?

The pop-sci reason that nothing can escape a black hole is that at the event horizon the escape velocity is faster than light and nothing can go faster than light. A warp drive can go faster than light.

The more detailed reason from general relativity is that at the event horizon space and time are rotated sufficiently that "inward" becomes "forward in time." You can no more go farther away from the singularity in a black hole once you are inside the event horizon than you can go backward in time. Any super-luminal travel can go backward in time, and warp drives can engage in super-luminal travel.

The Hawking area theorem that says that black holes can only grow, never shrink relies on the condition that the energy is everywhere positive. Warp drives have negative energy density.

So in principle, there's nothing that prevents a warp drive from taking a dip into a black hole and coming back out again.

Garattini and Zatrimaylov looked into the problem of a warp drive near (and in) a black hole<ref name="Garattini and Zatrimaylov">R. Garattini and K. Zatrimaylov, "Black holes, warp drives, and energy conditions", Physics Letters B 856 138910 (2024) https://doi.org/10.1016/j.physletb.2024.138910</ref>. The found that the black hole reduces the negative energy requirement for the warp bubble to move inward toward the black hole singularity, but increases the negative energy needed to move away from the center of the black hole. In addition, a warp drive parked at the event horizon would allow light from inside the horizon to pass through the bubble and escape back outside.

== Credit ==
Author: Luke Campbell

== References ==

[[Category:Physics]][[Category:Infrastructure]][[Category:Physics & Engineering‏‎]][[Category:Physics & Math & Engineering]][[Category:Transportation & Infrastructure‏‎]][[Category:Metric Engineering]]

Warp Drives

2026-07-26T23:06:53Z

Lwcamp: /* So what is going on? */

{{PageConstructionNotice}}

Science fiction often features spacecraft that can seemingly move across space and get between the place of departure and the destination much faster than light could have done. This appears to contradict the theory of relativity, which predicts unequivocally that nothing can move through space faster than light. Because relativity has been incredibly successful at describing nature, with its many other predictions regularly being confirmed to extraordinary accuracy and within the bounds of uncertainty of all the experiments that tested them, it gives confidence that relativity is a correct description of reality. Which seems to rather throw a wet towel on our hopes for rapid travel between stars.

However, while relativity does not allow things to move through space faster than light, it places no such restrictions on how fast space-time itself can expand, contract, or move around. This leads to the idea of a warp drive – the spacecraft remains stationary within a region of highly curved space-time, and that region moves at super-luminal speeds rather than the spacecraft.

== The Alcubierre warp drive ==

The first warp drive geometry that satisfied the Einstein field equations of relativity was proposed by Miguel Alcubierre<ref>M. Alcubierre, "The warp drive: hyper-fast travel within general relativity." Classical and Quantum Gravity. 11 (5): L73–L77 (1994). [https://arxiv.org/abs/gr-qc/0009013 arXiv:gr-qc/0009013]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1994CQGra..11L..73A 1994CQGra..11L..73A]. doi:[https://doi.org/10.1088%2F0264-9381%2F11%2F5%2F001 10.1088/0264-9381/11/5/001]. S2CID [https://api.semanticscholar.org/CorpusID:4797900 4797900].</ref>. In this geometry, a sphere of space-time moves at an arbitrary speed (potentially but not necessarily a speed much faster than light). Objects within the sphere are moved along with the sphere; an object at rest within the sphere would be moved along with the sphere indefinitely. Space is expanding at the rear boundary and contracting at the front boundary in order to keep the sphere moving. In order to satisfy the Einstein field equations, the boundary of the sphere must have a negative energy density. The challenges of space-time geometries with negative energy densities are described in our page on [[Wormholes#Exotic_energy_conditions|wormholes]], for our purposes it is enough to note that negative energy densities can pose problems if not handled carefully, there may be limits on how much negative energy you can have without nearby positive energy density, and it may not be possible to get enough negative energy to support a warp drive; although none of this is ruled out by physics – yet!

<table align=center>
<tr><td width=650>[[File:Energy_magnitude_of_Alcubierre_drive.png]]
<td width=650>[[File:Expansion_of_Alcubierre_drive.png]]
<tr><td>The magnitude of the energy of the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. (The energy density is negative everywhere in the shell, here only the magnitude is shown to aid visual comprehension.)
<td>The expansion of space in the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. Negative expansion means that space is contracting in that direction, positive expansion that it is expanding.
</table>

NOTE: What the? The energy distribution is symmetric! How does the drive know which way to go? There must be significant contributions of other components of the stress-energy tensor, and those have got to be asymmetric along the forward/backward axis. Check this when I get time.

Almost as soon as Alcubierre proposed his warp drive, others began picking it apart.
<ol>
<li>Pfenning and Ford<ref name="PfenningFord_2001">M. J. Pfenning and L. H. Ford, "The unphysical nature of 'Warp Drive'", Classical and Quantum Gravity. 14 (7): 1743–1751 (1997). arXiv:[https://arxiv.org/abs/gr-qc/9702026 gr-qc/9702026]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1997CQGra..14.1743P 1997CQGra..14.1743P]. doi:[https://doi.org/10.1088%2F0264-9381%2F14%2F7%2F011 10.1088/0264-9381/14/7/011]. S2CID [https://api.semanticscholar.org/CorpusID:15279207 15279207].</ref> found that if the negative energy density wall around the bubble obeys [[Wormholes#Quantum_energy_inequalities|quantum energy inequalities]], then for warp speeds of around light speed the shell thickness would be on the order of a hundred Planck lengths; a distance far smaller than any other known physical phenomenon or object.
<li>In the same work, Pfenning and Ford also found that for a bubble a hundred meters in radius, a warp speed of around the speed of light, and the required thickness of about 100 Planck lengths the energy in the bubble shell would be E ≈ -1063 kg c2. The magnitude of this latter value is ten orders of magnitude larger than the energy of the entire visible universe. They do note, however, that if quantum energy inequalities can be ignored then a warp drive with a hundred meters radius but a shell thickness of one meter would "only" have an energy magnitude of about a quarter solar mass.
<li>As with any method of faster than light travel, the warp drive could be used to make a time machine.
<li>Perhaps most seriously, if the warp drive is going faster than light speed, the negative energy regions on the outside of the shell won't be in the warp parts of the bubble. They will have to be moving through space at faster than light speed if the bubble is to maintain its integrity, which is the very problem that the warp drive was designed to avoid. The outside of the warp bubble shell that maintains the warp bubble would fall away and the warp bubble would quickly erode to sub-luminal speeds.<ref name="Van Den Broeck 2000">Chris Van Den Broeck, "On the (im)possibility of warp bubbles", arXiv:gr-qc/9906050 https://doi.org/10.48550/arXiv.gr-qc/9906050</ref>
<li>A more general problem that includes the previous one is that the geometry of the warp drive determines the distribution of energy, momentum, and stress (pressure, shear, and tension) by Einstein's field equation, which states that the stress energy tensor is directly proportional to a reduced order curvature tensor of space-time. However, the stress-energy tensor must also obey the continuity equation – basically, for the energy to change in a given volume of space that energy must come in or go out of the boundary of that volume and similarly for the momentum. So a warp drive can have the warp condition for one moment in time. But after that, the stress-energy that maintains the warp will start to move according to the forces on it and according to the natural paths that it takes through the warped space-time. There is no guarantee that the configuration of the stress-energy at later times will be sufficient to maintain the warp.
<blockquote>
From the perspective of simulating the warp drive dynamically, the key challenge is stability. One can straightforwardly generate the initial metric that describes the Alcubierre metric and infer the instantaneous matter configuration that must support it using the Einstein Equation, but a time-domain evolution of the coupled matter-gravity system necessitates that we specify the equation of state of the matter, which in turn determines how it evolves. In a perfect fluid, this involves specifying the relation between the fluid pressure and density, but in a more general case, the full relation between the
stress-energy components must be given. There is (to our knowledge) no known equation of state that would maintain the warp drive metric in a stable configuration over time - therefore, whilst one can require that initially, the warp bubble is constant, it will quickly evolve away from that state and, in most cases, the warp fluid and spacetime deformations will disperse or collapse into a central point.
 
– Clough, Dietrich, and Khan<ref>Katy Clough, Tim Dietrich, and Sebastian Khan, "What no one has seen before: gravitational waveforms from warp drive collapse", arXiv:2406.02466 [gr-qc] https://doi.org/10.48550/arXiv.2406.02466</ref>
</blockquote>
<li>In order to get the matter (or more generally, stress-energy) that forms the warp shell up to speeds that could allow the shell to exist, you need to be able to, well, accelerate that matter to warp speeds. Which gives a chicken-and-egg problem - in order to achieve superluminal travel you need to be able to get matter or energy moving superluminally in the first place<ref name="Bobrick and Martire">Alexey Bobrick and Gianni Martire, "Introducing Physical Warp Drives", arXiv:2102.06824 [gr-qc] https://doi.org/10.48550/arXiv.2102.06824</ref><ref>D. H. Coule, "No warp drive", Class. Quantum Grav. 15 (1998) 2523–2527. https://doi.org/10.1088/0264-9381/15/8/026</ref>, which rather makes warp drives unnecessary from the get-go.
<li>A warp bubble moving faster than light will not be able to get any signals in front of it as it will overtake its own signal. In fact, anyone in the interior of the bubble would not be able to send signals to the front surface of the bubble. In science-speak, a horizon forms at the front of the bubble<ref name="Natario">José Natário, "Warp drive with zero expansion", Classical and Quantum Gravity. 19 (6): 1157–1166 (2002). arXiv:[https://arxiv.org/abs/gr-qc/0110086 gr-qc/0110086]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2002CQGra..19.1157N 2002CQGra..19.1157N]. doi:[https://doi.org/10.1088%2F0264-9381%2F19%2F6%2F308 10.1088/0264-9381/19/6/308]. S2CID [https://api.semanticscholar.org/CorpusID:15859984 15859984].</ref>. As a consequence, the bubble cannot be controlled, steered, nor stopped from within the bubble itself.
<li>A Horizon will naturally emit Hawking radiation. It is estimated that the radiation will raise the temperature of the interior of the bubble to approximately 1032 K<ref name="Finazzi 2009">Stefano Finazzi, Stefano Liberati, and Carlos Barceló, "Semiclassical instability of dynamical warp drives", Phys. Rev. D 79, 124017 (12 June, 2009) https://doi.org/10.1103/PhysRevD.79.124017 , arXiv:0904.0141 [gr-qc] https://doi.org/10.48550/arXiv.0904.0141</ref> (at this level of accuracy, the result is the same if you replace temperatures in K by the same number in °C or °F). This temperature, however, may be lower if quantum inequalities do not have to be fulfilled.
<li>Semi-classical instabilities at the horizon are expected to lead to a divergence of the stress-energy tensor at the boundary in a short period of time, thus leading to the destruction of the warp bubble<ref name="Finazzi 2009"></ref>.
</ol>
Helpfully, Van Den Broeck<ref name="VanDenBroeck_2000">C. Van Den Broeck, "Alcubierre’s Warp Drive: Problems and Prospects" AIP Conference Proceedings. 504: 1105–1110 (2000). Bibcode:[https://ui.adsabs.harvard.edu/abs/2000AIPC..504.1105V 2000AIPC..504.1105V]. doi:[https://doi.org/10.1063%2F1.1290913 10.1063/1.1290913].</ref> suggested several solutions – or at least mitigations – for some of these problems
<ol>
<li>Quantum inequalities had not been shown to be true in general for highly curved space-times.
<li>A proposed warp drive geometry<ref name="VanDenBroeck_1999">C. Van Den Broeck, "A 'warp drive' with more reasonable total energy requirements". Classical and Quantum Gravity. 16 (12): 3973–3979 (1999). arXiv:[https://arxiv.org/abs/gr-qc/9905084 gr-qc/9905084]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1999CQGra..16.3973V 1999CQGra..16.3973V]. doi:[https://doi.org/10.1088%2F0264-9381%2F16%2F12%2F314 10.1088/0264-9381/16/12/314]. S2CID [https://api.semanticscholar.org/CorpusID:15466313 15466313]. </ref> (see the van Den Broeck drive, below) would be able to minimize the magnitude of the energy required to about that of the mass of our sun. While still large, it is not unphysically large.
<li>Time travel is always going to be a worry with faster than light travel. There's no neat solution to this one.
<li>It may be possible for the negative energy outside of the superluminal region to compress into the superluminal region to form a shock in space-time. The front surface of the bubble would then be a singularity. It is not clear if such a sharp jump in physical properties is possible, but neither is sure that it is impossible, either. However, while this might help with the negative energy in front of the bubble getting swept up because it cannot move faster than light, it does little to help with the material at the back of the bubble on the outside getting left behind because it cannot keep up.
</ol>
Other mitigating research has also shown
<ol start=9>
<li>Semi-classical divergence only occurs in warp drives with only one spatial dimensions. Those with two or more spatial dimensions (as in our universe) only contain points where divergences occur, and these divergences can be managed. In fact, their effects can be reduced by making the warp shape more elongated<ref>Carlos Barceló, Valentin Boyanov, Luis J. Garay, Eduardo Martín-Martínez,
and Jose M. Sánchez Velázquez, "Warp drive aerodynamics", JHEP08 (2022) 288 https://doi.org/10.1007/JHEP08(2022)288 , arXiv:2207.06458 [gr-qc] https://doi.org/10.48550/arXiv.2207.06458</ref>.
</ol>

(And what's with the ring in all the artwork, anyway?)

=== So what is going on? ===

What is it that makes a warp drive a warp drive, and how does it work?

A warp drive is a concept that comes out of the science of general relativity. General relativity collects space and time together into a single "structure" called spacetime. This structure is curved, and its curvature is determined by the material that is in it. Absent forces acting between matter, particles of matter move through spacetime on the shortest possible paths in this curved environment.

"Spacetime tells matter how to move; matter tells spacetime how to curve." - John Archibald Wheeler

All of this is encompassed in the Einstein field equation:
<div align=center>
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 8πG
<td rowspan="2" nowrap="nowrap" align="right"> T = R -  
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right"> R g
<tr>
<td style="border-top:solid 1px black;"> c4
<td style="border-top:solid 1px black;"> 2
</table>
</div>
Unfortunately, by convention, both the symbols G and R get used for a lot of different things. Here, symbols in bold are second rank tensors – constructs of arrays of 4×4 numbers where 4 is necessary because that is the number of dimensions of spacetime. Conventionally, these numbers are indexed by a value going from 0 to 3, with 0 being the part corresponding to the coordinate of time and 1 to 3 being the Cartesian coordinates that you might know as x, y, and z.

T is the stress-energy tensor: the (0,0) component represents the energy density at a given point; the (0,i) components (with going from 1 to 3) represent the momentum density; and the (i,j) components represent the stresses present, such as pressure or tension or shear.

The other term, R - R g, involves a partial description of the curvature. A full description of the curvature requires a construct of 4×4×4×4 components; too much for the stress energy tensor to fully constrain. Which is okay, because this allows curvature to exist away from where matter is, allowing gravity to have long range effects across empty space as well as for phenomena like gravitational waves. But it is enough that a given distribution of matter T gives enough constraints (along with specifying how the geometry behaves far away) that you can generally figure out what the actual gravity is.

Describing spacetime - lapse function & shift vectors (foliations, etc.)
Diagram of what the shift vector does.

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Lapse_shift.png]]
<tr><td>A folliation of space-time into slices of constant time coordinates (grey). At an event (black circle), the lapse function (yellow) shows how far into the future the point at that event will be projected in a unit amount of proper time experienced by an observer initially at the event. The shift vector (red) shows how that point changes coordinates after the unit amount of proper time. The cyan vector shows the new coordinates the point is projected onto. The lower black surface indicates a foliation of spacetime at the same time coordinate; after unit proper time, the events on the lower foliation get projected to the curved upper black foliation according to the shift vectors and lapse functions at each of their respective points.
</table>
</table>

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Warp_shift.png]]
<tr><td>The shift vector of a warp geometry, in red. At the edges there is no shift. At the center (inside the warp bubble) there is a region of uniform shift so that at a later time the inside of the warp bubble is rigidly translated to different spatial coordinates (in blue). The two foliations remain the same distance apart, because warp drives maintain a unit lapse function.
</table>
</table>

Solve backwards - assume a structure of space-time, figure out what distribution of matter is required for this from the EFE.

What sort of spacetime geometry produces a warp?

Unit lapse function <-> no time dilation

Region of near constant shift vector - that region is "moving" spacetime, the warp bubble.

Shift vector falls off to zero away from warp bubble -> normal flat spacetime.

=== Alcubierre warp interactions with light and matter ===

Space is not entirely empty. It is filled with a diffuse plasma in the form of the interstellar medium, as well as cosmic radiation, light from stars, and cosmic microwave background radiation. A warp drive propagating through space will encounter this stuff. What happens when this matter and radiation have a warp bubble pass across them?

The first analysis of matter encountering a warp bubble was performed by Pfenning and Ford<ref name="PfenningFord_2001"></ref>. They looked at the warp bubble interaction with a massive object at rest with the frame of reference of the warp drive (keep in mind that the rest frame is the same inside and outside the warp bubble; in this rest frame objects inside the bubble have no momentum even though they are, in some sense, changing location rapidly with time). When the warp bubble passes, the object experiences an acceleration in the direction of the warp bubble motion. When the warp shell passes and the object is inside the bubble, it will be moving with approximately the speed of the bubble. Pfenning and Ford analyzed this problem with a continuous distribution of shell energy that, strictly speaking, never falls to zero except at the bubble center and at spatial infinity, so unless the object passes through the center in Pfenning and Ford's description it will never quite get up to the bubble's speed. In this case, the object will move almost as fast as the object but will pass through the bubble in a finite time, after which it will again be at rest with respect to the reference frame but displaced along the direction of the warp bubble motion by some distance. With this description of the warp bubble, a spacecraft of finite size will always be moving a little slower than the warp bubble and would have to use rockets to keep up with it. in addition, the spacecraft would experience tidal forces that would cause stress on the spacecraft's structure.

Pfenning and Ford also examined cases where the shell is of a finite (possibly infinitesimal) width and falls to zero both inside and outside the bubble. In this case any matter encountering the bubble would be collected at the bow of the bubble and thereafter move along with it.

McMonigal et al. analyzed the situation for both massive particles and light moving along the axis of travel of the bubble<ref name="McMonigalEtAl_2012">B. McMonigal, G. F. Lewis, and P. O'Byrne, "Alcubierre warp drive: On the matter of matter". Physical Review D. 85 (6) 064024 (20 March 2012). arXiv:[https://arxiv.org/abs/1202.5708 1202.5708]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2012PhRvD..85f4024M 2012PhRvD..85f4024M]. doi:[https://doi.org/10.1103%2FPhysRevD.85.064024 10.1103/PhysRevD.85.064024]. S2CID [https://api.semanticscholar.org/CorpusID:3993148 3993148].</ref>. They found that light moving opposite the warp direction passed through the bubble without incident, being only somewhat delayed by passing through the bubble. A warp bubble that is warping at sub-luminal speeds can have light catch up from behind it. This light is able to pass through the bubble, and is somewhat advanced in its path by the speed of the bubble. For super-luminal warp bubbles, however, the situation is different. The bubble will catch up to light moving it its own direction that is originally in front of it. This light cannot escape forward, the bubble being too fast. Nor can it escape backward, as the light is propagating forward and the interior of the bubble is at rest. Thus, the light gets caught at the bow of the warp bubble, unable to escape for so long as the warp bubble is active. This light is strongly blue shifted to extremely energetic x-rays and gamma rays. The space behind the bubble is swept clear of forward-moving light.

The situation for matter over-run by a super-luminal warp bubble is similar. Matter moving backward passes through the bubble, being only swept a ways along the bubble's path as sign of its passing but otherwise continuing on their way. Matter at rest behaves the same way as Pfenning and Ford discovered. Meanwhile, matter moving in the same direction of the warp is overtaken and collects at the bow of the bubble, unable to leave for so long as the bubble is warping. This matter "surfing" on the bubble bow is highly accelerated to relativistic speeds, experiencing extreme time dilation.

For a sub-luminal bubble, there is a speed faster than the warp speed where particles coming up from behind overtake the bubble and pass through, out the front. Particles below this speed but still faster than the warp speed still overtake the bubble, but then bounce backwards out of the bubble from behind. They are still moving forward, but are now moving more slowly than the warp speed. Particles moving forward but slower than the bubble will be overtaken and then bounced out the front with a speed higher than the warp speed. Those particles moving backward will pass through the bubble and exit with their initial speed.

The spacecraft inside the bubble will observe that light moving backward compared to the warp direction is blue-shifted, and matter moving backward passes through with increased energy. Meanwhile, a sub-luminal bubble will see light that was moving forward as red-shifted and matter catching up to the bubble from behind will pass by with reduced energy (superluminal bubbles, of course, do not have any matter or radiation catching up to them from behind).

When a superluminal warp bubble that has been collecting matter and radiation for the duration of its journey and blue-shifting it to much higher energies is turned off, all that matter and energy is released as a blast of radiation in the direction the bubble was warping.

McMonigal et al. conclude by noting that any spacecraft in the warp bubble would need shielding to protect against the blue-shifted radiation and matter of increased energy. In addition the destination would be "blasted into oblivion" by the release of matter and radiation that had been caught in the bubble during the trip.

Clark et al.<ref>Chad Clark, William A. Hiscock, and Shane L. Larson, "Null geodesics in the Alcubierre warp drive spacetime: the view from the bridge", arXiv:gr-qc/9907019 (2018) https://doi.org/10.48550/arXiv.gr-qc/9907019</ref> examine what it would look like from within an operating warp drive. They find that light moving at 90 degrees to the direction of motion will look like it is still coming from 90 degrees. However, light from the front hemisphere is squished progressively toward the forward point the faster you go. For a warp bubble moving faster than the speed of light, there is a cone behind the warp bubble where no light can reach it. However, you don't see a black spot behind you – light from the regions in the backward hemisphere from outside that cone that can reach you is distorted toward the direction behind you to fill in that spot. Light from the forward direction is blue-shifted and light from behind is red-shifted. At speeds of around 200 times light speed, the cosmic microwave background would be blue shifted to energies similar to that of the solar photosphere, heating anything in the warp bubble as if they were close to the sun.

== Van Den Broeck warp drive ==

If an Alcubierre warp bubble a hundred meters across requires a magnitude of energy greater than the entire energy in the observable universe, one option to reduce the magnitude of energy used is to make the warp bubble smaller. Much smaller. Pfenning and Ford<ref name="PfenningFord_2001"></ref> suggested making the warp bubble smaller than an atom. Of course, that brings up the problem of how to stuff a spacecraft in there. Van Den Broeck proposed a solution<ref name="VanDenBroeck_1999"></ref>: make the warp bubble only about the size of an atomic nucleus but expand the space inside the bubble enormously so that a spacecraft could fit in. This is somewhat like a nuclear sized wormhole that leads to a pocket universe that holds the spacecraft. Unfortunately, the metric doesn't fit neatly into any [[Wormholes#Three_dimensions|embedding diagram]] that I can figure out, but the math works out so that spatial coordinates inside the bubble are expanded by a factor of 1017 and a spacecraft would have a few hundred meters of bubble interior to putz around in. This trick manages to reduce the magnitude of energy to only about the mass energy of a few stars similar to our own. Other than that, it is otherwise a normal Alcubierre drive.

Further work<ref>S. Krasnikov, "The quantum inequalities do not forbid spacetime shortcuts", Physical Review D. 67 (10) 104013. https://doi.org/10.1103%2FPhysRevD.67.104013, arXiv:gr-qc/0207057 (2003) https://doi.org/10.48550/arXiv.gr-qc/0207057 </ref> has managed to use Van Den Broeck's trick to reduce the magnitude of the negative energy to only about a milligram.

== Natário warp drive ==

The Alcubierre drive is not the only way to construct a warp drive. José Natário<ref name="Natario"></ref> showed that it was just one example of an entire class of warp drives. These are space-time geometries with a smooth and localized shift vector field a shift vector is a vector field that describes how one "slice" of space out of space-time connects to the next) and a unit lapse function (the lapse function describes how the time you experience changes when going from a "slice" of space-time to the next).

Natário then went on to find in that class a set of warp drives that do not have any expansion or contraction of space at all. Instead, space encountering the front boundary of the warp bubble instead "slides" around the outside of the bubble until it gets to the corresponding place on the back and is left behind there. Another way to think of it is that space entering the bubble shell is compressed in the radial direction of the shell but is simultaneously expanded in the tangential direction so that there is no net change in volume. When it gets to the back, the opposite occurs. In the language of mathematics, the shift vector field is divergenceless.

<table align=center>
<tr><td width=360>[[File:Natario_drive_flow_of_space.png]]
<td width=445>[[File:Natario_warp_drive_horizons.png]]
<tr><td style="vertical-align: top;">A representation of the motion of space in a zero-expansion Natário drive with a vanishingly thin warp shell around the bubble.
<td>The horizons of the generalized Natário class of warp drives (including the Alcubierre warp drive and all fell-Heisenberg drives) at superluminal speed in the direction of the green arrow. No event inside the bubble can affect events outside the bubble in front of the blue line. No events outside the bubble behind the red line can affect events inside the bubble. The lines have an angle α with respect to the direction of motion with sin(α) = 1/vs for vs the speed of the bubble relative to light speed. This is analogous to the Mach cone of supersonic objects.
</table>

NOTE: There are two different things that can be meant by the term "Natário warp drive". The generalized Natário warp drive is the general theory of all warp drives that includes within it the Alcubierre, Fell-Heisenberg, and Lentz warp drives, among infinitely many others. The zero-expansion Natário warp drive is the specific implementation of the generalized Natário warp drive that has a divergenceless shift vector field.

The generalized class of warp drives developed by Natário (including both the Alcubierre and this zero-expansion warp drive) are shown to always have regions in the warp shell where the energy density is negative to at least some observers. You can't get away from it – to warp, you need negative energy density.

The Natário class of warp drives blueshift light coming in from the front and redshift light catching up from the back. For the case of a super-luminal warp bubble, there will of course be a horizon at the back of the bubble and light will not be able to get through from behind. For a bubble speed vs relative to the speed of light, light coming from straight ahead will be blueshifted up in frequency and photon energy by a factor of 1 + vs. For sub-luminal travel, light coming from behind will be redshifted by a factor 1 - vs. In general for n as the unit direction along which the light is propagating, the blueshift factor will be 1 + vs ⋅ n. If the warp shell is infinitesimally thin, light from outside that reaches the center of the bubble will not be distorted in direction although it will be frequency shifted, but observers still inside the bubble but displaced from the center will see the field of view distorted as well as frequency shifted.

Most of the difficulties that apply to the Alcubierre warp drive also apply to the generalized Natário warp drive (and thus to the zero-expansion Natário drive and Fell-Heisenberg drive as well). In particular, quantum energy inequalities, high magnitudes of the energy required, formation of time machines, instabilities related to the stress-energy making up the warp shell being unable to keep up with the bubble, the chicken-and-egg problem of needing FTL to achieve FTL, the formation of horizons making warp bubbles uncontrollable, and Hawking radiation at the horizons, are all still concerns for generic warp drives.

== Fell-Heisenberg warp drives ==

S. D. B. Fell and L. Heisenberg investigated what sort of warp structure is necessary to produce a warp drive with positive energy density everywhere<ref name="Fell-Heisenberg">S. D. B. Fell and L> Heisenberg, "Positive energy warp drive from hidden geometric structures", Classical and Quantum Gravity 38 155020 (2021) https://doi.org/10.1088/1361-6382/ac0e47 https://arxiv.org/abs/2104.06488</ref>. They found that warp drives of the generic Natário form with irrotational shift vectors had a simple expression for the energy density and vanishing momentum (irrotational means that the shift vector can be described as the gradient of a scalar potential field or, equivalently, that the curl of the shift vector field vanishes everywhere). Fell and Heisenberg proposed several potential fields that either had positive energy density everywhere except at discontinuities, or that were continuous but had regions that violated the weak energy condition (i. e. negative energy densities in some frames of reference). One specific example they provided of a warp field had an energy that is about 10,000 times less than the mass-energy of our sun. Or only about half the mass-energy of Jupiter. This is a significant improvement over many other proposed drives.

Despite the introduction discussing warp drive configurations that satisfy the various energy conditions, the configurations described in the paper are shown to locally violate the weak and strong energy conditions. Nonetheless, the energy density is still mostly positive. Santiago, Schuster, and Visser's<ref name="Santiago Schuster Visser 2022">J. Santiago, S. Schuster, and M. Visser, "Generic warp drives violate the null energy condition", Physical Review D 105, 064038 (2022) https://doi.org/10.1103/PhysRevD.105.064038</ref> work shows that the various energy conditions must still be violated by this warp drive; to not violate these, energy density must be positive in all reference frames not just those of the co-moving observer.

Any vector field can be described by its divergence, its curl, and a constant. The constant part of a warp drive's shift vector can be discarded because then the warp drive would not be localized. It is interesting, therefore, that the Natário zero-expansion drive has zero divergence and can be described only by its curl, while the Fell-Heisenberg drive has zero curl and can be described only by its divergence. These two classes of drives thus rely on opposite choices of setting one of the two different parts describing a vector field to zero. As might be expected, the Fell-Heisenberg drive is not zero expansion – space-time will change in volume as it passes through the warp bubble (although after going all the way through it will return to its original volume). However, objects in space-time will retain their original orientation as they pass through the warp bubble.

Previous examples of warp drives, such as the Alcubierre and zero-expansion Natário drives, tend toward a flat space-time as you go far enough away faster than would occur if there were any mass present in the warp drive. That is, the ADM mass of such drives is zero. However, one of the Fell-Heisenberg examples described a spacetime where the geometry approaches that of what you would get for a massive object at the warp location, giving one of the first examples of a warp drive with mass.

In the Fell-Heisenberg drive the momentum vanishes everywhere. Yet the energy occupies regions where the shift vector is varying rapidly. The lack of momentum means that the energy will not move to keep up with the differential expansion and movement of the spacetime. As a consequence, at later times the energy will have a different distribution than what is necessary to maintain the given warp configuration. The exact time evolution is not solved but this likely leads to collapse of warp bubble. Note that this is not a problem unique to the Fell-Heisenberg drive. Assuming a geometry that describes a warp produces a specific distribution of the stress-energy tensor that satisfies Einstein's equation. But there is no guarantee that this stress energy tensor satisfies the continuity equation, which describes how the distribution of matter and momentum change with time. The issue of stability of any warp drive is not well understood, but most warp drive geometries will be unstable in the sense that their matter that generates the warp fields will not be able to keep up with the motion of the warp. The Fell-Heisenberg drive is unique only in that, with vanishing momentum, there is no configuration that allows stable propagation.

(n. b. The Heisenberg here is Lavinia Heisenberg, not the Werner Heisenberg of quantum physics and uncertainty principle fame.)

== Lentz warp drive ==

E. Lentz<ref name="Lentz">E. W. Lentz, "Breaking the warp barrier: hyper-fast solitons in Einstein–Maxwell-plasma theory", Classical and Quantum Gravity. 38 075015 (2021). arXiv:[https://arxiv.org/abs/2006.07125 2006.07125]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2021CQGra..38g5015L 2021CQGra..38g5015L]. doi:[https://doi.org/10.1088%2F1361-6382%2Fabe692 10.1088/1361-6382/abe692]. ISSN [https://search.worldcat.org/issn/0264-9381 0264-9381]. S2CID [https://api.semanticscholar.org/CorpusID:219635854 219635854].</ref> investigated a particular configuration of the Fell-Heisenberg class of warp drives, where the scalar field has a hyperbolic shape. The necessary conditions were found to be satisfied by a selection of rhomboid regions of the scalar field, resulting in an energy density that is everywhere positive and a shift vector with vanishing tidal force at the center for a comfortable ride.

However, Santiago, Schuster, and Visser<ref name="Santiago Schuster Visser 2022"></ref> dispute claims that the Lentz drive satisfies the energy conditions, noting that everywhere positive energy density in one frame of reference is insufficient to establish that the energy density is positive in all reference frames; knowledge of the Cauchy stress tensor is also needed. They show that any generic warp drive will violate the strong energy condition, null energy condition, and weak energy condition.

== Conservation laws ==

(Discussion of asymptotic flatness: what is it? Where does it apply?)

(The "big four" conservation laws and how they relate to asymptotic flatness.)

(ADM mass -> 0; conservation of energy/mass issues.)<ref>Sebastian Schuster, Jessica Santiago,and Matt Visser, "ADM mass in warp drive spacetimes" General Relativity and Gravitation (2023) 55:14 https://doi.org/10.1007/s10714-022-03061-9</ref>

(Discuss issues of conservation of angular momentum.)

(If angular momentum conservation is ignored, discuss how momentum & energy are affected by outside forces. In a gravitational field equivalent to inertial frame ... like being in an accelerated elevator; will acquire momentum buildup while staying "at rest".)

== Warp Railroads ==

Van Den Broeck<ref name="VanDenBroeck_2000"></ref> suggested a warp railroad,

You can set up devices ahead of time along the path of the bubble that produce the negative energy regions for the warp bubble at the appropriate time without needing the negative energy to ever move with the bubble at all. At least for bubbles on a predictable schedule and route.

helps with:

-stability (elaborate)

-conservation laws (elaborate)

Mention Krasnikov<ref name="Krasnikov 1998">S. V. Krasnikov, "Hyperfast travel in general relativity" Phys. Rev. D 57 4760</ref>

Hyperwave<ref name="Pieri 2024">Lorenzo Pieri, "Hyperwave: hyper-Fast Communication within General Relativity", arXiv:2311.12069 [gr-qc] https://doi.org/10.48550/arXiv.2311.12069</ref> makes use of this warp railroad idea.

== Quantum effects ==

(quantum stuff here<ref name="Hiscock1997">W. A. Hiscock, "Quantum effects in the Alcubierre warp drive spacetime", Classical and Quantum Gravity 14 L183 https://doi.org/10.1088/0264-9381/14/11/002 https://arxiv.org/abs/gr-qc/9707024</ref><ref name="Finazzi et al 2009">S. Finazzi, S. Liberati, C. Barceló, "Semiclassical instability of dynamical warp drives", Physical Review D 79, 124017 (2009)https://doi.org/10.1103/PhysRevD.79.124017 https://arxiv.org/abs/0904.0141</ref>)

From<ref name="Pieri 2024"></ref>
<blockquote>
It should be emphasised that Quantum Inequalities have not been proven for realistic interacting quantum fields, where the they are expected to be weaker if they exist [Cadamuro, 2019], and they often require Minkowskian conditions at infinity, a condition which is not appropriate for fields inside bounded regions, as in a Casimir devices.
</blockquote>

== Warp drives and black holes ==

What happens if you drive your warp drive into a black hole?

The pop-sci reason that nothing can escape a black hole is that at the event horizon the escape velocity is faster than light and nothing can go faster than light. A warp drive can go faster than light.

The more detailed reason from general relativity is that at the event horizon space and time are rotated sufficiently that "inward" becomes "forward in time." You can no more go farther away from the singularity in a black hole once you are inside the event horizon than you can go backward in time. Any super-luminal travel can go backward in time, and warp drives can engage in super-luminal travel.

The Hawking area theorem that says that black holes can only grow, never shrink relies on the condition that the energy is everywhere positive. Warp drives have negative energy density.

So in principle, there's nothing that prevents a warp drive from taking a dip into a black hole and coming back out again.

Garattini and Zatrimaylov looked into the problem of a warp drive near (and in) a black hole<ref name="Garattini and Zatrimaylov">R. Garattini and K. Zatrimaylov, "Black holes, warp drives, and energy conditions", Physics Letters B 856 138910 (2024) https://doi.org/10.1016/j.physletb.2024.138910</ref>. The found that the black hole reduces the negative energy requirement for the warp bubble to move inward toward the black hole singularity, but increases the negative energy needed to move away from the center of the black hole. In addition, a warp drive parked at the event horizon would allow light from inside the horizon to pass through the bubble and escape back outside.

== Credit ==
Author: Luke Campbell

== References ==

[[Category:Physics]][[Category:Infrastructure]][[Category:Physics & Engineering‏‎]][[Category:Physics & Math & Engineering]][[Category:Transportation & Infrastructure‏‎]][[Category:Metric Engineering]]

Warp Drives

2026-07-26T22:49:56Z

Lwcamp: /* So what is going on? */

{{PageConstructionNotice}}

Science fiction often features spacecraft that can seemingly move across space and get between the place of departure and the destination much faster than light could have done. This appears to contradict the theory of relativity, which predicts unequivocally that nothing can move through space faster than light. Because relativity has been incredibly successful at describing nature, with its many other predictions regularly being confirmed to extraordinary accuracy and within the bounds of uncertainty of all the experiments that tested them, it gives confidence that relativity is a correct description of reality. Which seems to rather throw a wet towel on our hopes for rapid travel between stars.

However, while relativity does not allow things to move through space faster than light, it places no such restrictions on how fast space-time itself can expand, contract, or move around. This leads to the idea of a warp drive – the spacecraft remains stationary within a region of highly curved space-time, and that region moves at super-luminal speeds rather than the spacecraft.

== The Alcubierre warp drive ==

The first warp drive geometry that satisfied the Einstein field equations of relativity was proposed by Miguel Alcubierre<ref>M. Alcubierre, "The warp drive: hyper-fast travel within general relativity." Classical and Quantum Gravity. 11 (5): L73–L77 (1994). [https://arxiv.org/abs/gr-qc/0009013 arXiv:gr-qc/0009013]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1994CQGra..11L..73A 1994CQGra..11L..73A]. doi:[https://doi.org/10.1088%2F0264-9381%2F11%2F5%2F001 10.1088/0264-9381/11/5/001]. S2CID [https://api.semanticscholar.org/CorpusID:4797900 4797900].</ref>. In this geometry, a sphere of space-time moves at an arbitrary speed (potentially but not necessarily a speed much faster than light). Objects within the sphere are moved along with the sphere; an object at rest within the sphere would be moved along with the sphere indefinitely. Space is expanding at the rear boundary and contracting at the front boundary in order to keep the sphere moving. In order to satisfy the Einstein field equations, the boundary of the sphere must have a negative energy density. The challenges of space-time geometries with negative energy densities are described in our page on [[Wormholes#Exotic_energy_conditions|wormholes]], for our purposes it is enough to note that negative energy densities can pose problems if not handled carefully, there may be limits on how much negative energy you can have without nearby positive energy density, and it may not be possible to get enough negative energy to support a warp drive; although none of this is ruled out by physics – yet!

<table align=center>
<tr><td width=650>[[File:Energy_magnitude_of_Alcubierre_drive.png]]
<td width=650>[[File:Expansion_of_Alcubierre_drive.png]]
<tr><td>The magnitude of the energy of the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. (The energy density is negative everywhere in the shell, here only the magnitude is shown to aid visual comprehension.)
<td>The expansion of space in the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. Negative expansion means that space is contracting in that direction, positive expansion that it is expanding.
</table>

NOTE: What the? The energy distribution is symmetric! How does the drive know which way to go? There must be significant contributions of other components of the stress-energy tensor, and those have got to be asymmetric along the forward/backward axis. Check this when I get time.

Almost as soon as Alcubierre proposed his warp drive, others began picking it apart.
<ol>
<li>Pfenning and Ford<ref name="PfenningFord_2001">M. J. Pfenning and L. H. Ford, "The unphysical nature of 'Warp Drive'", Classical and Quantum Gravity. 14 (7): 1743–1751 (1997). arXiv:[https://arxiv.org/abs/gr-qc/9702026 gr-qc/9702026]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1997CQGra..14.1743P 1997CQGra..14.1743P]. doi:[https://doi.org/10.1088%2F0264-9381%2F14%2F7%2F011 10.1088/0264-9381/14/7/011]. S2CID [https://api.semanticscholar.org/CorpusID:15279207 15279207].</ref> found that if the negative energy density wall around the bubble obeys [[Wormholes#Quantum_energy_inequalities|quantum energy inequalities]], then for warp speeds of around light speed the shell thickness would be on the order of a hundred Planck lengths; a distance far smaller than any other known physical phenomenon or object.
<li>In the same work, Pfenning and Ford also found that for a bubble a hundred meters in radius, a warp speed of around the speed of light, and the required thickness of about 100 Planck lengths the energy in the bubble shell would be E ≈ -1063 kg c2. The magnitude of this latter value is ten orders of magnitude larger than the energy of the entire visible universe. They do note, however, that if quantum energy inequalities can be ignored then a warp drive with a hundred meters radius but a shell thickness of one meter would "only" have an energy magnitude of about a quarter solar mass.
<li>As with any method of faster than light travel, the warp drive could be used to make a time machine.
<li>Perhaps most seriously, if the warp drive is going faster than light speed, the negative energy regions on the outside of the shell won't be in the warp parts of the bubble. They will have to be moving through space at faster than light speed if the bubble is to maintain its integrity, which is the very problem that the warp drive was designed to avoid. The outside of the warp bubble shell that maintains the warp bubble would fall away and the warp bubble would quickly erode to sub-luminal speeds.<ref name="Van Den Broeck 2000">Chris Van Den Broeck, "On the (im)possibility of warp bubbles", arXiv:gr-qc/9906050 https://doi.org/10.48550/arXiv.gr-qc/9906050</ref>
<li>A more general problem that includes the previous one is that the geometry of the warp drive determines the distribution of energy, momentum, and stress (pressure, shear, and tension) by Einstein's field equation, which states that the stress energy tensor is directly proportional to a reduced order curvature tensor of space-time. However, the stress-energy tensor must also obey the continuity equation – basically, for the energy to change in a given volume of space that energy must come in or go out of the boundary of that volume and similarly for the momentum. So a warp drive can have the warp condition for one moment in time. But after that, the stress-energy that maintains the warp will start to move according to the forces on it and according to the natural paths that it takes through the warped space-time. There is no guarantee that the configuration of the stress-energy at later times will be sufficient to maintain the warp.
<blockquote>
From the perspective of simulating the warp drive dynamically, the key challenge is stability. One can straightforwardly generate the initial metric that describes the Alcubierre metric and infer the instantaneous matter configuration that must support it using the Einstein Equation, but a time-domain evolution of the coupled matter-gravity system necessitates that we specify the equation of state of the matter, which in turn determines how it evolves. In a perfect fluid, this involves specifying the relation between the fluid pressure and density, but in a more general case, the full relation between the
stress-energy components must be given. There is (to our knowledge) no known equation of state that would maintain the warp drive metric in a stable configuration over time - therefore, whilst one can require that initially, the warp bubble is constant, it will quickly evolve away from that state and, in most cases, the warp fluid and spacetime deformations will disperse or collapse into a central point.
 
– Clough, Dietrich, and Khan<ref>Katy Clough, Tim Dietrich, and Sebastian Khan, "What no one has seen before: gravitational waveforms from warp drive collapse", arXiv:2406.02466 [gr-qc] https://doi.org/10.48550/arXiv.2406.02466</ref>
</blockquote>
<li>In order to get the matter (or more generally, stress-energy) that forms the warp shell up to speeds that could allow the shell to exist, you need to be able to, well, accelerate that matter to warp speeds. Which gives a chicken-and-egg problem - in order to achieve superluminal travel you need to be able to get matter or energy moving superluminally in the first place<ref name="Bobrick and Martire">Alexey Bobrick and Gianni Martire, "Introducing Physical Warp Drives", arXiv:2102.06824 [gr-qc] https://doi.org/10.48550/arXiv.2102.06824</ref><ref>D. H. Coule, "No warp drive", Class. Quantum Grav. 15 (1998) 2523–2527. https://doi.org/10.1088/0264-9381/15/8/026</ref>, which rather makes warp drives unnecessary from the get-go.
<li>A warp bubble moving faster than light will not be able to get any signals in front of it as it will overtake its own signal. In fact, anyone in the interior of the bubble would not be able to send signals to the front surface of the bubble. In science-speak, a horizon forms at the front of the bubble<ref name="Natario">José Natário, "Warp drive with zero expansion", Classical and Quantum Gravity. 19 (6): 1157–1166 (2002). arXiv:[https://arxiv.org/abs/gr-qc/0110086 gr-qc/0110086]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2002CQGra..19.1157N 2002CQGra..19.1157N]. doi:[https://doi.org/10.1088%2F0264-9381%2F19%2F6%2F308 10.1088/0264-9381/19/6/308]. S2CID [https://api.semanticscholar.org/CorpusID:15859984 15859984].</ref>. As a consequence, the bubble cannot be controlled, steered, nor stopped from within the bubble itself.
<li>A Horizon will naturally emit Hawking radiation. It is estimated that the radiation will raise the temperature of the interior of the bubble to approximately 1032 K<ref name="Finazzi 2009">Stefano Finazzi, Stefano Liberati, and Carlos Barceló, "Semiclassical instability of dynamical warp drives", Phys. Rev. D 79, 124017 (12 June, 2009) https://doi.org/10.1103/PhysRevD.79.124017 , arXiv:0904.0141 [gr-qc] https://doi.org/10.48550/arXiv.0904.0141</ref> (at this level of accuracy, the result is the same if you replace temperatures in K by the same number in °C or °F). This temperature, however, may be lower if quantum inequalities do not have to be fulfilled.
<li>Semi-classical instabilities at the horizon are expected to lead to a divergence of the stress-energy tensor at the boundary in a short period of time, thus leading to the destruction of the warp bubble<ref name="Finazzi 2009"></ref>.
</ol>
Helpfully, Van Den Broeck<ref name="VanDenBroeck_2000">C. Van Den Broeck, "Alcubierre’s Warp Drive: Problems and Prospects" AIP Conference Proceedings. 504: 1105–1110 (2000). Bibcode:[https://ui.adsabs.harvard.edu/abs/2000AIPC..504.1105V 2000AIPC..504.1105V]. doi:[https://doi.org/10.1063%2F1.1290913 10.1063/1.1290913].</ref> suggested several solutions – or at least mitigations – for some of these problems
<ol>
<li>Quantum inequalities had not been shown to be true in general for highly curved space-times.
<li>A proposed warp drive geometry<ref name="VanDenBroeck_1999">C. Van Den Broeck, "A 'warp drive' with more reasonable total energy requirements". Classical and Quantum Gravity. 16 (12): 3973–3979 (1999). arXiv:[https://arxiv.org/abs/gr-qc/9905084 gr-qc/9905084]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1999CQGra..16.3973V 1999CQGra..16.3973V]. doi:[https://doi.org/10.1088%2F0264-9381%2F16%2F12%2F314 10.1088/0264-9381/16/12/314]. S2CID [https://api.semanticscholar.org/CorpusID:15466313 15466313]. </ref> (see the van Den Broeck drive, below) would be able to minimize the magnitude of the energy required to about that of the mass of our sun. While still large, it is not unphysically large.
<li>Time travel is always going to be a worry with faster than light travel. There's no neat solution to this one.
<li>It may be possible for the negative energy outside of the superluminal region to compress into the superluminal region to form a shock in space-time. The front surface of the bubble would then be a singularity. It is not clear if such a sharp jump in physical properties is possible, but neither is sure that it is impossible, either. However, while this might help with the negative energy in front of the bubble getting swept up because it cannot move faster than light, it does little to help with the material at the back of the bubble on the outside getting left behind because it cannot keep up.
</ol>
Other mitigating research has also shown
<ol start=9>
<li>Semi-classical divergence only occurs in warp drives with only one spatial dimensions. Those with two or more spatial dimensions (as in our universe) only contain points where divergences occur, and these divergences can be managed. In fact, their effects can be reduced by making the warp shape more elongated<ref>Carlos Barceló, Valentin Boyanov, Luis J. Garay, Eduardo Martín-Martínez,
and Jose M. Sánchez Velázquez, "Warp drive aerodynamics", JHEP08 (2022) 288 https://doi.org/10.1007/JHEP08(2022)288 , arXiv:2207.06458 [gr-qc] https://doi.org/10.48550/arXiv.2207.06458</ref>.
</ol>

(And what's with the ring in all the artwork, anyway?)

=== So what is going on? ===

What is it that makes a warp drive a warp drive, and how does it work?

A warp drive is a concept that comes out of the science of general relativity. General relativity collects space and time together into a single "structure" called spacetime. This structure is curved, and its curvature is determined by the material that is in it. Absent forces acting between matter, particles of matter move through spacetime on the shortest possible paths in this curved environment.

"Spacetime tells matter how to move; matter tells spacetime how to curve." - John Archibald Wheeler

All of this is encompassed in the Einstein field equation

<div align=center>
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 8πG
<td rowspan="2" nowrap="nowrap" align="right"> G = R -  
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right"> R g
<tr>
<td style="border-top:solid 1px black;"> c4
<td style="border-top:solid 1px black;"> 2
</table>
</div>

(8 pi G / c^4) T = R - (1/2) R g [Einstein field equation]

Describing spacetime - lapse function & shift vectors (foliations, etc.)
Diagram of what the shift vector does.

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Lapse_shift.png]]
<tr><td>A folliation of space-time into slices of constant time coordinates (grey). At an event (black circle), the lapse function (yellow) shows how far into the future the point at that event will be projected in a unit amount of proper time experienced by an observer initially at the event. The shift vector (red) shows how that point changes coordinates after the unit amount of proper time. The cyan vector shows the new coordinates the point is projected onto. The lower black surface indicates a foliation of spacetime at the same time coordinate; after unit proper time, the events on the lower foliation get projected to the curved upper black foliation according to the shift vectors and lapse functions at each of their respective points.
</table>
</table>

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Warp_shift.png]]
<tr><td>The shift vector of a warp geometry, in red. At the edges there is no shift. At the center (inside the warp bubble) there is a region of uniform shift so that at a later time the inside of the warp bubble is rigidly translated to different spatial coordinates (in blue). The two foliations remain the same distance apart, because warp drives maintain a unit lapse function.
</table>
</table>

Solve backwards - assume a structure of space-time, figure out what distribution of matter is required for this from the EFE.

What sort of spacetime geometry produces a warp?

Unit lapse function <-> no time dilation

Region of near constant shift vector - that region is "moving" spacetime, the warp bubble.

Shift vector falls off to zero away from warp bubble -> normal flat spacetime.

=== Alcubierre warp interactions with light and matter ===

Space is not entirely empty. It is filled with a diffuse plasma in the form of the interstellar medium, as well as cosmic radiation, light from stars, and cosmic microwave background radiation. A warp drive propagating through space will encounter this stuff. What happens when this matter and radiation have a warp bubble pass across them?

The first analysis of matter encountering a warp bubble was performed by Pfenning and Ford<ref name="PfenningFord_2001"></ref>. They looked at the warp bubble interaction with a massive object at rest with the frame of reference of the warp drive (keep in mind that the rest frame is the same inside and outside the warp bubble; in this rest frame objects inside the bubble have no momentum even though they are, in some sense, changing location rapidly with time). When the warp bubble passes, the object experiences an acceleration in the direction of the warp bubble motion. When the warp shell passes and the object is inside the bubble, it will be moving with approximately the speed of the bubble. Pfenning and Ford analyzed this problem with a continuous distribution of shell energy that, strictly speaking, never falls to zero except at the bubble center and at spatial infinity, so unless the object passes through the center in Pfenning and Ford's description it will never quite get up to the bubble's speed. In this case, the object will move almost as fast as the object but will pass through the bubble in a finite time, after which it will again be at rest with respect to the reference frame but displaced along the direction of the warp bubble motion by some distance. With this description of the warp bubble, a spacecraft of finite size will always be moving a little slower than the warp bubble and would have to use rockets to keep up with it. in addition, the spacecraft would experience tidal forces that would cause stress on the spacecraft's structure.

Pfenning and Ford also examined cases where the shell is of a finite (possibly infinitesimal) width and falls to zero both inside and outside the bubble. In this case any matter encountering the bubble would be collected at the bow of the bubble and thereafter move along with it.

McMonigal et al. analyzed the situation for both massive particles and light moving along the axis of travel of the bubble<ref name="McMonigalEtAl_2012">B. McMonigal, G. F. Lewis, and P. O'Byrne, "Alcubierre warp drive: On the matter of matter". Physical Review D. 85 (6) 064024 (20 March 2012). arXiv:[https://arxiv.org/abs/1202.5708 1202.5708]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2012PhRvD..85f4024M 2012PhRvD..85f4024M]. doi:[https://doi.org/10.1103%2FPhysRevD.85.064024 10.1103/PhysRevD.85.064024]. S2CID [https://api.semanticscholar.org/CorpusID:3993148 3993148].</ref>. They found that light moving opposite the warp direction passed through the bubble without incident, being only somewhat delayed by passing through the bubble. A warp bubble that is warping at sub-luminal speeds can have light catch up from behind it. This light is able to pass through the bubble, and is somewhat advanced in its path by the speed of the bubble. For super-luminal warp bubbles, however, the situation is different. The bubble will catch up to light moving it its own direction that is originally in front of it. This light cannot escape forward, the bubble being too fast. Nor can it escape backward, as the light is propagating forward and the interior of the bubble is at rest. Thus, the light gets caught at the bow of the warp bubble, unable to escape for so long as the warp bubble is active. This light is strongly blue shifted to extremely energetic x-rays and gamma rays. The space behind the bubble is swept clear of forward-moving light.

The situation for matter over-run by a super-luminal warp bubble is similar. Matter moving backward passes through the bubble, being only swept a ways along the bubble's path as sign of its passing but otherwise continuing on their way. Matter at rest behaves the same way as Pfenning and Ford discovered. Meanwhile, matter moving in the same direction of the warp is overtaken and collects at the bow of the bubble, unable to leave for so long as the bubble is warping. This matter "surfing" on the bubble bow is highly accelerated to relativistic speeds, experiencing extreme time dilation.

For a sub-luminal bubble, there is a speed faster than the warp speed where particles coming up from behind overtake the bubble and pass through, out the front. Particles below this speed but still faster than the warp speed still overtake the bubble, but then bounce backwards out of the bubble from behind. They are still moving forward, but are now moving more slowly than the warp speed. Particles moving forward but slower than the bubble will be overtaken and then bounced out the front with a speed higher than the warp speed. Those particles moving backward will pass through the bubble and exit with their initial speed.

The spacecraft inside the bubble will observe that light moving backward compared to the warp direction is blue-shifted, and matter moving backward passes through with increased energy. Meanwhile, a sub-luminal bubble will see light that was moving forward as red-shifted and matter catching up to the bubble from behind will pass by with reduced energy (superluminal bubbles, of course, do not have any matter or radiation catching up to them from behind).

When a superluminal warp bubble that has been collecting matter and radiation for the duration of its journey and blue-shifting it to much higher energies is turned off, all that matter and energy is released as a blast of radiation in the direction the bubble was warping.

McMonigal et al. conclude by noting that any spacecraft in the warp bubble would need shielding to protect against the blue-shifted radiation and matter of increased energy. In addition the destination would be "blasted into oblivion" by the release of matter and radiation that had been caught in the bubble during the trip.

Clark et al.<ref>Chad Clark, William A. Hiscock, and Shane L. Larson, "Null geodesics in the Alcubierre warp drive spacetime: the view from the bridge", arXiv:gr-qc/9907019 (2018) https://doi.org/10.48550/arXiv.gr-qc/9907019</ref> examine what it would look like from within an operating warp drive. They find that light moving at 90 degrees to the direction of motion will look like it is still coming from 90 degrees. However, light from the front hemisphere is squished progressively toward the forward point the faster you go. For a warp bubble moving faster than the speed of light, there is a cone behind the warp bubble where no light can reach it. However, you don't see a black spot behind you – light from the regions in the backward hemisphere from outside that cone that can reach you is distorted toward the direction behind you to fill in that spot. Light from the forward direction is blue-shifted and light from behind is red-shifted. At speeds of around 200 times light speed, the cosmic microwave background would be blue shifted to energies similar to that of the solar photosphere, heating anything in the warp bubble as if they were close to the sun.

== Van Den Broeck warp drive ==

If an Alcubierre warp bubble a hundred meters across requires a magnitude of energy greater than the entire energy in the observable universe, one option to reduce the magnitude of energy used is to make the warp bubble smaller. Much smaller. Pfenning and Ford<ref name="PfenningFord_2001"></ref> suggested making the warp bubble smaller than an atom. Of course, that brings up the problem of how to stuff a spacecraft in there. Van Den Broeck proposed a solution<ref name="VanDenBroeck_1999"></ref>: make the warp bubble only about the size of an atomic nucleus but expand the space inside the bubble enormously so that a spacecraft could fit in. This is somewhat like a nuclear sized wormhole that leads to a pocket universe that holds the spacecraft. Unfortunately, the metric doesn't fit neatly into any [[Wormholes#Three_dimensions|embedding diagram]] that I can figure out, but the math works out so that spatial coordinates inside the bubble are expanded by a factor of 1017 and a spacecraft would have a few hundred meters of bubble interior to putz around in. This trick manages to reduce the magnitude of energy to only about the mass energy of a few stars similar to our own. Other than that, it is otherwise a normal Alcubierre drive.

Further work<ref>S. Krasnikov, "The quantum inequalities do not forbid spacetime shortcuts", Physical Review D. 67 (10) 104013. https://doi.org/10.1103%2FPhysRevD.67.104013, arXiv:gr-qc/0207057 (2003) https://doi.org/10.48550/arXiv.gr-qc/0207057 </ref> has managed to use Van Den Broeck's trick to reduce the magnitude of the negative energy to only about a milligram.

== Natário warp drive ==

The Alcubierre drive is not the only way to construct a warp drive. José Natário<ref name="Natario"></ref> showed that it was just one example of an entire class of warp drives. These are space-time geometries with a smooth and localized shift vector field a shift vector is a vector field that describes how one "slice" of space out of space-time connects to the next) and a unit lapse function (the lapse function describes how the time you experience changes when going from a "slice" of space-time to the next).

Natário then went on to find in that class a set of warp drives that do not have any expansion or contraction of space at all. Instead, space encountering the front boundary of the warp bubble instead "slides" around the outside of the bubble until it gets to the corresponding place on the back and is left behind there. Another way to think of it is that space entering the bubble shell is compressed in the radial direction of the shell but is simultaneously expanded in the tangential direction so that there is no net change in volume. When it gets to the back, the opposite occurs. In the language of mathematics, the shift vector field is divergenceless.

<table align=center>
<tr><td width=360>[[File:Natario_drive_flow_of_space.png]]
<td width=445>[[File:Natario_warp_drive_horizons.png]]
<tr><td style="vertical-align: top;">A representation of the motion of space in a zero-expansion Natário drive with a vanishingly thin warp shell around the bubble.
<td>The horizons of the generalized Natário class of warp drives (including the Alcubierre warp drive and all fell-Heisenberg drives) at superluminal speed in the direction of the green arrow. No event inside the bubble can affect events outside the bubble in front of the blue line. No events outside the bubble behind the red line can affect events inside the bubble. The lines have an angle α with respect to the direction of motion with sin(α) = 1/vs for vs the speed of the bubble relative to light speed. This is analogous to the Mach cone of supersonic objects.
</table>

NOTE: There are two different things that can be meant by the term "Natário warp drive". The generalized Natário warp drive is the general theory of all warp drives that includes within it the Alcubierre, Fell-Heisenberg, and Lentz warp drives, among infinitely many others. The zero-expansion Natário warp drive is the specific implementation of the generalized Natário warp drive that has a divergenceless shift vector field.

The generalized class of warp drives developed by Natário (including both the Alcubierre and this zero-expansion warp drive) are shown to always have regions in the warp shell where the energy density is negative to at least some observers. You can't get away from it – to warp, you need negative energy density.

The Natário class of warp drives blueshift light coming in from the front and redshift light catching up from the back. For the case of a super-luminal warp bubble, there will of course be a horizon at the back of the bubble and light will not be able to get through from behind. For a bubble speed vs relative to the speed of light, light coming from straight ahead will be blueshifted up in frequency and photon energy by a factor of 1 + vs. For sub-luminal travel, light coming from behind will be redshifted by a factor 1 - vs. In general for n as the unit direction along which the light is propagating, the blueshift factor will be 1 + vs ⋅ n. If the warp shell is infinitesimally thin, light from outside that reaches the center of the bubble will not be distorted in direction although it will be frequency shifted, but observers still inside the bubble but displaced from the center will see the field of view distorted as well as frequency shifted.

Most of the difficulties that apply to the Alcubierre warp drive also apply to the generalized Natário warp drive (and thus to the zero-expansion Natário drive and Fell-Heisenberg drive as well). In particular, quantum energy inequalities, high magnitudes of the energy required, formation of time machines, instabilities related to the stress-energy making up the warp shell being unable to keep up with the bubble, the chicken-and-egg problem of needing FTL to achieve FTL, the formation of horizons making warp bubbles uncontrollable, and Hawking radiation at the horizons, are all still concerns for generic warp drives.

== Fell-Heisenberg warp drives ==

S. D. B. Fell and L. Heisenberg investigated what sort of warp structure is necessary to produce a warp drive with positive energy density everywhere<ref name="Fell-Heisenberg">S. D. B. Fell and L> Heisenberg, "Positive energy warp drive from hidden geometric structures", Classical and Quantum Gravity 38 155020 (2021) https://doi.org/10.1088/1361-6382/ac0e47 https://arxiv.org/abs/2104.06488</ref>. They found that warp drives of the generic Natário form with irrotational shift vectors had a simple expression for the energy density and vanishing momentum (irrotational means that the shift vector can be described as the gradient of a scalar potential field or, equivalently, that the curl of the shift vector field vanishes everywhere). Fell and Heisenberg proposed several potential fields that either had positive energy density everywhere except at discontinuities, or that were continuous but had regions that violated the weak energy condition (i. e. negative energy densities in some frames of reference). One specific example they provided of a warp field had an energy that is about 10,000 times less than the mass-energy of our sun. Or only about half the mass-energy of Jupiter. This is a significant improvement over many other proposed drives.

Despite the introduction discussing warp drive configurations that satisfy the various energy conditions, the configurations described in the paper are shown to locally violate the weak and strong energy conditions. Nonetheless, the energy density is still mostly positive. Santiago, Schuster, and Visser's<ref name="Santiago Schuster Visser 2022">J. Santiago, S. Schuster, and M. Visser, "Generic warp drives violate the null energy condition", Physical Review D 105, 064038 (2022) https://doi.org/10.1103/PhysRevD.105.064038</ref> work shows that the various energy conditions must still be violated by this warp drive; to not violate these, energy density must be positive in all reference frames not just those of the co-moving observer.

Any vector field can be described by its divergence, its curl, and a constant. The constant part of a warp drive's shift vector can be discarded because then the warp drive would not be localized. It is interesting, therefore, that the Natário zero-expansion drive has zero divergence and can be described only by its curl, while the Fell-Heisenberg drive has zero curl and can be described only by its divergence. These two classes of drives thus rely on opposite choices of setting one of the two different parts describing a vector field to zero. As might be expected, the Fell-Heisenberg drive is not zero expansion – space-time will change in volume as it passes through the warp bubble (although after going all the way through it will return to its original volume). However, objects in space-time will retain their original orientation as they pass through the warp bubble.

Previous examples of warp drives, such as the Alcubierre and zero-expansion Natário drives, tend toward a flat space-time as you go far enough away faster than would occur if there were any mass present in the warp drive. That is, the ADM mass of such drives is zero. However, one of the Fell-Heisenberg examples described a spacetime where the geometry approaches that of what you would get for a massive object at the warp location, giving one of the first examples of a warp drive with mass.

In the Fell-Heisenberg drive the momentum vanishes everywhere. Yet the energy occupies regions where the shift vector is varying rapidly. The lack of momentum means that the energy will not move to keep up with the differential expansion and movement of the spacetime. As a consequence, at later times the energy will have a different distribution than what is necessary to maintain the given warp configuration. The exact time evolution is not solved but this likely leads to collapse of warp bubble. Note that this is not a problem unique to the Fell-Heisenberg drive. Assuming a geometry that describes a warp produces a specific distribution of the stress-energy tensor that satisfies Einstein's equation. But there is no guarantee that this stress energy tensor satisfies the continuity equation, which describes how the distribution of matter and momentum change with time. The issue of stability of any warp drive is not well understood, but most warp drive geometries will be unstable in the sense that their matter that generates the warp fields will not be able to keep up with the motion of the warp. The Fell-Heisenberg drive is unique only in that, with vanishing momentum, there is no configuration that allows stable propagation.

(n. b. The Heisenberg here is Lavinia Heisenberg, not the Werner Heisenberg of quantum physics and uncertainty principle fame.)

== Lentz warp drive ==

E. Lentz<ref name="Lentz">E. W. Lentz, "Breaking the warp barrier: hyper-fast solitons in Einstein–Maxwell-plasma theory", Classical and Quantum Gravity. 38 075015 (2021). arXiv:[https://arxiv.org/abs/2006.07125 2006.07125]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2021CQGra..38g5015L 2021CQGra..38g5015L]. doi:[https://doi.org/10.1088%2F1361-6382%2Fabe692 10.1088/1361-6382/abe692]. ISSN [https://search.worldcat.org/issn/0264-9381 0264-9381]. S2CID [https://api.semanticscholar.org/CorpusID:219635854 219635854].</ref> investigated a particular configuration of the Fell-Heisenberg class of warp drives, where the scalar field has a hyperbolic shape. The necessary conditions were found to be satisfied by a selection of rhomboid regions of the scalar field, resulting in an energy density that is everywhere positive and a shift vector with vanishing tidal force at the center for a comfortable ride.

However, Santiago, Schuster, and Visser<ref name="Santiago Schuster Visser 2022"></ref> dispute claims that the Lentz drive satisfies the energy conditions, noting that everywhere positive energy density in one frame of reference is insufficient to establish that the energy density is positive in all reference frames; knowledge of the Cauchy stress tensor is also needed. They show that any generic warp drive will violate the strong energy condition, null energy condition, and weak energy condition.

== Conservation laws ==

(Discussion of asymptotic flatness: what is it? Where does it apply?)

(The "big four" conservation laws and how they relate to asymptotic flatness.)

(ADM mass -> 0; conservation of energy/mass issues.)<ref>Sebastian Schuster, Jessica Santiago,and Matt Visser, "ADM mass in warp drive spacetimes" General Relativity and Gravitation (2023) 55:14 https://doi.org/10.1007/s10714-022-03061-9</ref>

(Discuss issues of conservation of angular momentum.)

(If angular momentum conservation is ignored, discuss how momentum & energy are affected by outside forces. In a gravitational field equivalent to inertial frame ... like being in an accelerated elevator; will acquire momentum buildup while staying "at rest".)

== Warp Railroads ==

Van Den Broeck<ref name="VanDenBroeck_2000"></ref> suggested a warp railroad,

You can set up devices ahead of time along the path of the bubble that produce the negative energy regions for the warp bubble at the appropriate time without needing the negative energy to ever move with the bubble at all. At least for bubbles on a predictable schedule and route.

helps with:

-stability (elaborate)

-conservation laws (elaborate)

Mention Krasnikov<ref name="Krasnikov 1998">S. V. Krasnikov, "Hyperfast travel in general relativity" Phys. Rev. D 57 4760</ref>

Hyperwave<ref name="Pieri 2024">Lorenzo Pieri, "Hyperwave: hyper-Fast Communication within General Relativity", arXiv:2311.12069 [gr-qc] https://doi.org/10.48550/arXiv.2311.12069</ref> makes use of this warp railroad idea.

== Quantum effects ==

(quantum stuff here<ref name="Hiscock1997">W. A. Hiscock, "Quantum effects in the Alcubierre warp drive spacetime", Classical and Quantum Gravity 14 L183 https://doi.org/10.1088/0264-9381/14/11/002 https://arxiv.org/abs/gr-qc/9707024</ref><ref name="Finazzi et al 2009">S. Finazzi, S. Liberati, C. Barceló, "Semiclassical instability of dynamical warp drives", Physical Review D 79, 124017 (2009)https://doi.org/10.1103/PhysRevD.79.124017 https://arxiv.org/abs/0904.0141</ref>)

From<ref name="Pieri 2024"></ref>
<blockquote>
It should be emphasised that Quantum Inequalities have not been proven for realistic interacting quantum fields, where the they are expected to be weaker if they exist [Cadamuro, 2019], and they often require Minkowskian conditions at infinity, a condition which is not appropriate for fields inside bounded regions, as in a Casimir devices.
</blockquote>

== Warp drives and black holes ==

What happens if you drive your warp drive into a black hole?

The pop-sci reason that nothing can escape a black hole is that at the event horizon the escape velocity is faster than light and nothing can go faster than light. A warp drive can go faster than light.

The more detailed reason from general relativity is that at the event horizon space and time are rotated sufficiently that "inward" becomes "forward in time." You can no more go farther away from the singularity in a black hole once you are inside the event horizon than you can go backward in time. Any super-luminal travel can go backward in time, and warp drives can engage in super-luminal travel.

The Hawking area theorem that says that black holes can only grow, never shrink relies on the condition that the energy is everywhere positive. Warp drives have negative energy density.

So in principle, there's nothing that prevents a warp drive from taking a dip into a black hole and coming back out again.

Garattini and Zatrimaylov looked into the problem of a warp drive near (and in) a black hole<ref name="Garattini and Zatrimaylov">R. Garattini and K. Zatrimaylov, "Black holes, warp drives, and energy conditions", Physics Letters B 856 138910 (2024) https://doi.org/10.1016/j.physletb.2024.138910</ref>. The found that the black hole reduces the negative energy requirement for the warp bubble to move inward toward the black hole singularity, but increases the negative energy needed to move away from the center of the black hole. In addition, a warp drive parked at the event horizon would allow light from inside the horizon to pass through the bubble and escape back outside.

== Credit ==
Author: Luke Campbell

== References ==

[[Category:Physics]][[Category:Infrastructure]][[Category:Physics & Engineering‏‎]][[Category:Physics & Math & Engineering]][[Category:Transportation & Infrastructure‏‎]][[Category:Metric Engineering]]

Warp Drives

2026-07-26T22:49:41Z

Lwcamp: /* So what is going on? */

{{PageConstructionNotice}}

Science fiction often features spacecraft that can seemingly move across space and get between the place of departure and the destination much faster than light could have done. This appears to contradict the theory of relativity, which predicts unequivocally that nothing can move through space faster than light. Because relativity has been incredibly successful at describing nature, with its many other predictions regularly being confirmed to extraordinary accuracy and within the bounds of uncertainty of all the experiments that tested them, it gives confidence that relativity is a correct description of reality. Which seems to rather throw a wet towel on our hopes for rapid travel between stars.

However, while relativity does not allow things to move through space faster than light, it places no such restrictions on how fast space-time itself can expand, contract, or move around. This leads to the idea of a warp drive – the spacecraft remains stationary within a region of highly curved space-time, and that region moves at super-luminal speeds rather than the spacecraft.

== The Alcubierre warp drive ==

The first warp drive geometry that satisfied the Einstein field equations of relativity was proposed by Miguel Alcubierre<ref>M. Alcubierre, "The warp drive: hyper-fast travel within general relativity." Classical and Quantum Gravity. 11 (5): L73–L77 (1994). [https://arxiv.org/abs/gr-qc/0009013 arXiv:gr-qc/0009013]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1994CQGra..11L..73A 1994CQGra..11L..73A]. doi:[https://doi.org/10.1088%2F0264-9381%2F11%2F5%2F001 10.1088/0264-9381/11/5/001]. S2CID [https://api.semanticscholar.org/CorpusID:4797900 4797900].</ref>. In this geometry, a sphere of space-time moves at an arbitrary speed (potentially but not necessarily a speed much faster than light). Objects within the sphere are moved along with the sphere; an object at rest within the sphere would be moved along with the sphere indefinitely. Space is expanding at the rear boundary and contracting at the front boundary in order to keep the sphere moving. In order to satisfy the Einstein field equations, the boundary of the sphere must have a negative energy density. The challenges of space-time geometries with negative energy densities are described in our page on [[Wormholes#Exotic_energy_conditions|wormholes]], for our purposes it is enough to note that negative energy densities can pose problems if not handled carefully, there may be limits on how much negative energy you can have without nearby positive energy density, and it may not be possible to get enough negative energy to support a warp drive; although none of this is ruled out by physics – yet!

<table align=center>
<tr><td width=650>[[File:Energy_magnitude_of_Alcubierre_drive.png]]
<td width=650>[[File:Expansion_of_Alcubierre_drive.png]]
<tr><td>The magnitude of the energy of the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. (The energy density is negative everywhere in the shell, here only the magnitude is shown to aid visual comprehension.)
<td>The expansion of space in the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. Negative expansion means that space is contracting in that direction, positive expansion that it is expanding.
</table>

NOTE: What the? The energy distribution is symmetric! How does the drive know which way to go? There must be significant contributions of other components of the stress-energy tensor, and those have got to be asymmetric along the forward/backward axis. Check this when I get time.

Almost as soon as Alcubierre proposed his warp drive, others began picking it apart.
<ol>
<li>Pfenning and Ford<ref name="PfenningFord_2001">M. J. Pfenning and L. H. Ford, "The unphysical nature of 'Warp Drive'", Classical and Quantum Gravity. 14 (7): 1743–1751 (1997). arXiv:[https://arxiv.org/abs/gr-qc/9702026 gr-qc/9702026]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1997CQGra..14.1743P 1997CQGra..14.1743P]. doi:[https://doi.org/10.1088%2F0264-9381%2F14%2F7%2F011 10.1088/0264-9381/14/7/011]. S2CID [https://api.semanticscholar.org/CorpusID:15279207 15279207].</ref> found that if the negative energy density wall around the bubble obeys [[Wormholes#Quantum_energy_inequalities|quantum energy inequalities]], then for warp speeds of around light speed the shell thickness would be on the order of a hundred Planck lengths; a distance far smaller than any other known physical phenomenon or object.
<li>In the same work, Pfenning and Ford also found that for a bubble a hundred meters in radius, a warp speed of around the speed of light, and the required thickness of about 100 Planck lengths the energy in the bubble shell would be E ≈ -1063 kg c2. The magnitude of this latter value is ten orders of magnitude larger than the energy of the entire visible universe. They do note, however, that if quantum energy inequalities can be ignored then a warp drive with a hundred meters radius but a shell thickness of one meter would "only" have an energy magnitude of about a quarter solar mass.
<li>As with any method of faster than light travel, the warp drive could be used to make a time machine.
<li>Perhaps most seriously, if the warp drive is going faster than light speed, the negative energy regions on the outside of the shell won't be in the warp parts of the bubble. They will have to be moving through space at faster than light speed if the bubble is to maintain its integrity, which is the very problem that the warp drive was designed to avoid. The outside of the warp bubble shell that maintains the warp bubble would fall away and the warp bubble would quickly erode to sub-luminal speeds.<ref name="Van Den Broeck 2000">Chris Van Den Broeck, "On the (im)possibility of warp bubbles", arXiv:gr-qc/9906050 https://doi.org/10.48550/arXiv.gr-qc/9906050</ref>
<li>A more general problem that includes the previous one is that the geometry of the warp drive determines the distribution of energy, momentum, and stress (pressure, shear, and tension) by Einstein's field equation, which states that the stress energy tensor is directly proportional to a reduced order curvature tensor of space-time. However, the stress-energy tensor must also obey the continuity equation – basically, for the energy to change in a given volume of space that energy must come in or go out of the boundary of that volume and similarly for the momentum. So a warp drive can have the warp condition for one moment in time. But after that, the stress-energy that maintains the warp will start to move according to the forces on it and according to the natural paths that it takes through the warped space-time. There is no guarantee that the configuration of the stress-energy at later times will be sufficient to maintain the warp.
<blockquote>
From the perspective of simulating the warp drive dynamically, the key challenge is stability. One can straightforwardly generate the initial metric that describes the Alcubierre metric and infer the instantaneous matter configuration that must support it using the Einstein Equation, but a time-domain evolution of the coupled matter-gravity system necessitates that we specify the equation of state of the matter, which in turn determines how it evolves. In a perfect fluid, this involves specifying the relation between the fluid pressure and density, but in a more general case, the full relation between the
stress-energy components must be given. There is (to our knowledge) no known equation of state that would maintain the warp drive metric in a stable configuration over time - therefore, whilst one can require that initially, the warp bubble is constant, it will quickly evolve away from that state and, in most cases, the warp fluid and spacetime deformations will disperse or collapse into a central point.
 
– Clough, Dietrich, and Khan<ref>Katy Clough, Tim Dietrich, and Sebastian Khan, "What no one has seen before: gravitational waveforms from warp drive collapse", arXiv:2406.02466 [gr-qc] https://doi.org/10.48550/arXiv.2406.02466</ref>
</blockquote>
<li>In order to get the matter (or more generally, stress-energy) that forms the warp shell up to speeds that could allow the shell to exist, you need to be able to, well, accelerate that matter to warp speeds. Which gives a chicken-and-egg problem - in order to achieve superluminal travel you need to be able to get matter or energy moving superluminally in the first place<ref name="Bobrick and Martire">Alexey Bobrick and Gianni Martire, "Introducing Physical Warp Drives", arXiv:2102.06824 [gr-qc] https://doi.org/10.48550/arXiv.2102.06824</ref><ref>D. H. Coule, "No warp drive", Class. Quantum Grav. 15 (1998) 2523–2527. https://doi.org/10.1088/0264-9381/15/8/026</ref>, which rather makes warp drives unnecessary from the get-go.
<li>A warp bubble moving faster than light will not be able to get any signals in front of it as it will overtake its own signal. In fact, anyone in the interior of the bubble would not be able to send signals to the front surface of the bubble. In science-speak, a horizon forms at the front of the bubble<ref name="Natario">José Natário, "Warp drive with zero expansion", Classical and Quantum Gravity. 19 (6): 1157–1166 (2002). arXiv:[https://arxiv.org/abs/gr-qc/0110086 gr-qc/0110086]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2002CQGra..19.1157N 2002CQGra..19.1157N]. doi:[https://doi.org/10.1088%2F0264-9381%2F19%2F6%2F308 10.1088/0264-9381/19/6/308]. S2CID [https://api.semanticscholar.org/CorpusID:15859984 15859984].</ref>. As a consequence, the bubble cannot be controlled, steered, nor stopped from within the bubble itself.
<li>A Horizon will naturally emit Hawking radiation. It is estimated that the radiation will raise the temperature of the interior of the bubble to approximately 1032 K<ref name="Finazzi 2009">Stefano Finazzi, Stefano Liberati, and Carlos Barceló, "Semiclassical instability of dynamical warp drives", Phys. Rev. D 79, 124017 (12 June, 2009) https://doi.org/10.1103/PhysRevD.79.124017 , arXiv:0904.0141 [gr-qc] https://doi.org/10.48550/arXiv.0904.0141</ref> (at this level of accuracy, the result is the same if you replace temperatures in K by the same number in °C or °F). This temperature, however, may be lower if quantum inequalities do not have to be fulfilled.
<li>Semi-classical instabilities at the horizon are expected to lead to a divergence of the stress-energy tensor at the boundary in a short period of time, thus leading to the destruction of the warp bubble<ref name="Finazzi 2009"></ref>.
</ol>
Helpfully, Van Den Broeck<ref name="VanDenBroeck_2000">C. Van Den Broeck, "Alcubierre’s Warp Drive: Problems and Prospects" AIP Conference Proceedings. 504: 1105–1110 (2000). Bibcode:[https://ui.adsabs.harvard.edu/abs/2000AIPC..504.1105V 2000AIPC..504.1105V]. doi:[https://doi.org/10.1063%2F1.1290913 10.1063/1.1290913].</ref> suggested several solutions – or at least mitigations – for some of these problems
<ol>
<li>Quantum inequalities had not been shown to be true in general for highly curved space-times.
<li>A proposed warp drive geometry<ref name="VanDenBroeck_1999">C. Van Den Broeck, "A 'warp drive' with more reasonable total energy requirements". Classical and Quantum Gravity. 16 (12): 3973–3979 (1999). arXiv:[https://arxiv.org/abs/gr-qc/9905084 gr-qc/9905084]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1999CQGra..16.3973V 1999CQGra..16.3973V]. doi:[https://doi.org/10.1088%2F0264-9381%2F16%2F12%2F314 10.1088/0264-9381/16/12/314]. S2CID [https://api.semanticscholar.org/CorpusID:15466313 15466313]. </ref> (see the van Den Broeck drive, below) would be able to minimize the magnitude of the energy required to about that of the mass of our sun. While still large, it is not unphysically large.
<li>Time travel is always going to be a worry with faster than light travel. There's no neat solution to this one.
<li>It may be possible for the negative energy outside of the superluminal region to compress into the superluminal region to form a shock in space-time. The front surface of the bubble would then be a singularity. It is not clear if such a sharp jump in physical properties is possible, but neither is sure that it is impossible, either. However, while this might help with the negative energy in front of the bubble getting swept up because it cannot move faster than light, it does little to help with the material at the back of the bubble on the outside getting left behind because it cannot keep up.
</ol>
Other mitigating research has also shown
<ol start=9>
<li>Semi-classical divergence only occurs in warp drives with only one spatial dimensions. Those with two or more spatial dimensions (as in our universe) only contain points where divergences occur, and these divergences can be managed. In fact, their effects can be reduced by making the warp shape more elongated<ref>Carlos Barceló, Valentin Boyanov, Luis J. Garay, Eduardo Martín-Martínez,
and Jose M. Sánchez Velázquez, "Warp drive aerodynamics", JHEP08 (2022) 288 https://doi.org/10.1007/JHEP08(2022)288 , arXiv:2207.06458 [gr-qc] https://doi.org/10.48550/arXiv.2207.06458</ref>.
</ol>

(And what's with the ring in all the artwork, anyway?)

=== So what is going on? ===

What is it that makes a warp drive a warp drive, and how does it work?

A warp drive is a concept that comes out of the science of general relativity. General relativity collects space and time together into a single "structure" called spacetime. This structure is curved, and its curvature is determined by the material that is in it. Absent forces acting between matter, particles of matter move through spacetime on the shortest possible paths in this curved environment.

"Spacetime tells matter how to move; matter tells spacetime how to curve." - John Archibald Wheeler

All of this is encompassed in the Einstein field equation

<div align=center>
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 8πG
<td rowspan="2" nowrap="nowrap" align="right"> G = R - &nbsp
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right"> R g
<tr>
<td style="border-top:solid 1px black;"> c4
<td style="border-top:solid 1px black;"> 2
</table>
</div>

(8 pi G / c^4) T = R - (1/2) R g [Einstein field equation]

Describing spacetime - lapse function & shift vectors (foliations, etc.)
Diagram of what the shift vector does.

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Lapse_shift.png]]
<tr><td>A folliation of space-time into slices of constant time coordinates (grey). At an event (black circle), the lapse function (yellow) shows how far into the future the point at that event will be projected in a unit amount of proper time experienced by an observer initially at the event. The shift vector (red) shows how that point changes coordinates after the unit amount of proper time. The cyan vector shows the new coordinates the point is projected onto. The lower black surface indicates a foliation of spacetime at the same time coordinate; after unit proper time, the events on the lower foliation get projected to the curved upper black foliation according to the shift vectors and lapse functions at each of their respective points.
</table>
</table>

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Warp_shift.png]]
<tr><td>The shift vector of a warp geometry, in red. At the edges there is no shift. At the center (inside the warp bubble) there is a region of uniform shift so that at a later time the inside of the warp bubble is rigidly translated to different spatial coordinates (in blue). The two foliations remain the same distance apart, because warp drives maintain a unit lapse function.
</table>
</table>

Solve backwards - assume a structure of space-time, figure out what distribution of matter is required for this from the EFE.

What sort of spacetime geometry produces a warp?

Unit lapse function <-> no time dilation

Region of near constant shift vector - that region is "moving" spacetime, the warp bubble.

Shift vector falls off to zero away from warp bubble -> normal flat spacetime.

=== Alcubierre warp interactions with light and matter ===

Space is not entirely empty. It is filled with a diffuse plasma in the form of the interstellar medium, as well as cosmic radiation, light from stars, and cosmic microwave background radiation. A warp drive propagating through space will encounter this stuff. What happens when this matter and radiation have a warp bubble pass across them?

The first analysis of matter encountering a warp bubble was performed by Pfenning and Ford<ref name="PfenningFord_2001"></ref>. They looked at the warp bubble interaction with a massive object at rest with the frame of reference of the warp drive (keep in mind that the rest frame is the same inside and outside the warp bubble; in this rest frame objects inside the bubble have no momentum even though they are, in some sense, changing location rapidly with time). When the warp bubble passes, the object experiences an acceleration in the direction of the warp bubble motion. When the warp shell passes and the object is inside the bubble, it will be moving with approximately the speed of the bubble. Pfenning and Ford analyzed this problem with a continuous distribution of shell energy that, strictly speaking, never falls to zero except at the bubble center and at spatial infinity, so unless the object passes through the center in Pfenning and Ford's description it will never quite get up to the bubble's speed. In this case, the object will move almost as fast as the object but will pass through the bubble in a finite time, after which it will again be at rest with respect to the reference frame but displaced along the direction of the warp bubble motion by some distance. With this description of the warp bubble, a spacecraft of finite size will always be moving a little slower than the warp bubble and would have to use rockets to keep up with it. in addition, the spacecraft would experience tidal forces that would cause stress on the spacecraft's structure.

Pfenning and Ford also examined cases where the shell is of a finite (possibly infinitesimal) width and falls to zero both inside and outside the bubble. In this case any matter encountering the bubble would be collected at the bow of the bubble and thereafter move along with it.

McMonigal et al. analyzed the situation for both massive particles and light moving along the axis of travel of the bubble<ref name="McMonigalEtAl_2012">B. McMonigal, G. F. Lewis, and P. O'Byrne, "Alcubierre warp drive: On the matter of matter". Physical Review D. 85 (6) 064024 (20 March 2012). arXiv:[https://arxiv.org/abs/1202.5708 1202.5708]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2012PhRvD..85f4024M 2012PhRvD..85f4024M]. doi:[https://doi.org/10.1103%2FPhysRevD.85.064024 10.1103/PhysRevD.85.064024]. S2CID [https://api.semanticscholar.org/CorpusID:3993148 3993148].</ref>. They found that light moving opposite the warp direction passed through the bubble without incident, being only somewhat delayed by passing through the bubble. A warp bubble that is warping at sub-luminal speeds can have light catch up from behind it. This light is able to pass through the bubble, and is somewhat advanced in its path by the speed of the bubble. For super-luminal warp bubbles, however, the situation is different. The bubble will catch up to light moving it its own direction that is originally in front of it. This light cannot escape forward, the bubble being too fast. Nor can it escape backward, as the light is propagating forward and the interior of the bubble is at rest. Thus, the light gets caught at the bow of the warp bubble, unable to escape for so long as the warp bubble is active. This light is strongly blue shifted to extremely energetic x-rays and gamma rays. The space behind the bubble is swept clear of forward-moving light.

The situation for matter over-run by a super-luminal warp bubble is similar. Matter moving backward passes through the bubble, being only swept a ways along the bubble's path as sign of its passing but otherwise continuing on their way. Matter at rest behaves the same way as Pfenning and Ford discovered. Meanwhile, matter moving in the same direction of the warp is overtaken and collects at the bow of the bubble, unable to leave for so long as the bubble is warping. This matter "surfing" on the bubble bow is highly accelerated to relativistic speeds, experiencing extreme time dilation.

For a sub-luminal bubble, there is a speed faster than the warp speed where particles coming up from behind overtake the bubble and pass through, out the front. Particles below this speed but still faster than the warp speed still overtake the bubble, but then bounce backwards out of the bubble from behind. They are still moving forward, but are now moving more slowly than the warp speed. Particles moving forward but slower than the bubble will be overtaken and then bounced out the front with a speed higher than the warp speed. Those particles moving backward will pass through the bubble and exit with their initial speed.

The spacecraft inside the bubble will observe that light moving backward compared to the warp direction is blue-shifted, and matter moving backward passes through with increased energy. Meanwhile, a sub-luminal bubble will see light that was moving forward as red-shifted and matter catching up to the bubble from behind will pass by with reduced energy (superluminal bubbles, of course, do not have any matter or radiation catching up to them from behind).

When a superluminal warp bubble that has been collecting matter and radiation for the duration of its journey and blue-shifting it to much higher energies is turned off, all that matter and energy is released as a blast of radiation in the direction the bubble was warping.

McMonigal et al. conclude by noting that any spacecraft in the warp bubble would need shielding to protect against the blue-shifted radiation and matter of increased energy. In addition the destination would be "blasted into oblivion" by the release of matter and radiation that had been caught in the bubble during the trip.

Clark et al.<ref>Chad Clark, William A. Hiscock, and Shane L. Larson, "Null geodesics in the Alcubierre warp drive spacetime: the view from the bridge", arXiv:gr-qc/9907019 (2018) https://doi.org/10.48550/arXiv.gr-qc/9907019</ref> examine what it would look like from within an operating warp drive. They find that light moving at 90 degrees to the direction of motion will look like it is still coming from 90 degrees. However, light from the front hemisphere is squished progressively toward the forward point the faster you go. For a warp bubble moving faster than the speed of light, there is a cone behind the warp bubble where no light can reach it. However, you don't see a black spot behind you – light from the regions in the backward hemisphere from outside that cone that can reach you is distorted toward the direction behind you to fill in that spot. Light from the forward direction is blue-shifted and light from behind is red-shifted. At speeds of around 200 times light speed, the cosmic microwave background would be blue shifted to energies similar to that of the solar photosphere, heating anything in the warp bubble as if they were close to the sun.

== Van Den Broeck warp drive ==

If an Alcubierre warp bubble a hundred meters across requires a magnitude of energy greater than the entire energy in the observable universe, one option to reduce the magnitude of energy used is to make the warp bubble smaller. Much smaller. Pfenning and Ford<ref name="PfenningFord_2001"></ref> suggested making the warp bubble smaller than an atom. Of course, that brings up the problem of how to stuff a spacecraft in there. Van Den Broeck proposed a solution<ref name="VanDenBroeck_1999"></ref>: make the warp bubble only about the size of an atomic nucleus but expand the space inside the bubble enormously so that a spacecraft could fit in. This is somewhat like a nuclear sized wormhole that leads to a pocket universe that holds the spacecraft. Unfortunately, the metric doesn't fit neatly into any [[Wormholes#Three_dimensions|embedding diagram]] that I can figure out, but the math works out so that spatial coordinates inside the bubble are expanded by a factor of 1017 and a spacecraft would have a few hundred meters of bubble interior to putz around in. This trick manages to reduce the magnitude of energy to only about the mass energy of a few stars similar to our own. Other than that, it is otherwise a normal Alcubierre drive.

Further work<ref>S. Krasnikov, "The quantum inequalities do not forbid spacetime shortcuts", Physical Review D. 67 (10) 104013. https://doi.org/10.1103%2FPhysRevD.67.104013, arXiv:gr-qc/0207057 (2003) https://doi.org/10.48550/arXiv.gr-qc/0207057 </ref> has managed to use Van Den Broeck's trick to reduce the magnitude of the negative energy to only about a milligram.

== Natário warp drive ==

The Alcubierre drive is not the only way to construct a warp drive. José Natário<ref name="Natario"></ref> showed that it was just one example of an entire class of warp drives. These are space-time geometries with a smooth and localized shift vector field a shift vector is a vector field that describes how one "slice" of space out of space-time connects to the next) and a unit lapse function (the lapse function describes how the time you experience changes when going from a "slice" of space-time to the next).

Natário then went on to find in that class a set of warp drives that do not have any expansion or contraction of space at all. Instead, space encountering the front boundary of the warp bubble instead "slides" around the outside of the bubble until it gets to the corresponding place on the back and is left behind there. Another way to think of it is that space entering the bubble shell is compressed in the radial direction of the shell but is simultaneously expanded in the tangential direction so that there is no net change in volume. When it gets to the back, the opposite occurs. In the language of mathematics, the shift vector field is divergenceless.

<table align=center>
<tr><td width=360>[[File:Natario_drive_flow_of_space.png]]
<td width=445>[[File:Natario_warp_drive_horizons.png]]
<tr><td style="vertical-align: top;">A representation of the motion of space in a zero-expansion Natário drive with a vanishingly thin warp shell around the bubble.
<td>The horizons of the generalized Natário class of warp drives (including the Alcubierre warp drive and all fell-Heisenberg drives) at superluminal speed in the direction of the green arrow. No event inside the bubble can affect events outside the bubble in front of the blue line. No events outside the bubble behind the red line can affect events inside the bubble. The lines have an angle α with respect to the direction of motion with sin(α) = 1/vs for vs the speed of the bubble relative to light speed. This is analogous to the Mach cone of supersonic objects.
</table>

NOTE: There are two different things that can be meant by the term "Natário warp drive". The generalized Natário warp drive is the general theory of all warp drives that includes within it the Alcubierre, Fell-Heisenberg, and Lentz warp drives, among infinitely many others. The zero-expansion Natário warp drive is the specific implementation of the generalized Natário warp drive that has a divergenceless shift vector field.

The generalized class of warp drives developed by Natário (including both the Alcubierre and this zero-expansion warp drive) are shown to always have regions in the warp shell where the energy density is negative to at least some observers. You can't get away from it – to warp, you need negative energy density.

The Natário class of warp drives blueshift light coming in from the front and redshift light catching up from the back. For the case of a super-luminal warp bubble, there will of course be a horizon at the back of the bubble and light will not be able to get through from behind. For a bubble speed vs relative to the speed of light, light coming from straight ahead will be blueshifted up in frequency and photon energy by a factor of 1 + vs. For sub-luminal travel, light coming from behind will be redshifted by a factor 1 - vs. In general for n as the unit direction along which the light is propagating, the blueshift factor will be 1 + vs ⋅ n. If the warp shell is infinitesimally thin, light from outside that reaches the center of the bubble will not be distorted in direction although it will be frequency shifted, but observers still inside the bubble but displaced from the center will see the field of view distorted as well as frequency shifted.

Most of the difficulties that apply to the Alcubierre warp drive also apply to the generalized Natário warp drive (and thus to the zero-expansion Natário drive and Fell-Heisenberg drive as well). In particular, quantum energy inequalities, high magnitudes of the energy required, formation of time machines, instabilities related to the stress-energy making up the warp shell being unable to keep up with the bubble, the chicken-and-egg problem of needing FTL to achieve FTL, the formation of horizons making warp bubbles uncontrollable, and Hawking radiation at the horizons, are all still concerns for generic warp drives.

== Fell-Heisenberg warp drives ==

S. D. B. Fell and L. Heisenberg investigated what sort of warp structure is necessary to produce a warp drive with positive energy density everywhere<ref name="Fell-Heisenberg">S. D. B. Fell and L> Heisenberg, "Positive energy warp drive from hidden geometric structures", Classical and Quantum Gravity 38 155020 (2021) https://doi.org/10.1088/1361-6382/ac0e47 https://arxiv.org/abs/2104.06488</ref>. They found that warp drives of the generic Natário form with irrotational shift vectors had a simple expression for the energy density and vanishing momentum (irrotational means that the shift vector can be described as the gradient of a scalar potential field or, equivalently, that the curl of the shift vector field vanishes everywhere). Fell and Heisenberg proposed several potential fields that either had positive energy density everywhere except at discontinuities, or that were continuous but had regions that violated the weak energy condition (i. e. negative energy densities in some frames of reference). One specific example they provided of a warp field had an energy that is about 10,000 times less than the mass-energy of our sun. Or only about half the mass-energy of Jupiter. This is a significant improvement over many other proposed drives.

Despite the introduction discussing warp drive configurations that satisfy the various energy conditions, the configurations described in the paper are shown to locally violate the weak and strong energy conditions. Nonetheless, the energy density is still mostly positive. Santiago, Schuster, and Visser's<ref name="Santiago Schuster Visser 2022">J. Santiago, S. Schuster, and M. Visser, "Generic warp drives violate the null energy condition", Physical Review D 105, 064038 (2022) https://doi.org/10.1103/PhysRevD.105.064038</ref> work shows that the various energy conditions must still be violated by this warp drive; to not violate these, energy density must be positive in all reference frames not just those of the co-moving observer.

Any vector field can be described by its divergence, its curl, and a constant. The constant part of a warp drive's shift vector can be discarded because then the warp drive would not be localized. It is interesting, therefore, that the Natário zero-expansion drive has zero divergence and can be described only by its curl, while the Fell-Heisenberg drive has zero curl and can be described only by its divergence. These two classes of drives thus rely on opposite choices of setting one of the two different parts describing a vector field to zero. As might be expected, the Fell-Heisenberg drive is not zero expansion – space-time will change in volume as it passes through the warp bubble (although after going all the way through it will return to its original volume). However, objects in space-time will retain their original orientation as they pass through the warp bubble.

Previous examples of warp drives, such as the Alcubierre and zero-expansion Natário drives, tend toward a flat space-time as you go far enough away faster than would occur if there were any mass present in the warp drive. That is, the ADM mass of such drives is zero. However, one of the Fell-Heisenberg examples described a spacetime where the geometry approaches that of what you would get for a massive object at the warp location, giving one of the first examples of a warp drive with mass.

In the Fell-Heisenberg drive the momentum vanishes everywhere. Yet the energy occupies regions where the shift vector is varying rapidly. The lack of momentum means that the energy will not move to keep up with the differential expansion and movement of the spacetime. As a consequence, at later times the energy will have a different distribution than what is necessary to maintain the given warp configuration. The exact time evolution is not solved but this likely leads to collapse of warp bubble. Note that this is not a problem unique to the Fell-Heisenberg drive. Assuming a geometry that describes a warp produces a specific distribution of the stress-energy tensor that satisfies Einstein's equation. But there is no guarantee that this stress energy tensor satisfies the continuity equation, which describes how the distribution of matter and momentum change with time. The issue of stability of any warp drive is not well understood, but most warp drive geometries will be unstable in the sense that their matter that generates the warp fields will not be able to keep up with the motion of the warp. The Fell-Heisenberg drive is unique only in that, with vanishing momentum, there is no configuration that allows stable propagation.

(n. b. The Heisenberg here is Lavinia Heisenberg, not the Werner Heisenberg of quantum physics and uncertainty principle fame.)

== Lentz warp drive ==

E. Lentz<ref name="Lentz">E. W. Lentz, "Breaking the warp barrier: hyper-fast solitons in Einstein–Maxwell-plasma theory", Classical and Quantum Gravity. 38 075015 (2021). arXiv:[https://arxiv.org/abs/2006.07125 2006.07125]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2021CQGra..38g5015L 2021CQGra..38g5015L]. doi:[https://doi.org/10.1088%2F1361-6382%2Fabe692 10.1088/1361-6382/abe692]. ISSN [https://search.worldcat.org/issn/0264-9381 0264-9381]. S2CID [https://api.semanticscholar.org/CorpusID:219635854 219635854].</ref> investigated a particular configuration of the Fell-Heisenberg class of warp drives, where the scalar field has a hyperbolic shape. The necessary conditions were found to be satisfied by a selection of rhomboid regions of the scalar field, resulting in an energy density that is everywhere positive and a shift vector with vanishing tidal force at the center for a comfortable ride.

However, Santiago, Schuster, and Visser<ref name="Santiago Schuster Visser 2022"></ref> dispute claims that the Lentz drive satisfies the energy conditions, noting that everywhere positive energy density in one frame of reference is insufficient to establish that the energy density is positive in all reference frames; knowledge of the Cauchy stress tensor is also needed. They show that any generic warp drive will violate the strong energy condition, null energy condition, and weak energy condition.

== Conservation laws ==

(Discussion of asymptotic flatness: what is it? Where does it apply?)

(The "big four" conservation laws and how they relate to asymptotic flatness.)

(ADM mass -> 0; conservation of energy/mass issues.)<ref>Sebastian Schuster, Jessica Santiago,and Matt Visser, "ADM mass in warp drive spacetimes" General Relativity and Gravitation (2023) 55:14 https://doi.org/10.1007/s10714-022-03061-9</ref>

(Discuss issues of conservation of angular momentum.)

(If angular momentum conservation is ignored, discuss how momentum & energy are affected by outside forces. In a gravitational field equivalent to inertial frame ... like being in an accelerated elevator; will acquire momentum buildup while staying "at rest".)

== Warp Railroads ==

Van Den Broeck<ref name="VanDenBroeck_2000"></ref> suggested a warp railroad,

You can set up devices ahead of time along the path of the bubble that produce the negative energy regions for the warp bubble at the appropriate time without needing the negative energy to ever move with the bubble at all. At least for bubbles on a predictable schedule and route.

helps with:

-stability (elaborate)

-conservation laws (elaborate)

Mention Krasnikov<ref name="Krasnikov 1998">S. V. Krasnikov, "Hyperfast travel in general relativity" Phys. Rev. D 57 4760</ref>

Hyperwave<ref name="Pieri 2024">Lorenzo Pieri, "Hyperwave: hyper-Fast Communication within General Relativity", arXiv:2311.12069 [gr-qc] https://doi.org/10.48550/arXiv.2311.12069</ref> makes use of this warp railroad idea.

== Quantum effects ==

(quantum stuff here<ref name="Hiscock1997">W. A. Hiscock, "Quantum effects in the Alcubierre warp drive spacetime", Classical and Quantum Gravity 14 L183 https://doi.org/10.1088/0264-9381/14/11/002 https://arxiv.org/abs/gr-qc/9707024</ref><ref name="Finazzi et al 2009">S. Finazzi, S. Liberati, C. Barceló, "Semiclassical instability of dynamical warp drives", Physical Review D 79, 124017 (2009)https://doi.org/10.1103/PhysRevD.79.124017 https://arxiv.org/abs/0904.0141</ref>)

From<ref name="Pieri 2024"></ref>
<blockquote>
It should be emphasised that Quantum Inequalities have not been proven for realistic interacting quantum fields, where the they are expected to be weaker if they exist [Cadamuro, 2019], and they often require Minkowskian conditions at infinity, a condition which is not appropriate for fields inside bounded regions, as in a Casimir devices.
</blockquote>

== Warp drives and black holes ==

What happens if you drive your warp drive into a black hole?

The pop-sci reason that nothing can escape a black hole is that at the event horizon the escape velocity is faster than light and nothing can go faster than light. A warp drive can go faster than light.

The more detailed reason from general relativity is that at the event horizon space and time are rotated sufficiently that "inward" becomes "forward in time." You can no more go farther away from the singularity in a black hole once you are inside the event horizon than you can go backward in time. Any super-luminal travel can go backward in time, and warp drives can engage in super-luminal travel.

The Hawking area theorem that says that black holes can only grow, never shrink relies on the condition that the energy is everywhere positive. Warp drives have negative energy density.

So in principle, there's nothing that prevents a warp drive from taking a dip into a black hole and coming back out again.

Garattini and Zatrimaylov looked into the problem of a warp drive near (and in) a black hole<ref name="Garattini and Zatrimaylov">R. Garattini and K. Zatrimaylov, "Black holes, warp drives, and energy conditions", Physics Letters B 856 138910 (2024) https://doi.org/10.1016/j.physletb.2024.138910</ref>. The found that the black hole reduces the negative energy requirement for the warp bubble to move inward toward the black hole singularity, but increases the negative energy needed to move away from the center of the black hole. In addition, a warp drive parked at the event horizon would allow light from inside the horizon to pass through the bubble and escape back outside.

== Credit ==
Author: Luke Campbell

== References ==

[[Category:Physics]][[Category:Infrastructure]][[Category:Physics & Engineering‏‎]][[Category:Physics & Math & Engineering]][[Category:Transportation & Infrastructure‏‎]][[Category:Metric Engineering]]

Warp Drives

2026-07-26T18:31:54Z

Lwcamp: /* So what is going on? */

{{PageConstructionNotice}}

Science fiction often features spacecraft that can seemingly move across space and get between the place of departure and the destination much faster than light could have done. This appears to contradict the theory of relativity, which predicts unequivocally that nothing can move through space faster than light. Because relativity has been incredibly successful at describing nature, with its many other predictions regularly being confirmed to extraordinary accuracy and within the bounds of uncertainty of all the experiments that tested them, it gives confidence that relativity is a correct description of reality. Which seems to rather throw a wet towel on our hopes for rapid travel between stars.

However, while relativity does not allow things to move through space faster than light, it places no such restrictions on how fast space-time itself can expand, contract, or move around. This leads to the idea of a warp drive – the spacecraft remains stationary within a region of highly curved space-time, and that region moves at super-luminal speeds rather than the spacecraft.

== The Alcubierre warp drive ==

The first warp drive geometry that satisfied the Einstein field equations of relativity was proposed by Miguel Alcubierre<ref>M. Alcubierre, "The warp drive: hyper-fast travel within general relativity." Classical and Quantum Gravity. 11 (5): L73–L77 (1994). [https://arxiv.org/abs/gr-qc/0009013 arXiv:gr-qc/0009013]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1994CQGra..11L..73A 1994CQGra..11L..73A]. doi:[https://doi.org/10.1088%2F0264-9381%2F11%2F5%2F001 10.1088/0264-9381/11/5/001]. S2CID [https://api.semanticscholar.org/CorpusID:4797900 4797900].</ref>. In this geometry, a sphere of space-time moves at an arbitrary speed (potentially but not necessarily a speed much faster than light). Objects within the sphere are moved along with the sphere; an object at rest within the sphere would be moved along with the sphere indefinitely. Space is expanding at the rear boundary and contracting at the front boundary in order to keep the sphere moving. In order to satisfy the Einstein field equations, the boundary of the sphere must have a negative energy density. The challenges of space-time geometries with negative energy densities are described in our page on [[Wormholes#Exotic_energy_conditions|wormholes]], for our purposes it is enough to note that negative energy densities can pose problems if not handled carefully, there may be limits on how much negative energy you can have without nearby positive energy density, and it may not be possible to get enough negative energy to support a warp drive; although none of this is ruled out by physics – yet!

<table align=center>
<tr><td width=650>[[File:Energy_magnitude_of_Alcubierre_drive.png]]
<td width=650>[[File:Expansion_of_Alcubierre_drive.png]]
<tr><td>The magnitude of the energy of the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. (The energy density is negative everywhere in the shell, here only the magnitude is shown to aid visual comprehension.)
<td>The expansion of space in the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. Negative expansion means that space is contracting in that direction, positive expansion that it is expanding.
</table>

NOTE: What the? The energy distribution is symmetric! How does the drive know which way to go? There must be significant contributions of other components of the stress-energy tensor, and those have got to be asymmetric along the forward/backward axis. Check this when I get time.

Almost as soon as Alcubierre proposed his warp drive, others began picking it apart.
<ol>
<li>Pfenning and Ford<ref name="PfenningFord_2001">M. J. Pfenning and L. H. Ford, "The unphysical nature of 'Warp Drive'", Classical and Quantum Gravity. 14 (7): 1743–1751 (1997). arXiv:[https://arxiv.org/abs/gr-qc/9702026 gr-qc/9702026]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1997CQGra..14.1743P 1997CQGra..14.1743P]. doi:[https://doi.org/10.1088%2F0264-9381%2F14%2F7%2F011 10.1088/0264-9381/14/7/011]. S2CID [https://api.semanticscholar.org/CorpusID:15279207 15279207].</ref> found that if the negative energy density wall around the bubble obeys [[Wormholes#Quantum_energy_inequalities|quantum energy inequalities]], then for warp speeds of around light speed the shell thickness would be on the order of a hundred Planck lengths; a distance far smaller than any other known physical phenomenon or object.
<li>In the same work, Pfenning and Ford also found that for a bubble a hundred meters in radius, a warp speed of around the speed of light, and the required thickness of about 100 Planck lengths the energy in the bubble shell would be E ≈ -1063 kg c2. The magnitude of this latter value is ten orders of magnitude larger than the energy of the entire visible universe. They do note, however, that if quantum energy inequalities can be ignored then a warp drive with a hundred meters radius but a shell thickness of one meter would "only" have an energy magnitude of about a quarter solar mass.
<li>As with any method of faster than light travel, the warp drive could be used to make a time machine.
<li>Perhaps most seriously, if the warp drive is going faster than light speed, the negative energy regions on the outside of the shell won't be in the warp parts of the bubble. They will have to be moving through space at faster than light speed if the bubble is to maintain its integrity, which is the very problem that the warp drive was designed to avoid. The outside of the warp bubble shell that maintains the warp bubble would fall away and the warp bubble would quickly erode to sub-luminal speeds.<ref name="Van Den Broeck 2000">Chris Van Den Broeck, "On the (im)possibility of warp bubbles", arXiv:gr-qc/9906050 https://doi.org/10.48550/arXiv.gr-qc/9906050</ref>
<li>A more general problem that includes the previous one is that the geometry of the warp drive determines the distribution of energy, momentum, and stress (pressure, shear, and tension) by Einstein's field equation, which states that the stress energy tensor is directly proportional to a reduced order curvature tensor of space-time. However, the stress-energy tensor must also obey the continuity equation – basically, for the energy to change in a given volume of space that energy must come in or go out of the boundary of that volume and similarly for the momentum. So a warp drive can have the warp condition for one moment in time. But after that, the stress-energy that maintains the warp will start to move according to the forces on it and according to the natural paths that it takes through the warped space-time. There is no guarantee that the configuration of the stress-energy at later times will be sufficient to maintain the warp.
<blockquote>
From the perspective of simulating the warp drive dynamically, the key challenge is stability. One can straightforwardly generate the initial metric that describes the Alcubierre metric and infer the instantaneous matter configuration that must support it using the Einstein Equation, but a time-domain evolution of the coupled matter-gravity system necessitates that we specify the equation of state of the matter, which in turn determines how it evolves. In a perfect fluid, this involves specifying the relation between the fluid pressure and density, but in a more general case, the full relation between the
stress-energy components must be given. There is (to our knowledge) no known equation of state that would maintain the warp drive metric in a stable configuration over time - therefore, whilst one can require that initially, the warp bubble is constant, it will quickly evolve away from that state and, in most cases, the warp fluid and spacetime deformations will disperse or collapse into a central point.
 
– Clough, Dietrich, and Khan<ref>Katy Clough, Tim Dietrich, and Sebastian Khan, "What no one has seen before: gravitational waveforms from warp drive collapse", arXiv:2406.02466 [gr-qc] https://doi.org/10.48550/arXiv.2406.02466</ref>
</blockquote>
<li>In order to get the matter (or more generally, stress-energy) that forms the warp shell up to speeds that could allow the shell to exist, you need to be able to, well, accelerate that matter to warp speeds. Which gives a chicken-and-egg problem - in order to achieve superluminal travel you need to be able to get matter or energy moving superluminally in the first place<ref name="Bobrick and Martire">Alexey Bobrick and Gianni Martire, "Introducing Physical Warp Drives", arXiv:2102.06824 [gr-qc] https://doi.org/10.48550/arXiv.2102.06824</ref><ref>D. H. Coule, "No warp drive", Class. Quantum Grav. 15 (1998) 2523–2527. https://doi.org/10.1088/0264-9381/15/8/026</ref>, which rather makes warp drives unnecessary from the get-go.
<li>A warp bubble moving faster than light will not be able to get any signals in front of it as it will overtake its own signal. In fact, anyone in the interior of the bubble would not be able to send signals to the front surface of the bubble. In science-speak, a horizon forms at the front of the bubble<ref name="Natario">José Natário, "Warp drive with zero expansion", Classical and Quantum Gravity. 19 (6): 1157–1166 (2002). arXiv:[https://arxiv.org/abs/gr-qc/0110086 gr-qc/0110086]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2002CQGra..19.1157N 2002CQGra..19.1157N]. doi:[https://doi.org/10.1088%2F0264-9381%2F19%2F6%2F308 10.1088/0264-9381/19/6/308]. S2CID [https://api.semanticscholar.org/CorpusID:15859984 15859984].</ref>. As a consequence, the bubble cannot be controlled, steered, nor stopped from within the bubble itself.
<li>A Horizon will naturally emit Hawking radiation. It is estimated that the radiation will raise the temperature of the interior of the bubble to approximately 1032 K<ref name="Finazzi 2009">Stefano Finazzi, Stefano Liberati, and Carlos Barceló, "Semiclassical instability of dynamical warp drives", Phys. Rev. D 79, 124017 (12 June, 2009) https://doi.org/10.1103/PhysRevD.79.124017 , arXiv:0904.0141 [gr-qc] https://doi.org/10.48550/arXiv.0904.0141</ref> (at this level of accuracy, the result is the same if you replace temperatures in K by the same number in °C or °F). This temperature, however, may be lower if quantum inequalities do not have to be fulfilled.
<li>Semi-classical instabilities at the horizon are expected to lead to a divergence of the stress-energy tensor at the boundary in a short period of time, thus leading to the destruction of the warp bubble<ref name="Finazzi 2009"></ref>.
</ol>
Helpfully, Van Den Broeck<ref name="VanDenBroeck_2000">C. Van Den Broeck, "Alcubierre’s Warp Drive: Problems and Prospects" AIP Conference Proceedings. 504: 1105–1110 (2000). Bibcode:[https://ui.adsabs.harvard.edu/abs/2000AIPC..504.1105V 2000AIPC..504.1105V]. doi:[https://doi.org/10.1063%2F1.1290913 10.1063/1.1290913].</ref> suggested several solutions – or at least mitigations – for some of these problems
<ol>
<li>Quantum inequalities had not been shown to be true in general for highly curved space-times.
<li>A proposed warp drive geometry<ref name="VanDenBroeck_1999">C. Van Den Broeck, "A 'warp drive' with more reasonable total energy requirements". Classical and Quantum Gravity. 16 (12): 3973–3979 (1999). arXiv:[https://arxiv.org/abs/gr-qc/9905084 gr-qc/9905084]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1999CQGra..16.3973V 1999CQGra..16.3973V]. doi:[https://doi.org/10.1088%2F0264-9381%2F16%2F12%2F314 10.1088/0264-9381/16/12/314]. S2CID [https://api.semanticscholar.org/CorpusID:15466313 15466313]. </ref> (see the van Den Broeck drive, below) would be able to minimize the magnitude of the energy required to about that of the mass of our sun. While still large, it is not unphysically large.
<li>Time travel is always going to be a worry with faster than light travel. There's no neat solution to this one.
<li>It may be possible for the negative energy outside of the superluminal region to compress into the superluminal region to form a shock in space-time. The front surface of the bubble would then be a singularity. It is not clear if such a sharp jump in physical properties is possible, but neither is sure that it is impossible, either. However, while this might help with the negative energy in front of the bubble getting swept up because it cannot move faster than light, it does little to help with the material at the back of the bubble on the outside getting left behind because it cannot keep up.
</ol>
Other mitigating research has also shown
<ol start=9>
<li>Semi-classical divergence only occurs in warp drives with only one spatial dimensions. Those with two or more spatial dimensions (as in our universe) only contain points where divergences occur, and these divergences can be managed. In fact, their effects can be reduced by making the warp shape more elongated<ref>Carlos Barceló, Valentin Boyanov, Luis J. Garay, Eduardo Martín-Martínez,
and Jose M. Sánchez Velázquez, "Warp drive aerodynamics", JHEP08 (2022) 288 https://doi.org/10.1007/JHEP08(2022)288 , arXiv:2207.06458 [gr-qc] https://doi.org/10.48550/arXiv.2207.06458</ref>.
</ol>

(And what's with the ring in all the artwork, anyway?)

=== So what is going on? ===

What is it that makes a warp drive a warp drive, and how does it work?

(8 pi G / c^4) T = R - (1/2) R g [Einstein field equation]

"Spacetime tells matter how to move; matter tells spacetime how to curve." - John Archibald Wheeler

Describing spacetime - lapse function & shift vectors (foliations, etc.)
Diagram of what the shift vector does.

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Lapse_shift.png]]
<tr><td>A folliation of space-time into slices of constant time coordinates (grey). At an event (black circle), the lapse function (yellow) shows how far into the future the point at that event will be projected in a unit amount of proper time experienced by an observer initially at the event. The shift vector (red) shows how that point changes coordinates after the unit amount of proper time. The cyan vector shows the new coordinates the point is projected onto. The lower black surface indicates a foliation of spacetime at the same time coordinate; after unit proper time, the events on the lower foliation get projected to the curved upper black foliation according to the shift vectors and lapse functions at each of their respective points.
</table>
</table>

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Warp_shift.png]]
<tr><td>The shift vector of a warp geometry, in red. At the edges there is no shift. At the center (inside the warp bubble) there is a region of uniform shift so that at a later time the inside of the warp bubble is rigidly translated to different spatial coordinates (in blue). The two foliations remain the same distance apart, because warp drives maintain a unit lapse function.
</table>
</table>

Solve backwards - assume a structure of space-time, figure out what distribution of matter is required for this from the EFE.

What sort of spacetime geometry produces a warp?

Unit lapse function <-> no time dilation

Region of near constant shift vector - that region is "moving" spacetime, the warp bubble.

Shift vector falls off to zero away from warp bubble -> normal flat spacetime.

=== Alcubierre warp interactions with light and matter ===

Space is not entirely empty. It is filled with a diffuse plasma in the form of the interstellar medium, as well as cosmic radiation, light from stars, and cosmic microwave background radiation. A warp drive propagating through space will encounter this stuff. What happens when this matter and radiation have a warp bubble pass across them?

The first analysis of matter encountering a warp bubble was performed by Pfenning and Ford<ref name="PfenningFord_2001"></ref>. They looked at the warp bubble interaction with a massive object at rest with the frame of reference of the warp drive (keep in mind that the rest frame is the same inside and outside the warp bubble; in this rest frame objects inside the bubble have no momentum even though they are, in some sense, changing location rapidly with time). When the warp bubble passes, the object experiences an acceleration in the direction of the warp bubble motion. When the warp shell passes and the object is inside the bubble, it will be moving with approximately the speed of the bubble. Pfenning and Ford analyzed this problem with a continuous distribution of shell energy that, strictly speaking, never falls to zero except at the bubble center and at spatial infinity, so unless the object passes through the center in Pfenning and Ford's description it will never quite get up to the bubble's speed. In this case, the object will move almost as fast as the object but will pass through the bubble in a finite time, after which it will again be at rest with respect to the reference frame but displaced along the direction of the warp bubble motion by some distance. With this description of the warp bubble, a spacecraft of finite size will always be moving a little slower than the warp bubble and would have to use rockets to keep up with it. in addition, the spacecraft would experience tidal forces that would cause stress on the spacecraft's structure.

Pfenning and Ford also examined cases where the shell is of a finite (possibly infinitesimal) width and falls to zero both inside and outside the bubble. In this case any matter encountering the bubble would be collected at the bow of the bubble and thereafter move along with it.

McMonigal et al. analyzed the situation for both massive particles and light moving along the axis of travel of the bubble<ref name="McMonigalEtAl_2012">B. McMonigal, G. F. Lewis, and P. O'Byrne, "Alcubierre warp drive: On the matter of matter". Physical Review D. 85 (6) 064024 (20 March 2012). arXiv:[https://arxiv.org/abs/1202.5708 1202.5708]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2012PhRvD..85f4024M 2012PhRvD..85f4024M]. doi:[https://doi.org/10.1103%2FPhysRevD.85.064024 10.1103/PhysRevD.85.064024]. S2CID [https://api.semanticscholar.org/CorpusID:3993148 3993148].</ref>. They found that light moving opposite the warp direction passed through the bubble without incident, being only somewhat delayed by passing through the bubble. A warp bubble that is warping at sub-luminal speeds can have light catch up from behind it. This light is able to pass through the bubble, and is somewhat advanced in its path by the speed of the bubble. For super-luminal warp bubbles, however, the situation is different. The bubble will catch up to light moving it its own direction that is originally in front of it. This light cannot escape forward, the bubble being too fast. Nor can it escape backward, as the light is propagating forward and the interior of the bubble is at rest. Thus, the light gets caught at the bow of the warp bubble, unable to escape for so long as the warp bubble is active. This light is strongly blue shifted to extremely energetic x-rays and gamma rays. The space behind the bubble is swept clear of forward-moving light.

The situation for matter over-run by a super-luminal warp bubble is similar. Matter moving backward passes through the bubble, being only swept a ways along the bubble's path as sign of its passing but otherwise continuing on their way. Matter at rest behaves the same way as Pfenning and Ford discovered. Meanwhile, matter moving in the same direction of the warp is overtaken and collects at the bow of the bubble, unable to leave for so long as the bubble is warping. This matter "surfing" on the bubble bow is highly accelerated to relativistic speeds, experiencing extreme time dilation.

For a sub-luminal bubble, there is a speed faster than the warp speed where particles coming up from behind overtake the bubble and pass through, out the front. Particles below this speed but still faster than the warp speed still overtake the bubble, but then bounce backwards out of the bubble from behind. They are still moving forward, but are now moving more slowly than the warp speed. Particles moving forward but slower than the bubble will be overtaken and then bounced out the front with a speed higher than the warp speed. Those particles moving backward will pass through the bubble and exit with their initial speed.

The spacecraft inside the bubble will observe that light moving backward compared to the warp direction is blue-shifted, and matter moving backward passes through with increased energy. Meanwhile, a sub-luminal bubble will see light that was moving forward as red-shifted and matter catching up to the bubble from behind will pass by with reduced energy (superluminal bubbles, of course, do not have any matter or radiation catching up to them from behind).

When a superluminal warp bubble that has been collecting matter and radiation for the duration of its journey and blue-shifting it to much higher energies is turned off, all that matter and energy is released as a blast of radiation in the direction the bubble was warping.

McMonigal et al. conclude by noting that any spacecraft in the warp bubble would need shielding to protect against the blue-shifted radiation and matter of increased energy. In addition the destination would be "blasted into oblivion" by the release of matter and radiation that had been caught in the bubble during the trip.

Clark et al.<ref>Chad Clark, William A. Hiscock, and Shane L. Larson, "Null geodesics in the Alcubierre warp drive spacetime: the view from the bridge", arXiv:gr-qc/9907019 (2018) https://doi.org/10.48550/arXiv.gr-qc/9907019</ref> examine what it would look like from within an operating warp drive. They find that light moving at 90 degrees to the direction of motion will look like it is still coming from 90 degrees. However, light from the front hemisphere is squished progressively toward the forward point the faster you go. For a warp bubble moving faster than the speed of light, there is a cone behind the warp bubble where no light can reach it. However, you don't see a black spot behind you – light from the regions in the backward hemisphere from outside that cone that can reach you is distorted toward the direction behind you to fill in that spot. Light from the forward direction is blue-shifted and light from behind is red-shifted. At speeds of around 200 times light speed, the cosmic microwave background would be blue shifted to energies similar to that of the solar photosphere, heating anything in the warp bubble as if they were close to the sun.

== Van Den Broeck warp drive ==

If an Alcubierre warp bubble a hundred meters across requires a magnitude of energy greater than the entire energy in the observable universe, one option to reduce the magnitude of energy used is to make the warp bubble smaller. Much smaller. Pfenning and Ford<ref name="PfenningFord_2001"></ref> suggested making the warp bubble smaller than an atom. Of course, that brings up the problem of how to stuff a spacecraft in there. Van Den Broeck proposed a solution<ref name="VanDenBroeck_1999"></ref>: make the warp bubble only about the size of an atomic nucleus but expand the space inside the bubble enormously so that a spacecraft could fit in. This is somewhat like a nuclear sized wormhole that leads to a pocket universe that holds the spacecraft. Unfortunately, the metric doesn't fit neatly into any [[Wormholes#Three_dimensions|embedding diagram]] that I can figure out, but the math works out so that spatial coordinates inside the bubble are expanded by a factor of 1017 and a spacecraft would have a few hundred meters of bubble interior to putz around in. This trick manages to reduce the magnitude of energy to only about the mass energy of a few stars similar to our own. Other than that, it is otherwise a normal Alcubierre drive.

Further work<ref>S. Krasnikov, "The quantum inequalities do not forbid spacetime shortcuts", Physical Review D. 67 (10) 104013. https://doi.org/10.1103%2FPhysRevD.67.104013, arXiv:gr-qc/0207057 (2003) https://doi.org/10.48550/arXiv.gr-qc/0207057 </ref> has managed to use Van Den Broeck's trick to reduce the magnitude of the negative energy to only about a milligram.

== Natário warp drive ==

The Alcubierre drive is not the only way to construct a warp drive. José Natário<ref name="Natario"></ref> showed that it was just one example of an entire class of warp drives. These are space-time geometries with a smooth and localized shift vector field a shift vector is a vector field that describes how one "slice" of space out of space-time connects to the next) and a unit lapse function (the lapse function describes how the time you experience changes when going from a "slice" of space-time to the next).

Natário then went on to find in that class a set of warp drives that do not have any expansion or contraction of space at all. Instead, space encountering the front boundary of the warp bubble instead "slides" around the outside of the bubble until it gets to the corresponding place on the back and is left behind there. Another way to think of it is that space entering the bubble shell is compressed in the radial direction of the shell but is simultaneously expanded in the tangential direction so that there is no net change in volume. When it gets to the back, the opposite occurs. In the language of mathematics, the shift vector field is divergenceless.

<table align=center>
<tr><td width=360>[[File:Natario_drive_flow_of_space.png]]
<td width=445>[[File:Natario_warp_drive_horizons.png]]
<tr><td style="vertical-align: top;">A representation of the motion of space in a zero-expansion Natário drive with a vanishingly thin warp shell around the bubble.
<td>The horizons of the generalized Natário class of warp drives (including the Alcubierre warp drive and all fell-Heisenberg drives) at superluminal speed in the direction of the green arrow. No event inside the bubble can affect events outside the bubble in front of the blue line. No events outside the bubble behind the red line can affect events inside the bubble. The lines have an angle α with respect to the direction of motion with sin(α) = 1/vs for vs the speed of the bubble relative to light speed. This is analogous to the Mach cone of supersonic objects.
</table>

NOTE: There are two different things that can be meant by the term "Natário warp drive". The generalized Natário warp drive is the general theory of all warp drives that includes within it the Alcubierre, Fell-Heisenberg, and Lentz warp drives, among infinitely many others. The zero-expansion Natário warp drive is the specific implementation of the generalized Natário warp drive that has a divergenceless shift vector field.

The generalized class of warp drives developed by Natário (including both the Alcubierre and this zero-expansion warp drive) are shown to always have regions in the warp shell where the energy density is negative to at least some observers. You can't get away from it – to warp, you need negative energy density.

The Natário class of warp drives blueshift light coming in from the front and redshift light catching up from the back. For the case of a super-luminal warp bubble, there will of course be a horizon at the back of the bubble and light will not be able to get through from behind. For a bubble speed vs relative to the speed of light, light coming from straight ahead will be blueshifted up in frequency and photon energy by a factor of 1 + vs. For sub-luminal travel, light coming from behind will be redshifted by a factor 1 - vs. In general for n as the unit direction along which the light is propagating, the blueshift factor will be 1 + vs ⋅ n. If the warp shell is infinitesimally thin, light from outside that reaches the center of the bubble will not be distorted in direction although it will be frequency shifted, but observers still inside the bubble but displaced from the center will see the field of view distorted as well as frequency shifted.

Most of the difficulties that apply to the Alcubierre warp drive also apply to the generalized Natário warp drive (and thus to the zero-expansion Natário drive and Fell-Heisenberg drive as well). In particular, quantum energy inequalities, high magnitudes of the energy required, formation of time machines, instabilities related to the stress-energy making up the warp shell being unable to keep up with the bubble, the chicken-and-egg problem of needing FTL to achieve FTL, the formation of horizons making warp bubbles uncontrollable, and Hawking radiation at the horizons, are all still concerns for generic warp drives.

== Fell-Heisenberg warp drives ==

S. D. B. Fell and L. Heisenberg investigated what sort of warp structure is necessary to produce a warp drive with positive energy density everywhere<ref name="Fell-Heisenberg">S. D. B. Fell and L> Heisenberg, "Positive energy warp drive from hidden geometric structures", Classical and Quantum Gravity 38 155020 (2021) https://doi.org/10.1088/1361-6382/ac0e47 https://arxiv.org/abs/2104.06488</ref>. They found that warp drives of the generic Natário form with irrotational shift vectors had a simple expression for the energy density and vanishing momentum (irrotational means that the shift vector can be described as the gradient of a scalar potential field or, equivalently, that the curl of the shift vector field vanishes everywhere). Fell and Heisenberg proposed several potential fields that either had positive energy density everywhere except at discontinuities, or that were continuous but had regions that violated the weak energy condition (i. e. negative energy densities in some frames of reference). One specific example they provided of a warp field had an energy that is about 10,000 times less than the mass-energy of our sun. Or only about half the mass-energy of Jupiter. This is a significant improvement over many other proposed drives.

Despite the introduction discussing warp drive configurations that satisfy the various energy conditions, the configurations described in the paper are shown to locally violate the weak and strong energy conditions. Nonetheless, the energy density is still mostly positive. Santiago, Schuster, and Visser's<ref name="Santiago Schuster Visser 2022">J. Santiago, S. Schuster, and M. Visser, "Generic warp drives violate the null energy condition", Physical Review D 105, 064038 (2022) https://doi.org/10.1103/PhysRevD.105.064038</ref> work shows that the various energy conditions must still be violated by this warp drive; to not violate these, energy density must be positive in all reference frames not just those of the co-moving observer.

Any vector field can be described by its divergence, its curl, and a constant. The constant part of a warp drive's shift vector can be discarded because then the warp drive would not be localized. It is interesting, therefore, that the Natário zero-expansion drive has zero divergence and can be described only by its curl, while the Fell-Heisenberg drive has zero curl and can be described only by its divergence. These two classes of drives thus rely on opposite choices of setting one of the two different parts describing a vector field to zero. As might be expected, the Fell-Heisenberg drive is not zero expansion – space-time will change in volume as it passes through the warp bubble (although after going all the way through it will return to its original volume). However, objects in space-time will retain their original orientation as they pass through the warp bubble.

Previous examples of warp drives, such as the Alcubierre and zero-expansion Natário drives, tend toward a flat space-time as you go far enough away faster than would occur if there were any mass present in the warp drive. That is, the ADM mass of such drives is zero. However, one of the Fell-Heisenberg examples described a spacetime where the geometry approaches that of what you would get for a massive object at the warp location, giving one of the first examples of a warp drive with mass.

In the Fell-Heisenberg drive the momentum vanishes everywhere. Yet the energy occupies regions where the shift vector is varying rapidly. The lack of momentum means that the energy will not move to keep up with the differential expansion and movement of the spacetime. As a consequence, at later times the energy will have a different distribution than what is necessary to maintain the given warp configuration. The exact time evolution is not solved but this likely leads to collapse of warp bubble. Note that this is not a problem unique to the Fell-Heisenberg drive. Assuming a geometry that describes a warp produces a specific distribution of the stress-energy tensor that satisfies Einstein's equation. But there is no guarantee that this stress energy tensor satisfies the continuity equation, which describes how the distribution of matter and momentum change with time. The issue of stability of any warp drive is not well understood, but most warp drive geometries will be unstable in the sense that their matter that generates the warp fields will not be able to keep up with the motion of the warp. The Fell-Heisenberg drive is unique only in that, with vanishing momentum, there is no configuration that allows stable propagation.

(n. b. The Heisenberg here is Lavinia Heisenberg, not the Werner Heisenberg of quantum physics and uncertainty principle fame.)

== Lentz warp drive ==

E. Lentz<ref name="Lentz">E. W. Lentz, "Breaking the warp barrier: hyper-fast solitons in Einstein–Maxwell-plasma theory", Classical and Quantum Gravity. 38 075015 (2021). arXiv:[https://arxiv.org/abs/2006.07125 2006.07125]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2021CQGra..38g5015L 2021CQGra..38g5015L]. doi:[https://doi.org/10.1088%2F1361-6382%2Fabe692 10.1088/1361-6382/abe692]. ISSN [https://search.worldcat.org/issn/0264-9381 0264-9381]. S2CID [https://api.semanticscholar.org/CorpusID:219635854 219635854].</ref> investigated a particular configuration of the Fell-Heisenberg class of warp drives, where the scalar field has a hyperbolic shape. The necessary conditions were found to be satisfied by a selection of rhomboid regions of the scalar field, resulting in an energy density that is everywhere positive and a shift vector with vanishing tidal force at the center for a comfortable ride.

However, Santiago, Schuster, and Visser<ref name="Santiago Schuster Visser 2022"></ref> dispute claims that the Lentz drive satisfies the energy conditions, noting that everywhere positive energy density in one frame of reference is insufficient to establish that the energy density is positive in all reference frames; knowledge of the Cauchy stress tensor is also needed. They show that any generic warp drive will violate the strong energy condition, null energy condition, and weak energy condition.

== Conservation laws ==

(Discussion of asymptotic flatness: what is it? Where does it apply?)

(The "big four" conservation laws and how they relate to asymptotic flatness.)

(ADM mass -> 0; conservation of energy/mass issues.)<ref>Sebastian Schuster, Jessica Santiago,and Matt Visser, "ADM mass in warp drive spacetimes" General Relativity and Gravitation (2023) 55:14 https://doi.org/10.1007/s10714-022-03061-9</ref>

(Discuss issues of conservation of angular momentum.)

(If angular momentum conservation is ignored, discuss how momentum & energy are affected by outside forces. In a gravitational field equivalent to inertial frame ... like being in an accelerated elevator; will acquire momentum buildup while staying "at rest".)

== Warp Railroads ==

Van Den Broeck<ref name="VanDenBroeck_2000"></ref> suggested a warp railroad,

You can set up devices ahead of time along the path of the bubble that produce the negative energy regions for the warp bubble at the appropriate time without needing the negative energy to ever move with the bubble at all. At least for bubbles on a predictable schedule and route.

helps with:

-stability (elaborate)

-conservation laws (elaborate)

Mention Krasnikov<ref name="Krasnikov 1998">S. V. Krasnikov, "Hyperfast travel in general relativity" Phys. Rev. D 57 4760</ref>

Hyperwave<ref name="Pieri 2024">Lorenzo Pieri, "Hyperwave: hyper-Fast Communication within General Relativity", arXiv:2311.12069 [gr-qc] https://doi.org/10.48550/arXiv.2311.12069</ref> makes use of this warp railroad idea.

== Quantum effects ==

(quantum stuff here<ref name="Hiscock1997">W. A. Hiscock, "Quantum effects in the Alcubierre warp drive spacetime", Classical and Quantum Gravity 14 L183 https://doi.org/10.1088/0264-9381/14/11/002 https://arxiv.org/abs/gr-qc/9707024</ref><ref name="Finazzi et al 2009">S. Finazzi, S. Liberati, C. Barceló, "Semiclassical instability of dynamical warp drives", Physical Review D 79, 124017 (2009)https://doi.org/10.1103/PhysRevD.79.124017 https://arxiv.org/abs/0904.0141</ref>)

From<ref name="Pieri 2024"></ref>
<blockquote>
It should be emphasised that Quantum Inequalities have not been proven for realistic interacting quantum fields, where the they are expected to be weaker if they exist [Cadamuro, 2019], and they often require Minkowskian conditions at infinity, a condition which is not appropriate for fields inside bounded regions, as in a Casimir devices.
</blockquote>

== Warp drives and black holes ==

What happens if you drive your warp drive into a black hole?

The pop-sci reason that nothing can escape a black hole is that at the event horizon the escape velocity is faster than light and nothing can go faster than light. A warp drive can go faster than light.

The more detailed reason from general relativity is that at the event horizon space and time are rotated sufficiently that "inward" becomes "forward in time." You can no more go farther away from the singularity in a black hole once you are inside the event horizon than you can go backward in time. Any super-luminal travel can go backward in time, and warp drives can engage in super-luminal travel.

The Hawking area theorem that says that black holes can only grow, never shrink relies on the condition that the energy is everywhere positive. Warp drives have negative energy density.

So in principle, there's nothing that prevents a warp drive from taking a dip into a black hole and coming back out again.

Garattini and Zatrimaylov looked into the problem of a warp drive near (and in) a black hole<ref name="Garattini and Zatrimaylov">R. Garattini and K. Zatrimaylov, "Black holes, warp drives, and energy conditions", Physics Letters B 856 138910 (2024) https://doi.org/10.1016/j.physletb.2024.138910</ref>. The found that the black hole reduces the negative energy requirement for the warp bubble to move inward toward the black hole singularity, but increases the negative energy needed to move away from the center of the black hole. In addition, a warp drive parked at the event horizon would allow light from inside the horizon to pass through the bubble and escape back outside.

== Credit ==
Author: Luke Campbell

== References ==

[[Category:Physics]][[Category:Infrastructure]][[Category:Physics & Engineering‏‎]][[Category:Physics & Math & Engineering]][[Category:Transportation & Infrastructure‏‎]][[Category:Metric Engineering]]

File:Lapse shift.png

2026-07-26T18:31:12Z

Lwcamp: /* Summary */

== Summary ==
A folliation of space-time into slices of constant time coordinates (grey). At an event (black circle), the lapse function (yellow) shows how far into the future the point at that event will be projected in a unit amount of proper time experienced by an observer initially at the event. The shift vector (red) shows how that point changes coordinates after the unit amount of proper time. The cyan vector shows the new coordinates the point is projected onto. The lower black surface indicates a foliation of spacetime at the same time coordinate; after unit proper time, the events on the lower foliation get projected to the curved upper black foliation according to the shift vectors and lapse functions at each of their respective points.

Warp Drives

2026-07-26T18:30:44Z

Lwcamp: /* So what is going on? */

{{PageConstructionNotice}}

Science fiction often features spacecraft that can seemingly move across space and get between the place of departure and the destination much faster than light could have done. This appears to contradict the theory of relativity, which predicts unequivocally that nothing can move through space faster than light. Because relativity has been incredibly successful at describing nature, with its many other predictions regularly being confirmed to extraordinary accuracy and within the bounds of uncertainty of all the experiments that tested them, it gives confidence that relativity is a correct description of reality. Which seems to rather throw a wet towel on our hopes for rapid travel between stars.

However, while relativity does not allow things to move through space faster than light, it places no such restrictions on how fast space-time itself can expand, contract, or move around. This leads to the idea of a warp drive – the spacecraft remains stationary within a region of highly curved space-time, and that region moves at super-luminal speeds rather than the spacecraft.

== The Alcubierre warp drive ==

The first warp drive geometry that satisfied the Einstein field equations of relativity was proposed by Miguel Alcubierre<ref>M. Alcubierre, "The warp drive: hyper-fast travel within general relativity." Classical and Quantum Gravity. 11 (5): L73–L77 (1994). [https://arxiv.org/abs/gr-qc/0009013 arXiv:gr-qc/0009013]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1994CQGra..11L..73A 1994CQGra..11L..73A]. doi:[https://doi.org/10.1088%2F0264-9381%2F11%2F5%2F001 10.1088/0264-9381/11/5/001]. S2CID [https://api.semanticscholar.org/CorpusID:4797900 4797900].</ref>. In this geometry, a sphere of space-time moves at an arbitrary speed (potentially but not necessarily a speed much faster than light). Objects within the sphere are moved along with the sphere; an object at rest within the sphere would be moved along with the sphere indefinitely. Space is expanding at the rear boundary and contracting at the front boundary in order to keep the sphere moving. In order to satisfy the Einstein field equations, the boundary of the sphere must have a negative energy density. The challenges of space-time geometries with negative energy densities are described in our page on [[Wormholes#Exotic_energy_conditions|wormholes]], for our purposes it is enough to note that negative energy densities can pose problems if not handled carefully, there may be limits on how much negative energy you can have without nearby positive energy density, and it may not be possible to get enough negative energy to support a warp drive; although none of this is ruled out by physics – yet!

<table align=center>
<tr><td width=650>[[File:Energy_magnitude_of_Alcubierre_drive.png]]
<td width=650>[[File:Expansion_of_Alcubierre_drive.png]]
<tr><td>The magnitude of the energy of the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. (The energy density is negative everywhere in the shell, here only the magnitude is shown to aid visual comprehension.)
<td>The expansion of space in the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. Negative expansion means that space is contracting in that direction, positive expansion that it is expanding.
</table>

NOTE: What the? The energy distribution is symmetric! How does the drive know which way to go? There must be significant contributions of other components of the stress-energy tensor, and those have got to be asymmetric along the forward/backward axis. Check this when I get time.

Almost as soon as Alcubierre proposed his warp drive, others began picking it apart.
<ol>
<li>Pfenning and Ford<ref name="PfenningFord_2001">M. J. Pfenning and L. H. Ford, "The unphysical nature of 'Warp Drive'", Classical and Quantum Gravity. 14 (7): 1743–1751 (1997). arXiv:[https://arxiv.org/abs/gr-qc/9702026 gr-qc/9702026]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1997CQGra..14.1743P 1997CQGra..14.1743P]. doi:[https://doi.org/10.1088%2F0264-9381%2F14%2F7%2F011 10.1088/0264-9381/14/7/011]. S2CID [https://api.semanticscholar.org/CorpusID:15279207 15279207].</ref> found that if the negative energy density wall around the bubble obeys [[Wormholes#Quantum_energy_inequalities|quantum energy inequalities]], then for warp speeds of around light speed the shell thickness would be on the order of a hundred Planck lengths; a distance far smaller than any other known physical phenomenon or object.
<li>In the same work, Pfenning and Ford also found that for a bubble a hundred meters in radius, a warp speed of around the speed of light, and the required thickness of about 100 Planck lengths the energy in the bubble shell would be E ≈ -1063 kg c2. The magnitude of this latter value is ten orders of magnitude larger than the energy of the entire visible universe. They do note, however, that if quantum energy inequalities can be ignored then a warp drive with a hundred meters radius but a shell thickness of one meter would "only" have an energy magnitude of about a quarter solar mass.
<li>As with any method of faster than light travel, the warp drive could be used to make a time machine.
<li>Perhaps most seriously, if the warp drive is going faster than light speed, the negative energy regions on the outside of the shell won't be in the warp parts of the bubble. They will have to be moving through space at faster than light speed if the bubble is to maintain its integrity, which is the very problem that the warp drive was designed to avoid. The outside of the warp bubble shell that maintains the warp bubble would fall away and the warp bubble would quickly erode to sub-luminal speeds.<ref name="Van Den Broeck 2000">Chris Van Den Broeck, "On the (im)possibility of warp bubbles", arXiv:gr-qc/9906050 https://doi.org/10.48550/arXiv.gr-qc/9906050</ref>
<li>A more general problem that includes the previous one is that the geometry of the warp drive determines the distribution of energy, momentum, and stress (pressure, shear, and tension) by Einstein's field equation, which states that the stress energy tensor is directly proportional to a reduced order curvature tensor of space-time. However, the stress-energy tensor must also obey the continuity equation – basically, for the energy to change in a given volume of space that energy must come in or go out of the boundary of that volume and similarly for the momentum. So a warp drive can have the warp condition for one moment in time. But after that, the stress-energy that maintains the warp will start to move according to the forces on it and according to the natural paths that it takes through the warped space-time. There is no guarantee that the configuration of the stress-energy at later times will be sufficient to maintain the warp.
<blockquote>
From the perspective of simulating the warp drive dynamically, the key challenge is stability. One can straightforwardly generate the initial metric that describes the Alcubierre metric and infer the instantaneous matter configuration that must support it using the Einstein Equation, but a time-domain evolution of the coupled matter-gravity system necessitates that we specify the equation of state of the matter, which in turn determines how it evolves. In a perfect fluid, this involves specifying the relation between the fluid pressure and density, but in a more general case, the full relation between the
stress-energy components must be given. There is (to our knowledge) no known equation of state that would maintain the warp drive metric in a stable configuration over time - therefore, whilst one can require that initially, the warp bubble is constant, it will quickly evolve away from that state and, in most cases, the warp fluid and spacetime deformations will disperse or collapse into a central point.
 
– Clough, Dietrich, and Khan<ref>Katy Clough, Tim Dietrich, and Sebastian Khan, "What no one has seen before: gravitational waveforms from warp drive collapse", arXiv:2406.02466 [gr-qc] https://doi.org/10.48550/arXiv.2406.02466</ref>
</blockquote>
<li>In order to get the matter (or more generally, stress-energy) that forms the warp shell up to speeds that could allow the shell to exist, you need to be able to, well, accelerate that matter to warp speeds. Which gives a chicken-and-egg problem - in order to achieve superluminal travel you need to be able to get matter or energy moving superluminally in the first place<ref name="Bobrick and Martire">Alexey Bobrick and Gianni Martire, "Introducing Physical Warp Drives", arXiv:2102.06824 [gr-qc] https://doi.org/10.48550/arXiv.2102.06824</ref><ref>D. H. Coule, "No warp drive", Class. Quantum Grav. 15 (1998) 2523–2527. https://doi.org/10.1088/0264-9381/15/8/026</ref>, which rather makes warp drives unnecessary from the get-go.
<li>A warp bubble moving faster than light will not be able to get any signals in front of it as it will overtake its own signal. In fact, anyone in the interior of the bubble would not be able to send signals to the front surface of the bubble. In science-speak, a horizon forms at the front of the bubble<ref name="Natario">José Natário, "Warp drive with zero expansion", Classical and Quantum Gravity. 19 (6): 1157–1166 (2002). arXiv:[https://arxiv.org/abs/gr-qc/0110086 gr-qc/0110086]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2002CQGra..19.1157N 2002CQGra..19.1157N]. doi:[https://doi.org/10.1088%2F0264-9381%2F19%2F6%2F308 10.1088/0264-9381/19/6/308]. S2CID [https://api.semanticscholar.org/CorpusID:15859984 15859984].</ref>. As a consequence, the bubble cannot be controlled, steered, nor stopped from within the bubble itself.
<li>A Horizon will naturally emit Hawking radiation. It is estimated that the radiation will raise the temperature of the interior of the bubble to approximately 1032 K<ref name="Finazzi 2009">Stefano Finazzi, Stefano Liberati, and Carlos Barceló, "Semiclassical instability of dynamical warp drives", Phys. Rev. D 79, 124017 (12 June, 2009) https://doi.org/10.1103/PhysRevD.79.124017 , arXiv:0904.0141 [gr-qc] https://doi.org/10.48550/arXiv.0904.0141</ref> (at this level of accuracy, the result is the same if you replace temperatures in K by the same number in °C or °F). This temperature, however, may be lower if quantum inequalities do not have to be fulfilled.
<li>Semi-classical instabilities at the horizon are expected to lead to a divergence of the stress-energy tensor at the boundary in a short period of time, thus leading to the destruction of the warp bubble<ref name="Finazzi 2009"></ref>.
</ol>
Helpfully, Van Den Broeck<ref name="VanDenBroeck_2000">C. Van Den Broeck, "Alcubierre’s Warp Drive: Problems and Prospects" AIP Conference Proceedings. 504: 1105–1110 (2000). Bibcode:[https://ui.adsabs.harvard.edu/abs/2000AIPC..504.1105V 2000AIPC..504.1105V]. doi:[https://doi.org/10.1063%2F1.1290913 10.1063/1.1290913].</ref> suggested several solutions – or at least mitigations – for some of these problems
<ol>
<li>Quantum inequalities had not been shown to be true in general for highly curved space-times.
<li>A proposed warp drive geometry<ref name="VanDenBroeck_1999">C. Van Den Broeck, "A 'warp drive' with more reasonable total energy requirements". Classical and Quantum Gravity. 16 (12): 3973–3979 (1999). arXiv:[https://arxiv.org/abs/gr-qc/9905084 gr-qc/9905084]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1999CQGra..16.3973V 1999CQGra..16.3973V]. doi:[https://doi.org/10.1088%2F0264-9381%2F16%2F12%2F314 10.1088/0264-9381/16/12/314]. S2CID [https://api.semanticscholar.org/CorpusID:15466313 15466313]. </ref> (see the van Den Broeck drive, below) would be able to minimize the magnitude of the energy required to about that of the mass of our sun. While still large, it is not unphysically large.
<li>Time travel is always going to be a worry with faster than light travel. There's no neat solution to this one.
<li>It may be possible for the negative energy outside of the superluminal region to compress into the superluminal region to form a shock in space-time. The front surface of the bubble would then be a singularity. It is not clear if such a sharp jump in physical properties is possible, but neither is sure that it is impossible, either. However, while this might help with the negative energy in front of the bubble getting swept up because it cannot move faster than light, it does little to help with the material at the back of the bubble on the outside getting left behind because it cannot keep up.
</ol>
Other mitigating research has also shown
<ol start=9>
<li>Semi-classical divergence only occurs in warp drives with only one spatial dimensions. Those with two or more spatial dimensions (as in our universe) only contain points where divergences occur, and these divergences can be managed. In fact, their effects can be reduced by making the warp shape more elongated<ref>Carlos Barceló, Valentin Boyanov, Luis J. Garay, Eduardo Martín-Martínez,
and Jose M. Sánchez Velázquez, "Warp drive aerodynamics", JHEP08 (2022) 288 https://doi.org/10.1007/JHEP08(2022)288 , arXiv:2207.06458 [gr-qc] https://doi.org/10.48550/arXiv.2207.06458</ref>.
</ol>

(And what's with the ring in all the artwork, anyway?)

=== So what is going on? ===

What is it that makes a warp drive a warp drive, and how does it work?

(8 pi G / c^4) G = R - (1/2) R g [Einstein field equation]

"Spacetime tells matter how to move; matter tells spacetime how to curve." - John Archibald Wheeler

Describing spacetime - lapse function & shift vectors (foliations, etc.)
Diagram of what the shift vector does.

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Lapse_shift.png]]
<tr><td>A folliation of space-time into slices of constant time coordinates (grey). At an event (black circle), the lapse function (yellow) shows how far into the future the point at that event will be projected in a unit amount of proper time experienced by an observer initially at the event. The shift vector (red) shows how that point changes coordinates after the unit amount of proper time. The cyan vector shows the new coordinates the point is projected onto. The lower black surface indicates a foliation of spacetime at the same time coordinate; after unit proper time, the events on the lower foliation get projected to the curved upper black foliation according to the shift vectors and lapse functions at each of their respective points.
</table>
</table>

<table border=1>
<tr><td><table>
<tr><td align=center>[[File:Warp_shift.png]]
<tr><td>The shift vector of a warp geometry, in red. At the edges there is no shift. At the center (inside the warp bubble) there is a region of uniform shift so that at a later time the inside of the warp bubble is rigidly translated to different spatial coordinates (in blue). The two foliations remain the same distance apart, because warp drives maintain a unit lapse function.
</table>
</table>

Solve backwards - assume a structure of space-time, figure out what distribution of matter is required for this from the EFE.

What sort of spacetime geometry produces a warp?

Unit lapse function <-> no time dilation

Region of near constant shift vector - that region is "moving" spacetime, the warp bubble.

Shift vector falls off to zero away from warp bubble -> normal flat spacetime.

=== Alcubierre warp interactions with light and matter ===

Space is not entirely empty. It is filled with a diffuse plasma in the form of the interstellar medium, as well as cosmic radiation, light from stars, and cosmic microwave background radiation. A warp drive propagating through space will encounter this stuff. What happens when this matter and radiation have a warp bubble pass across them?

The first analysis of matter encountering a warp bubble was performed by Pfenning and Ford<ref name="PfenningFord_2001"></ref>. They looked at the warp bubble interaction with a massive object at rest with the frame of reference of the warp drive (keep in mind that the rest frame is the same inside and outside the warp bubble; in this rest frame objects inside the bubble have no momentum even though they are, in some sense, changing location rapidly with time). When the warp bubble passes, the object experiences an acceleration in the direction of the warp bubble motion. When the warp shell passes and the object is inside the bubble, it will be moving with approximately the speed of the bubble. Pfenning and Ford analyzed this problem with a continuous distribution of shell energy that, strictly speaking, never falls to zero except at the bubble center and at spatial infinity, so unless the object passes through the center in Pfenning and Ford's description it will never quite get up to the bubble's speed. In this case, the object will move almost as fast as the object but will pass through the bubble in a finite time, after which it will again be at rest with respect to the reference frame but displaced along the direction of the warp bubble motion by some distance. With this description of the warp bubble, a spacecraft of finite size will always be moving a little slower than the warp bubble and would have to use rockets to keep up with it. in addition, the spacecraft would experience tidal forces that would cause stress on the spacecraft's structure.

Pfenning and Ford also examined cases where the shell is of a finite (possibly infinitesimal) width and falls to zero both inside and outside the bubble. In this case any matter encountering the bubble would be collected at the bow of the bubble and thereafter move along with it.

McMonigal et al. analyzed the situation for both massive particles and light moving along the axis of travel of the bubble<ref name="McMonigalEtAl_2012">B. McMonigal, G. F. Lewis, and P. O'Byrne, "Alcubierre warp drive: On the matter of matter". Physical Review D. 85 (6) 064024 (20 March 2012). arXiv:[https://arxiv.org/abs/1202.5708 1202.5708]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2012PhRvD..85f4024M 2012PhRvD..85f4024M]. doi:[https://doi.org/10.1103%2FPhysRevD.85.064024 10.1103/PhysRevD.85.064024]. S2CID [https://api.semanticscholar.org/CorpusID:3993148 3993148].</ref>. They found that light moving opposite the warp direction passed through the bubble without incident, being only somewhat delayed by passing through the bubble. A warp bubble that is warping at sub-luminal speeds can have light catch up from behind it. This light is able to pass through the bubble, and is somewhat advanced in its path by the speed of the bubble. For super-luminal warp bubbles, however, the situation is different. The bubble will catch up to light moving it its own direction that is originally in front of it. This light cannot escape forward, the bubble being too fast. Nor can it escape backward, as the light is propagating forward and the interior of the bubble is at rest. Thus, the light gets caught at the bow of the warp bubble, unable to escape for so long as the warp bubble is active. This light is strongly blue shifted to extremely energetic x-rays and gamma rays. The space behind the bubble is swept clear of forward-moving light.

The situation for matter over-run by a super-luminal warp bubble is similar. Matter moving backward passes through the bubble, being only swept a ways along the bubble's path as sign of its passing but otherwise continuing on their way. Matter at rest behaves the same way as Pfenning and Ford discovered. Meanwhile, matter moving in the same direction of the warp is overtaken and collects at the bow of the bubble, unable to leave for so long as the bubble is warping. This matter "surfing" on the bubble bow is highly accelerated to relativistic speeds, experiencing extreme time dilation.

For a sub-luminal bubble, there is a speed faster than the warp speed where particles coming up from behind overtake the bubble and pass through, out the front. Particles below this speed but still faster than the warp speed still overtake the bubble, but then bounce backwards out of the bubble from behind. They are still moving forward, but are now moving more slowly than the warp speed. Particles moving forward but slower than the bubble will be overtaken and then bounced out the front with a speed higher than the warp speed. Those particles moving backward will pass through the bubble and exit with their initial speed.

The spacecraft inside the bubble will observe that light moving backward compared to the warp direction is blue-shifted, and matter moving backward passes through with increased energy. Meanwhile, a sub-luminal bubble will see light that was moving forward as red-shifted and matter catching up to the bubble from behind will pass by with reduced energy (superluminal bubbles, of course, do not have any matter or radiation catching up to them from behind).

When a superluminal warp bubble that has been collecting matter and radiation for the duration of its journey and blue-shifting it to much higher energies is turned off, all that matter and energy is released as a blast of radiation in the direction the bubble was warping.

McMonigal et al. conclude by noting that any spacecraft in the warp bubble would need shielding to protect against the blue-shifted radiation and matter of increased energy. In addition the destination would be "blasted into oblivion" by the release of matter and radiation that had been caught in the bubble during the trip.

Clark et al.<ref>Chad Clark, William A. Hiscock, and Shane L. Larson, "Null geodesics in the Alcubierre warp drive spacetime: the view from the bridge", arXiv:gr-qc/9907019 (2018) https://doi.org/10.48550/arXiv.gr-qc/9907019</ref> examine what it would look like from within an operating warp drive. They find that light moving at 90 degrees to the direction of motion will look like it is still coming from 90 degrees. However, light from the front hemisphere is squished progressively toward the forward point the faster you go. For a warp bubble moving faster than the speed of light, there is a cone behind the warp bubble where no light can reach it. However, you don't see a black spot behind you – light from the regions in the backward hemisphere from outside that cone that can reach you is distorted toward the direction behind you to fill in that spot. Light from the forward direction is blue-shifted and light from behind is red-shifted. At speeds of around 200 times light speed, the cosmic microwave background would be blue shifted to energies similar to that of the solar photosphere, heating anything in the warp bubble as if they were close to the sun.

== Van Den Broeck warp drive ==

If an Alcubierre warp bubble a hundred meters across requires a magnitude of energy greater than the entire energy in the observable universe, one option to reduce the magnitude of energy used is to make the warp bubble smaller. Much smaller. Pfenning and Ford<ref name="PfenningFord_2001"></ref> suggested making the warp bubble smaller than an atom. Of course, that brings up the problem of how to stuff a spacecraft in there. Van Den Broeck proposed a solution<ref name="VanDenBroeck_1999"></ref>: make the warp bubble only about the size of an atomic nucleus but expand the space inside the bubble enormously so that a spacecraft could fit in. This is somewhat like a nuclear sized wormhole that leads to a pocket universe that holds the spacecraft. Unfortunately, the metric doesn't fit neatly into any [[Wormholes#Three_dimensions|embedding diagram]] that I can figure out, but the math works out so that spatial coordinates inside the bubble are expanded by a factor of 1017 and a spacecraft would have a few hundred meters of bubble interior to putz around in. This trick manages to reduce the magnitude of energy to only about the mass energy of a few stars similar to our own. Other than that, it is otherwise a normal Alcubierre drive.

Further work<ref>S. Krasnikov, "The quantum inequalities do not forbid spacetime shortcuts", Physical Review D. 67 (10) 104013. https://doi.org/10.1103%2FPhysRevD.67.104013, arXiv:gr-qc/0207057 (2003) https://doi.org/10.48550/arXiv.gr-qc/0207057 </ref> has managed to use Van Den Broeck's trick to reduce the magnitude of the negative energy to only about a milligram.

== Natário warp drive ==

The Alcubierre drive is not the only way to construct a warp drive. José Natário<ref name="Natario"></ref> showed that it was just one example of an entire class of warp drives. These are space-time geometries with a smooth and localized shift vector field a shift vector is a vector field that describes how one "slice" of space out of space-time connects to the next) and a unit lapse function (the lapse function describes how the time you experience changes when going from a "slice" of space-time to the next).

Natário then went on to find in that class a set of warp drives that do not have any expansion or contraction of space at all. Instead, space encountering the front boundary of the warp bubble instead "slides" around the outside of the bubble until it gets to the corresponding place on the back and is left behind there. Another way to think of it is that space entering the bubble shell is compressed in the radial direction of the shell but is simultaneously expanded in the tangential direction so that there is no net change in volume. When it gets to the back, the opposite occurs. In the language of mathematics, the shift vector field is divergenceless.

<table align=center>
<tr><td width=360>[[File:Natario_drive_flow_of_space.png]]
<td width=445>[[File:Natario_warp_drive_horizons.png]]
<tr><td style="vertical-align: top;">A representation of the motion of space in a zero-expansion Natário drive with a vanishingly thin warp shell around the bubble.
<td>The horizons of the generalized Natário class of warp drives (including the Alcubierre warp drive and all fell-Heisenberg drives) at superluminal speed in the direction of the green arrow. No event inside the bubble can affect events outside the bubble in front of the blue line. No events outside the bubble behind the red line can affect events inside the bubble. The lines have an angle α with respect to the direction of motion with sin(α) = 1/vs for vs the speed of the bubble relative to light speed. This is analogous to the Mach cone of supersonic objects.
</table>

NOTE: There are two different things that can be meant by the term "Natário warp drive". The generalized Natário warp drive is the general theory of all warp drives that includes within it the Alcubierre, Fell-Heisenberg, and Lentz warp drives, among infinitely many others. The zero-expansion Natário warp drive is the specific implementation of the generalized Natário warp drive that has a divergenceless shift vector field.

The generalized class of warp drives developed by Natário (including both the Alcubierre and this zero-expansion warp drive) are shown to always have regions in the warp shell where the energy density is negative to at least some observers. You can't get away from it – to warp, you need negative energy density.

The Natário class of warp drives blueshift light coming in from the front and redshift light catching up from the back. For the case of a super-luminal warp bubble, there will of course be a horizon at the back of the bubble and light will not be able to get through from behind. For a bubble speed vs relative to the speed of light, light coming from straight ahead will be blueshifted up in frequency and photon energy by a factor of 1 + vs. For sub-luminal travel, light coming from behind will be redshifted by a factor 1 - vs. In general for n as the unit direction along which the light is propagating, the blueshift factor will be 1 + vs ⋅ n. If the warp shell is infinitesimally thin, light from outside that reaches the center of the bubble will not be distorted in direction although it will be frequency shifted, but observers still inside the bubble but displaced from the center will see the field of view distorted as well as frequency shifted.

Most of the difficulties that apply to the Alcubierre warp drive also apply to the generalized Natário warp drive (and thus to the zero-expansion Natário drive and Fell-Heisenberg drive as well). In particular, quantum energy inequalities, high magnitudes of the energy required, formation of time machines, instabilities related to the stress-energy making up the warp shell being unable to keep up with the bubble, the chicken-and-egg problem of needing FTL to achieve FTL, the formation of horizons making warp bubbles uncontrollable, and Hawking radiation at the horizons, are all still concerns for generic warp drives.

== Fell-Heisenberg warp drives ==

S. D. B. Fell and L. Heisenberg investigated what sort of warp structure is necessary to produce a warp drive with positive energy density everywhere<ref name="Fell-Heisenberg">S. D. B. Fell and L> Heisenberg, "Positive energy warp drive from hidden geometric structures", Classical and Quantum Gravity 38 155020 (2021) https://doi.org/10.1088/1361-6382/ac0e47 https://arxiv.org/abs/2104.06488</ref>. They found that warp drives of the generic Natário form with irrotational shift vectors had a simple expression for the energy density and vanishing momentum (irrotational means that the shift vector can be described as the gradient of a scalar potential field or, equivalently, that the curl of the shift vector field vanishes everywhere). Fell and Heisenberg proposed several potential fields that either had positive energy density everywhere except at discontinuities, or that were continuous but had regions that violated the weak energy condition (i. e. negative energy densities in some frames of reference). One specific example they provided of a warp field had an energy that is about 10,000 times less than the mass-energy of our sun. Or only about half the mass-energy of Jupiter. This is a significant improvement over many other proposed drives.

Despite the introduction discussing warp drive configurations that satisfy the various energy conditions, the configurations described in the paper are shown to locally violate the weak and strong energy conditions. Nonetheless, the energy density is still mostly positive. Santiago, Schuster, and Visser's<ref name="Santiago Schuster Visser 2022">J. Santiago, S. Schuster, and M. Visser, "Generic warp drives violate the null energy condition", Physical Review D 105, 064038 (2022) https://doi.org/10.1103/PhysRevD.105.064038</ref> work shows that the various energy conditions must still be violated by this warp drive; to not violate these, energy density must be positive in all reference frames not just those of the co-moving observer.

Any vector field can be described by its divergence, its curl, and a constant. The constant part of a warp drive's shift vector can be discarded because then the warp drive would not be localized. It is interesting, therefore, that the Natário zero-expansion drive has zero divergence and can be described only by its curl, while the Fell-Heisenberg drive has zero curl and can be described only by its divergence. These two classes of drives thus rely on opposite choices of setting one of the two different parts describing a vector field to zero. As might be expected, the Fell-Heisenberg drive is not zero expansion – space-time will change in volume as it passes through the warp bubble (although after going all the way through it will return to its original volume). However, objects in space-time will retain their original orientation as they pass through the warp bubble.

Previous examples of warp drives, such as the Alcubierre and zero-expansion Natário drives, tend toward a flat space-time as you go far enough away faster than would occur if there were any mass present in the warp drive. That is, the ADM mass of such drives is zero. However, one of the Fell-Heisenberg examples described a spacetime where the geometry approaches that of what you would get for a massive object at the warp location, giving one of the first examples of a warp drive with mass.

In the Fell-Heisenberg drive the momentum vanishes everywhere. Yet the energy occupies regions where the shift vector is varying rapidly. The lack of momentum means that the energy will not move to keep up with the differential expansion and movement of the spacetime. As a consequence, at later times the energy will have a different distribution than what is necessary to maintain the given warp configuration. The exact time evolution is not solved but this likely leads to collapse of warp bubble. Note that this is not a problem unique to the Fell-Heisenberg drive. Assuming a geometry that describes a warp produces a specific distribution of the stress-energy tensor that satisfies Einstein's equation. But there is no guarantee that this stress energy tensor satisfies the continuity equation, which describes how the distribution of matter and momentum change with time. The issue of stability of any warp drive is not well understood, but most warp drive geometries will be unstable in the sense that their matter that generates the warp fields will not be able to keep up with the motion of the warp. The Fell-Heisenberg drive is unique only in that, with vanishing momentum, there is no configuration that allows stable propagation.

(n. b. The Heisenberg here is Lavinia Heisenberg, not the Werner Heisenberg of quantum physics and uncertainty principle fame.)

== Lentz warp drive ==

E. Lentz<ref name="Lentz">E. W. Lentz, "Breaking the warp barrier: hyper-fast solitons in Einstein–Maxwell-plasma theory", Classical and Quantum Gravity. 38 075015 (2021). arXiv:[https://arxiv.org/abs/2006.07125 2006.07125]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2021CQGra..38g5015L 2021CQGra..38g5015L]. doi:[https://doi.org/10.1088%2F1361-6382%2Fabe692 10.1088/1361-6382/abe692]. ISSN [https://search.worldcat.org/issn/0264-9381 0264-9381]. S2CID [https://api.semanticscholar.org/CorpusID:219635854 219635854].</ref> investigated a particular configuration of the Fell-Heisenberg class of warp drives, where the scalar field has a hyperbolic shape. The necessary conditions were found to be satisfied by a selection of rhomboid regions of the scalar field, resulting in an energy density that is everywhere positive and a shift vector with vanishing tidal force at the center for a comfortable ride.

However, Santiago, Schuster, and Visser<ref name="Santiago Schuster Visser 2022"></ref> dispute claims that the Lentz drive satisfies the energy conditions, noting that everywhere positive energy density in one frame of reference is insufficient to establish that the energy density is positive in all reference frames; knowledge of the Cauchy stress tensor is also needed. They show that any generic warp drive will violate the strong energy condition, null energy condition, and weak energy condition.

== Conservation laws ==

(Discussion of asymptotic flatness: what is it? Where does it apply?)

(The "big four" conservation laws and how they relate to asymptotic flatness.)

(ADM mass -> 0; conservation of energy/mass issues.)<ref>Sebastian Schuster, Jessica Santiago,and Matt Visser, "ADM mass in warp drive spacetimes" General Relativity and Gravitation (2023) 55:14 https://doi.org/10.1007/s10714-022-03061-9</ref>

(Discuss issues of conservation of angular momentum.)

(If angular momentum conservation is ignored, discuss how momentum & energy are affected by outside forces. In a gravitational field equivalent to inertial frame ... like being in an accelerated elevator; will acquire momentum buildup while staying "at rest".)

== Warp Railroads ==

Van Den Broeck<ref name="VanDenBroeck_2000"></ref> suggested a warp railroad,

You can set up devices ahead of time along the path of the bubble that produce the negative energy regions for the warp bubble at the appropriate time without needing the negative energy to ever move with the bubble at all. At least for bubbles on a predictable schedule and route.

helps with:

-stability (elaborate)

-conservation laws (elaborate)

Mention Krasnikov<ref name="Krasnikov 1998">S. V. Krasnikov, "Hyperfast travel in general relativity" Phys. Rev. D 57 4760</ref>

Hyperwave<ref name="Pieri 2024">Lorenzo Pieri, "Hyperwave: hyper-Fast Communication within General Relativity", arXiv:2311.12069 [gr-qc] https://doi.org/10.48550/arXiv.2311.12069</ref> makes use of this warp railroad idea.

== Quantum effects ==

(quantum stuff here<ref name="Hiscock1997">W. A. Hiscock, "Quantum effects in the Alcubierre warp drive spacetime", Classical and Quantum Gravity 14 L183 https://doi.org/10.1088/0264-9381/14/11/002 https://arxiv.org/abs/gr-qc/9707024</ref><ref name="Finazzi et al 2009">S. Finazzi, S. Liberati, C. Barceló, "Semiclassical instability of dynamical warp drives", Physical Review D 79, 124017 (2009)https://doi.org/10.1103/PhysRevD.79.124017 https://arxiv.org/abs/0904.0141</ref>)

From<ref name="Pieri 2024"></ref>
<blockquote>
It should be emphasised that Quantum Inequalities have not been proven for realistic interacting quantum fields, where the they are expected to be weaker if they exist [Cadamuro, 2019], and they often require Minkowskian conditions at infinity, a condition which is not appropriate for fields inside bounded regions, as in a Casimir devices.
</blockquote>

== Warp drives and black holes ==

What happens if you drive your warp drive into a black hole?

The pop-sci reason that nothing can escape a black hole is that at the event horizon the escape velocity is faster than light and nothing can go faster than light. A warp drive can go faster than light.

The more detailed reason from general relativity is that at the event horizon space and time are rotated sufficiently that "inward" becomes "forward in time." You can no more go farther away from the singularity in a black hole once you are inside the event horizon than you can go backward in time. Any super-luminal travel can go backward in time, and warp drives can engage in super-luminal travel.

The Hawking area theorem that says that black holes can only grow, never shrink relies on the condition that the energy is everywhere positive. Warp drives have negative energy density.

So in principle, there's nothing that prevents a warp drive from taking a dip into a black hole and coming back out again.

Garattini and Zatrimaylov looked into the problem of a warp drive near (and in) a black hole<ref name="Garattini and Zatrimaylov">R. Garattini and K. Zatrimaylov, "Black holes, warp drives, and energy conditions", Physics Letters B 856 138910 (2024) https://doi.org/10.1016/j.physletb.2024.138910</ref>. The found that the black hole reduces the negative energy requirement for the warp bubble to move inward toward the black hole singularity, but increases the negative energy needed to move away from the center of the black hole. In addition, a warp drive parked at the event horizon would allow light from inside the horizon to pass through the bubble and escape back outside.

== Credit ==
Author: Luke Campbell

== References ==

[[Category:Physics]][[Category:Infrastructure]][[Category:Physics & Engineering‏‎]][[Category:Physics & Math & Engineering]][[Category:Transportation & Infrastructure‏‎]][[Category:Metric Engineering]]

Warp Drives

2026-07-26T18:26:20Z

Lwcamp: /* So what is going on? */

{{PageConstructionNotice}}

Science fiction often features spacecraft that can seemingly move across space and get between the place of departure and the destination much faster than light could have done. This appears to contradict the theory of relativity, which predicts unequivocally that nothing can move through space faster than light. Because relativity has been incredibly successful at describing nature, with its many other predictions regularly being confirmed to extraordinary accuracy and within the bounds of uncertainty of all the experiments that tested them, it gives confidence that relativity is a correct description of reality. Which seems to rather throw a wet towel on our hopes for rapid travel between stars.

However, while relativity does not allow things to move through space faster than light, it places no such restrictions on how fast space-time itself can expand, contract, or move around. This leads to the idea of a warp drive – the spacecraft remains stationary within a region of highly curved space-time, and that region moves at super-luminal speeds rather than the spacecraft.

== The Alcubierre warp drive ==

The first warp drive geometry that satisfied the Einstein field equations of relativity was proposed by Miguel Alcubierre<ref>M. Alcubierre, "The warp drive: hyper-fast travel within general relativity." Classical and Quantum Gravity. 11 (5): L73–L77 (1994). [https://arxiv.org/abs/gr-qc/0009013 arXiv:gr-qc/0009013]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1994CQGra..11L..73A 1994CQGra..11L..73A]. doi:[https://doi.org/10.1088%2F0264-9381%2F11%2F5%2F001 10.1088/0264-9381/11/5/001]. S2CID [https://api.semanticscholar.org/CorpusID:4797900 4797900].</ref>. In this geometry, a sphere of space-time moves at an arbitrary speed (potentially but not necessarily a speed much faster than light). Objects within the sphere are moved along with the sphere; an object at rest within the sphere would be moved along with the sphere indefinitely. Space is expanding at the rear boundary and contracting at the front boundary in order to keep the sphere moving. In order to satisfy the Einstein field equations, the boundary of the sphere must have a negative energy density. The challenges of space-time geometries with negative energy densities are described in our page on [[Wormholes#Exotic_energy_conditions|wormholes]], for our purposes it is enough to note that negative energy densities can pose problems if not handled carefully, there may be limits on how much negative energy you can have without nearby positive energy density, and it may not be possible to get enough negative energy to support a warp drive; although none of this is ruled out by physics – yet!

<table align=center>
<tr><td width=650>[[File:Energy_magnitude_of_Alcubierre_drive.png]]
<td width=650>[[File:Expansion_of_Alcubierre_drive.png]]
<tr><td>The magnitude of the energy of the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. (The energy density is negative everywhere in the shell, here only the magnitude is shown to aid visual comprehension.)
<td>The expansion of space in the warp shell of an Alcubierre drive, assuming an infinitesimally thin shell. Negative expansion means that space is contracting in that direction, positive expansion that it is expanding.
</table>

NOTE: What the? The energy distribution is symmetric! How does the drive know which way to go? There must be significant contributions of other components of the stress-energy tensor, and those have got to be asymmetric along the forward/backward axis. Check this when I get time.

Almost as soon as Alcubierre proposed his warp drive, others began picking it apart.
<ol>
<li>Pfenning and Ford<ref name="PfenningFord_2001">M. J. Pfenning and L. H. Ford, "The unphysical nature of 'Warp Drive'", Classical and Quantum Gravity. 14 (7): 1743–1751 (1997). arXiv:[https://arxiv.org/abs/gr-qc/9702026 gr-qc/9702026]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1997CQGra..14.1743P 1997CQGra..14.1743P]. doi:[https://doi.org/10.1088%2F0264-9381%2F14%2F7%2F011 10.1088/0264-9381/14/7/011]. S2CID [https://api.semanticscholar.org/CorpusID:15279207 15279207].</ref> found that if the negative energy density wall around the bubble obeys [[Wormholes#Quantum_energy_inequalities|quantum energy inequalities]], then for warp speeds of around light speed the shell thickness would be on the order of a hundred Planck lengths; a distance far smaller than any other known physical phenomenon or object.
<li>In the same work, Pfenning and Ford also found that for a bubble a hundred meters in radius, a warp speed of around the speed of light, and the required thickness of about 100 Planck lengths the energy in the bubble shell would be E ≈ -1063 kg c2. The magnitude of this latter value is ten orders of magnitude larger than the energy of the entire visible universe. They do note, however, that if quantum energy inequalities can be ignored then a warp drive with a hundred meters radius but a shell thickness of one meter would "only" have an energy magnitude of about a quarter solar mass.
<li>As with any method of faster than light travel, the warp drive could be used to make a time machine.
<li>Perhaps most seriously, if the warp drive is going faster than light speed, the negative energy regions on the outside of the shell won't be in the warp parts of the bubble. They will have to be moving through space at faster than light speed if the bubble is to maintain its integrity, which is the very problem that the warp drive was designed to avoid. The outside of the warp bubble shell that maintains the warp bubble would fall away and the warp bubble would quickly erode to sub-luminal speeds.<ref name="Van Den Broeck 2000">Chris Van Den Broeck, "On the (im)possibility of warp bubbles", arXiv:gr-qc/9906050 https://doi.org/10.48550/arXiv.gr-qc/9906050</ref>
<li>A more general problem that includes the previous one is that the geometry of the warp drive determines the distribution of energy, momentum, and stress (pressure, shear, and tension) by Einstein's field equation, which states that the stress energy tensor is directly proportional to a reduced order curvature tensor of space-time. However, the stress-energy tensor must also obey the continuity equation – basically, for the energy to change in a given volume of space that energy must come in or go out of the boundary of that volume and similarly for the momentum. So a warp drive can have the warp condition for one moment in time. But after that, the stress-energy that maintains the warp will start to move according to the forces on it and according to the natural paths that it takes through the warped space-time. There is no guarantee that the configuration of the stress-energy at later times will be sufficient to maintain the warp.
<blockquote>
From the perspective of simulating the warp drive dynamically, the key challenge is stability. One can straightforwardly generate the initial metric that describes the Alcubierre metric and infer the instantaneous matter configuration that must support it using the Einstein Equation, but a time-domain evolution of the coupled matter-gravity system necessitates that we specify the equation of state of the matter, which in turn determines how it evolves. In a perfect fluid, this involves specifying the relation between the fluid pressure and density, but in a more general case, the full relation between the
stress-energy components must be given. There is (to our knowledge) no known equation of state that would maintain the warp drive metric in a stable configuration over time - therefore, whilst one can require that initially, the warp bubble is constant, it will quickly evolve away from that state and, in most cases, the warp fluid and spacetime deformations will disperse or collapse into a central point.
 
– Clough, Dietrich, and Khan<ref>Katy Clough, Tim Dietrich, and Sebastian Khan, "What no one has seen before: gravitational waveforms from warp drive collapse", arXiv:2406.02466 [gr-qc] https://doi.org/10.48550/arXiv.2406.02466</ref>
</blockquote>
<li>In order to get the matter (or more generally, stress-energy) that forms the warp shell up to speeds that could allow the shell to exist, you need to be able to, well, accelerate that matter to warp speeds. Which gives a chicken-and-egg problem - in order to achieve superluminal travel you need to be able to get matter or energy moving superluminally in the first place<ref name="Bobrick and Martire">Alexey Bobrick and Gianni Martire, "Introducing Physical Warp Drives", arXiv:2102.06824 [gr-qc] https://doi.org/10.48550/arXiv.2102.06824</ref><ref>D. H. Coule, "No warp drive", Class. Quantum Grav. 15 (1998) 2523–2527. https://doi.org/10.1088/0264-9381/15/8/026</ref>, which rather makes warp drives unnecessary from the get-go.
<li>A warp bubble moving faster than light will not be able to get any signals in front of it as it will overtake its own signal. In fact, anyone in the interior of the bubble would not be able to send signals to the front surface of the bubble. In science-speak, a horizon forms at the front of the bubble<ref name="Natario">José Natário, "Warp drive with zero expansion", Classical and Quantum Gravity. 19 (6): 1157–1166 (2002). arXiv:[https://arxiv.org/abs/gr-qc/0110086 gr-qc/0110086]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2002CQGra..19.1157N 2002CQGra..19.1157N]. doi:[https://doi.org/10.1088%2F0264-9381%2F19%2F6%2F308 10.1088/0264-9381/19/6/308]. S2CID [https://api.semanticscholar.org/CorpusID:15859984 15859984].</ref>. As a consequence, the bubble cannot be controlled, steered, nor stopped from within the bubble itself.
<li>A Horizon will naturally emit Hawking radiation. It is estimated that the radiation will raise the temperature of the interior of the bubble to approximately 1032 K<ref name="Finazzi 2009">Stefano Finazzi, Stefano Liberati, and Carlos Barceló, "Semiclassical instability of dynamical warp drives", Phys. Rev. D 79, 124017 (12 June, 2009) https://doi.org/10.1103/PhysRevD.79.124017 , arXiv:0904.0141 [gr-qc] https://doi.org/10.48550/arXiv.0904.0141</ref> (at this level of accuracy, the result is the same if you replace temperatures in K by the same number in °C or °F). This temperature, however, may be lower if quantum inequalities do not have to be fulfilled.
<li>Semi-classical instabilities at the horizon are expected to lead to a divergence of the stress-energy tensor at the boundary in a short period of time, thus leading to the destruction of the warp bubble<ref name="Finazzi 2009"></ref>.
</ol>
Helpfully, Van Den Broeck<ref name="VanDenBroeck_2000">C. Van Den Broeck, "Alcubierre’s Warp Drive: Problems and Prospects" AIP Conference Proceedings. 504: 1105–1110 (2000). Bibcode:[https://ui.adsabs.harvard.edu/abs/2000AIPC..504.1105V 2000AIPC..504.1105V]. doi:[https://doi.org/10.1063%2F1.1290913 10.1063/1.1290913].</ref> suggested several solutions – or at least mitigations – for some of these problems
<ol>
<li>Quantum inequalities had not been shown to be true in general for highly curved space-times.
<li>A proposed warp drive geometry<ref name="VanDenBroeck_1999">C. Van Den Broeck, "A 'warp drive' with more reasonable total energy requirements". Classical and Quantum Gravity. 16 (12): 3973–3979 (1999). arXiv:[https://arxiv.org/abs/gr-qc/9905084 gr-qc/9905084]. Bibcode:[https://ui.adsabs.harvard.edu/abs/1999CQGra..16.3973V 1999CQGra..16.3973V]. doi:[https://doi.org/10.1088%2F0264-9381%2F16%2F12%2F314 10.1088/0264-9381/16/12/314]. S2CID [https://api.semanticscholar.org/CorpusID:15466313 15466313]. </ref> (see the van Den Broeck drive, below) would be able to minimize the magnitude of the energy required to about that of the mass of our sun. While still large, it is not unphysically large.
<li>Time travel is always going to be a worry with faster than light travel. There's no neat solution to this one.
<li>It may be possible for the negative energy outside of the superluminal region to compress into the superluminal region to form a shock in space-time. The front surface of the bubble would then be a singularity. It is not clear if such a sharp jump in physical properties is possible, but neither is sure that it is impossible, either. However, while this might help with the negative energy in front of the bubble getting swept up because it cannot move faster than light, it does little to help with the material at the back of the bubble on the outside getting left behind because it cannot keep up.
</ol>
Other mitigating research has also shown
<ol start=9>
<li>Semi-classical divergence only occurs in warp drives with only one spatial dimensions. Those with two or more spatial dimensions (as in our universe) only contain points where divergences occur, and these divergences can be managed. In fact, their effects can be reduced by making the warp shape more elongated<ref>Carlos Barceló, Valentin Boyanov, Luis J. Garay, Eduardo Martín-Martínez,
and Jose M. Sánchez Velázquez, "Warp drive aerodynamics", JHEP08 (2022) 288 https://doi.org/10.1007/JHEP08(2022)288 , arXiv:2207.06458 [gr-qc] https://doi.org/10.48550/arXiv.2207.06458</ref>.
</ol>

(And what's with the ring in all the artwork, anyway?)

=== So what is going on? ===

What is it that makes a warp drive a warp drive, and how does it work?

(8 pi G / c^4) G = R - (1/2) R g [Einstein field equation]

"Spacetime tells matter how to move; matter tells spacetime how to curve." - John Archibald Wheeler

Describing spacetime - lapse function & shift vectors (foliations, etc.)
Diagram of what the shift vector does.

<table align=center>
<tr><td width=389>[[File:Lapse_shift.png]]
<td width=649>[[File:Warp_shift.png]]
<tr><td style="vertical-align: top;">A folliation of space-time into slices of constant time coordinates (grey). At a point (black circle), the lapse function (yellow) shows how far into the future that point will be projected in a unit amount of proper time experienced by an observer initially at the point. The shift vector (red) shows how that point changes coordinates after the unit amount of proper time. The cyan vector shows the new coordinates the point is projected onto. The lower black surface indicates a set of points all at the same time coordinate; after unit proper time, they get projected to the curved upper black surface according to the shift vectors and lapse functions at each of their respective points.
<td>The shift vector of a warp geometry, in red. At the edges there is no shift. At the center (inside the warp bubble) there is a region of uniform shift so that at a later time the inside of the warp bubble is rigidly translated to different spatial coordinates (in blue). The two foliations remain the same distance apart, because warp drives maintain a unit lapse function.
</table>

Solve backwards - assume a structure of space-time, figure out what distribution of matter is required for this from the EFE.

What sort of spacetime geometry produces a warp?

Unit lapse function <-> no time dilation

Region of near constant shift vector - that region is "moving" spacetime, the warp bubble.

Shift vector falls off to zero away from warp bubble -> normal flat spacetime.

=== Alcubierre warp interactions with light and matter ===

Space is not entirely empty. It is filled with a diffuse plasma in the form of the interstellar medium, as well as cosmic radiation, light from stars, and cosmic microwave background radiation. A warp drive propagating through space will encounter this stuff. What happens when this matter and radiation have a warp bubble pass across them?

The first analysis of matter encountering a warp bubble was performed by Pfenning and Ford<ref name="PfenningFord_2001"></ref>. They looked at the warp bubble interaction with a massive object at rest with the frame of reference of the warp drive (keep in mind that the rest frame is the same inside and outside the warp bubble; in this rest frame objects inside the bubble have no momentum even though they are, in some sense, changing location rapidly with time). When the warp bubble passes, the object experiences an acceleration in the direction of the warp bubble motion. When the warp shell passes and the object is inside the bubble, it will be moving with approximately the speed of the bubble. Pfenning and Ford analyzed this problem with a continuous distribution of shell energy that, strictly speaking, never falls to zero except at the bubble center and at spatial infinity, so unless the object passes through the center in Pfenning and Ford's description it will never quite get up to the bubble's speed. In this case, the object will move almost as fast as the object but will pass through the bubble in a finite time, after which it will again be at rest with respect to the reference frame but displaced along the direction of the warp bubble motion by some distance. With this description of the warp bubble, a spacecraft of finite size will always be moving a little slower than the warp bubble and would have to use rockets to keep up with it. in addition, the spacecraft would experience tidal forces that would cause stress on the spacecraft's structure.

Pfenning and Ford also examined cases where the shell is of a finite (possibly infinitesimal) width and falls to zero both inside and outside the bubble. In this case any matter encountering the bubble would be collected at the bow of the bubble and thereafter move along with it.

McMonigal et al. analyzed the situation for both massive particles and light moving along the axis of travel of the bubble<ref name="McMonigalEtAl_2012">B. McMonigal, G. F. Lewis, and P. O'Byrne, "Alcubierre warp drive: On the matter of matter". Physical Review D. 85 (6) 064024 (20 March 2012). arXiv:[https://arxiv.org/abs/1202.5708 1202.5708]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2012PhRvD..85f4024M 2012PhRvD..85f4024M]. doi:[https://doi.org/10.1103%2FPhysRevD.85.064024 10.1103/PhysRevD.85.064024]. S2CID [https://api.semanticscholar.org/CorpusID:3993148 3993148].</ref>. They found that light moving opposite the warp direction passed through the bubble without incident, being only somewhat delayed by passing through the bubble. A warp bubble that is warping at sub-luminal speeds can have light catch up from behind it. This light is able to pass through the bubble, and is somewhat advanced in its path by the speed of the bubble. For super-luminal warp bubbles, however, the situation is different. The bubble will catch up to light moving it its own direction that is originally in front of it. This light cannot escape forward, the bubble being too fast. Nor can it escape backward, as the light is propagating forward and the interior of the bubble is at rest. Thus, the light gets caught at the bow of the warp bubble, unable to escape for so long as the warp bubble is active. This light is strongly blue shifted to extremely energetic x-rays and gamma rays. The space behind the bubble is swept clear of forward-moving light.

The situation for matter over-run by a super-luminal warp bubble is similar. Matter moving backward passes through the bubble, being only swept a ways along the bubble's path as sign of its passing but otherwise continuing on their way. Matter at rest behaves the same way as Pfenning and Ford discovered. Meanwhile, matter moving in the same direction of the warp is overtaken and collects at the bow of the bubble, unable to leave for so long as the bubble is warping. This matter "surfing" on the bubble bow is highly accelerated to relativistic speeds, experiencing extreme time dilation.

For a sub-luminal bubble, there is a speed faster than the warp speed where particles coming up from behind overtake the bubble and pass through, out the front. Particles below this speed but still faster than the warp speed still overtake the bubble, but then bounce backwards out of the bubble from behind. They are still moving forward, but are now moving more slowly than the warp speed. Particles moving forward but slower than the bubble will be overtaken and then bounced out the front with a speed higher than the warp speed. Those particles moving backward will pass through the bubble and exit with their initial speed.

The spacecraft inside the bubble will observe that light moving backward compared to the warp direction is blue-shifted, and matter moving backward passes through with increased energy. Meanwhile, a sub-luminal bubble will see light that was moving forward as red-shifted and matter catching up to the bubble from behind will pass by with reduced energy (superluminal bubbles, of course, do not have any matter or radiation catching up to them from behind).

When a superluminal warp bubble that has been collecting matter and radiation for the duration of its journey and blue-shifting it to much higher energies is turned off, all that matter and energy is released as a blast of radiation in the direction the bubble was warping.

McMonigal et al. conclude by noting that any spacecraft in the warp bubble would need shielding to protect against the blue-shifted radiation and matter of increased energy. In addition the destination would be "blasted into oblivion" by the release of matter and radiation that had been caught in the bubble during the trip.

Clark et al.<ref>Chad Clark, William A. Hiscock, and Shane L. Larson, "Null geodesics in the Alcubierre warp drive spacetime: the view from the bridge", arXiv:gr-qc/9907019 (2018) https://doi.org/10.48550/arXiv.gr-qc/9907019</ref> examine what it would look like from within an operating warp drive. They find that light moving at 90 degrees to the direction of motion will look like it is still coming from 90 degrees. However, light from the front hemisphere is squished progressively toward the forward point the faster you go. For a warp bubble moving faster than the speed of light, there is a cone behind the warp bubble where no light can reach it. However, you don't see a black spot behind you – light from the regions in the backward hemisphere from outside that cone that can reach you is distorted toward the direction behind you to fill in that spot. Light from the forward direction is blue-shifted and light from behind is red-shifted. At speeds of around 200 times light speed, the cosmic microwave background would be blue shifted to energies similar to that of the solar photosphere, heating anything in the warp bubble as if they were close to the sun.

== Van Den Broeck warp drive ==

If an Alcubierre warp bubble a hundred meters across requires a magnitude of energy greater than the entire energy in the observable universe, one option to reduce the magnitude of energy used is to make the warp bubble smaller. Much smaller. Pfenning and Ford<ref name="PfenningFord_2001"></ref> suggested making the warp bubble smaller than an atom. Of course, that brings up the problem of how to stuff a spacecraft in there. Van Den Broeck proposed a solution<ref name="VanDenBroeck_1999"></ref>: make the warp bubble only about the size of an atomic nucleus but expand the space inside the bubble enormously so that a spacecraft could fit in. This is somewhat like a nuclear sized wormhole that leads to a pocket universe that holds the spacecraft. Unfortunately, the metric doesn't fit neatly into any [[Wormholes#Three_dimensions|embedding diagram]] that I can figure out, but the math works out so that spatial coordinates inside the bubble are expanded by a factor of 1017 and a spacecraft would have a few hundred meters of bubble interior to putz around in. This trick manages to reduce the magnitude of energy to only about the mass energy of a few stars similar to our own. Other than that, it is otherwise a normal Alcubierre drive.

Further work<ref>S. Krasnikov, "The quantum inequalities do not forbid spacetime shortcuts", Physical Review D. 67 (10) 104013. https://doi.org/10.1103%2FPhysRevD.67.104013, arXiv:gr-qc/0207057 (2003) https://doi.org/10.48550/arXiv.gr-qc/0207057 </ref> has managed to use Van Den Broeck's trick to reduce the magnitude of the negative energy to only about a milligram.

== Natário warp drive ==

The Alcubierre drive is not the only way to construct a warp drive. José Natário<ref name="Natario"></ref> showed that it was just one example of an entire class of warp drives. These are space-time geometries with a smooth and localized shift vector field a shift vector is a vector field that describes how one "slice" of space out of space-time connects to the next) and a unit lapse function (the lapse function describes how the time you experience changes when going from a "slice" of space-time to the next).

Natário then went on to find in that class a set of warp drives that do not have any expansion or contraction of space at all. Instead, space encountering the front boundary of the warp bubble instead "slides" around the outside of the bubble until it gets to the corresponding place on the back and is left behind there. Another way to think of it is that space entering the bubble shell is compressed in the radial direction of the shell but is simultaneously expanded in the tangential direction so that there is no net change in volume. When it gets to the back, the opposite occurs. In the language of mathematics, the shift vector field is divergenceless.

<table align=center>
<tr><td width=360>[[File:Natario_drive_flow_of_space.png]]
<td width=445>[[File:Natario_warp_drive_horizons.png]]
<tr><td style="vertical-align: top;">A representation of the motion of space in a zero-expansion Natário drive with a vanishingly thin warp shell around the bubble.
<td>The horizons of the generalized Natário class of warp drives (including the Alcubierre warp drive and all fell-Heisenberg drives) at superluminal speed in the direction of the green arrow. No event inside the bubble can affect events outside the bubble in front of the blue line. No events outside the bubble behind the red line can affect events inside the bubble. The lines have an angle α with respect to the direction of motion with sin(α) = 1/vs for vs the speed of the bubble relative to light speed. This is analogous to the Mach cone of supersonic objects.
</table>

NOTE: There are two different things that can be meant by the term "Natário warp drive". The generalized Natário warp drive is the general theory of all warp drives that includes within it the Alcubierre, Fell-Heisenberg, and Lentz warp drives, among infinitely many others. The zero-expansion Natário warp drive is the specific implementation of the generalized Natário warp drive that has a divergenceless shift vector field.

The generalized class of warp drives developed by Natário (including both the Alcubierre and this zero-expansion warp drive) are shown to always have regions in the warp shell where the energy density is negative to at least some observers. You can't get away from it – to warp, you need negative energy density.

The Natário class of warp drives blueshift light coming in from the front and redshift light catching up from the back. For the case of a super-luminal warp bubble, there will of course be a horizon at the back of the bubble and light will not be able to get through from behind. For a bubble speed vs relative to the speed of light, light coming from straight ahead will be blueshifted up in frequency and photon energy by a factor of 1 + vs. For sub-luminal travel, light coming from behind will be redshifted by a factor 1 - vs. In general for n as the unit direction along which the light is propagating, the blueshift factor will be 1 + vs ⋅ n. If the warp shell is infinitesimally thin, light from outside that reaches the center of the bubble will not be distorted in direction although it will be frequency shifted, but observers still inside the bubble but displaced from the center will see the field of view distorted as well as frequency shifted.

Most of the difficulties that apply to the Alcubierre warp drive also apply to the generalized Natário warp drive (and thus to the zero-expansion Natário drive and Fell-Heisenberg drive as well). In particular, quantum energy inequalities, high magnitudes of the energy required, formation of time machines, instabilities related to the stress-energy making up the warp shell being unable to keep up with the bubble, the chicken-and-egg problem of needing FTL to achieve FTL, the formation of horizons making warp bubbles uncontrollable, and Hawking radiation at the horizons, are all still concerns for generic warp drives.

== Fell-Heisenberg warp drives ==

S. D. B. Fell and L. Heisenberg investigated what sort of warp structure is necessary to produce a warp drive with positive energy density everywhere<ref name="Fell-Heisenberg">S. D. B. Fell and L> Heisenberg, "Positive energy warp drive from hidden geometric structures", Classical and Quantum Gravity 38 155020 (2021) https://doi.org/10.1088/1361-6382/ac0e47 https://arxiv.org/abs/2104.06488</ref>. They found that warp drives of the generic Natário form with irrotational shift vectors had a simple expression for the energy density and vanishing momentum (irrotational means that the shift vector can be described as the gradient of a scalar potential field or, equivalently, that the curl of the shift vector field vanishes everywhere). Fell and Heisenberg proposed several potential fields that either had positive energy density everywhere except at discontinuities, or that were continuous but had regions that violated the weak energy condition (i. e. negative energy densities in some frames of reference). One specific example they provided of a warp field had an energy that is about 10,000 times less than the mass-energy of our sun. Or only about half the mass-energy of Jupiter. This is a significant improvement over many other proposed drives.

Despite the introduction discussing warp drive configurations that satisfy the various energy conditions, the configurations described in the paper are shown to locally violate the weak and strong energy conditions. Nonetheless, the energy density is still mostly positive. Santiago, Schuster, and Visser's<ref name="Santiago Schuster Visser 2022">J. Santiago, S. Schuster, and M. Visser, "Generic warp drives violate the null energy condition", Physical Review D 105, 064038 (2022) https://doi.org/10.1103/PhysRevD.105.064038</ref> work shows that the various energy conditions must still be violated by this warp drive; to not violate these, energy density must be positive in all reference frames not just those of the co-moving observer.

Any vector field can be described by its divergence, its curl, and a constant. The constant part of a warp drive's shift vector can be discarded because then the warp drive would not be localized. It is interesting, therefore, that the Natário zero-expansion drive has zero divergence and can be described only by its curl, while the Fell-Heisenberg drive has zero curl and can be described only by its divergence. These two classes of drives thus rely on opposite choices of setting one of the two different parts describing a vector field to zero. As might be expected, the Fell-Heisenberg drive is not zero expansion – space-time will change in volume as it passes through the warp bubble (although after going all the way through it will return to its original volume). However, objects in space-time will retain their original orientation as they pass through the warp bubble.

Previous examples of warp drives, such as the Alcubierre and zero-expansion Natário drives, tend toward a flat space-time as you go far enough away faster than would occur if there were any mass present in the warp drive. That is, the ADM mass of such drives is zero. However, one of the Fell-Heisenberg examples described a spacetime where the geometry approaches that of what you would get for a massive object at the warp location, giving one of the first examples of a warp drive with mass.

In the Fell-Heisenberg drive the momentum vanishes everywhere. Yet the energy occupies regions where the shift vector is varying rapidly. The lack of momentum means that the energy will not move to keep up with the differential expansion and movement of the spacetime. As a consequence, at later times the energy will have a different distribution than what is necessary to maintain the given warp configuration. The exact time evolution is not solved but this likely leads to collapse of warp bubble. Note that this is not a problem unique to the Fell-Heisenberg drive. Assuming a geometry that describes a warp produces a specific distribution of the stress-energy tensor that satisfies Einstein's equation. But there is no guarantee that this stress energy tensor satisfies the continuity equation, which describes how the distribution of matter and momentum change with time. The issue of stability of any warp drive is not well understood, but most warp drive geometries will be unstable in the sense that their matter that generates the warp fields will not be able to keep up with the motion of the warp. The Fell-Heisenberg drive is unique only in that, with vanishing momentum, there is no configuration that allows stable propagation.

(n. b. The Heisenberg here is Lavinia Heisenberg, not the Werner Heisenberg of quantum physics and uncertainty principle fame.)

== Lentz warp drive ==

E. Lentz<ref name="Lentz">E. W. Lentz, "Breaking the warp barrier: hyper-fast solitons in Einstein–Maxwell-plasma theory", Classical and Quantum Gravity. 38 075015 (2021). arXiv:[https://arxiv.org/abs/2006.07125 2006.07125]. Bibcode:[https://ui.adsabs.harvard.edu/abs/2021CQGra..38g5015L 2021CQGra..38g5015L]. doi:[https://doi.org/10.1088%2F1361-6382%2Fabe692 10.1088/1361-6382/abe692]. ISSN [https://search.worldcat.org/issn/0264-9381 0264-9381]. S2CID [https://api.semanticscholar.org/CorpusID:219635854 219635854].</ref> investigated a particular configuration of the Fell-Heisenberg class of warp drives, where the scalar field has a hyperbolic shape. The necessary conditions were found to be satisfied by a selection of rhomboid regions of the scalar field, resulting in an energy density that is everywhere positive and a shift vector with vanishing tidal force at the center for a comfortable ride.

However, Santiago, Schuster, and Visser<ref name="Santiago Schuster Visser 2022"></ref> dispute claims that the Lentz drive satisfies the energy conditions, noting that everywhere positive energy density in one frame of reference is insufficient to establish that the energy density is positive in all reference frames; knowledge of the Cauchy stress tensor is also needed. They show that any generic warp drive will violate the strong energy condition, null energy condition, and weak energy condition.

== Conservation laws ==

(Discussion of asymptotic flatness: what is it? Where does it apply?)

(The "big four" conservation laws and how they relate to asymptotic flatness.)

(ADM mass -> 0; conservation of energy/mass issues.)<ref>Sebastian Schuster, Jessica Santiago,and Matt Visser, "ADM mass in warp drive spacetimes" General Relativity and Gravitation (2023) 55:14 https://doi.org/10.1007/s10714-022-03061-9</ref>

(Discuss issues of conservation of angular momentum.)

(If angular momentum conservation is ignored, discuss how momentum & energy are affected by outside forces. In a gravitational field equivalent to inertial frame ... like being in an accelerated elevator; will acquire momentum buildup while staying "at rest".)

== Warp Railroads ==

Van Den Broeck<ref name="VanDenBroeck_2000"></ref> suggested a warp railroad,

You can set up devices ahead of time along the path of the bubble that produce the negative energy regions for the warp bubble at the appropriate time without needing the negative energy to ever move with the bubble at all. At least for bubbles on a predictable schedule and route.

helps with:

-stability (elaborate)

-conservation laws (elaborate)

Mention Krasnikov<ref name="Krasnikov 1998">S. V. Krasnikov, "Hyperfast travel in general relativity" Phys. Rev. D 57 4760</ref>

Hyperwave<ref name="Pieri 2024">Lorenzo Pieri, "Hyperwave: hyper-Fast Communication within General Relativity", arXiv:2311.12069 [gr-qc] https://doi.org/10.48550/arXiv.2311.12069</ref> makes use of this warp railroad idea.

== Quantum effects ==

(quantum stuff here<ref name="Hiscock1997">W. A. Hiscock, "Quantum effects in the Alcubierre warp drive spacetime", Classical and Quantum Gravity 14 L183 https://doi.org/10.1088/0264-9381/14/11/002 https://arxiv.org/abs/gr-qc/9707024</ref><ref name="Finazzi et al 2009">S. Finazzi, S. Liberati, C. Barceló, "Semiclassical instability of dynamical warp drives", Physical Review D 79, 124017 (2009)https://doi.org/10.1103/PhysRevD.79.124017 https://arxiv.org/abs/0904.0141</ref>)

From<ref name="Pieri 2024"></ref>
<blockquote>
It should be emphasised that Quantum Inequalities have not been proven for realistic interacting quantum fields, where the they are expected to be weaker if they exist [Cadamuro, 2019], and they often require Minkowskian conditions at infinity, a condition which is not appropriate for fields inside bounded regions, as in a Casimir devices.
</blockquote>

== Warp drives and black holes ==

What happens if you drive your warp drive into a black hole?

The pop-sci reason that nothing can escape a black hole is that at the event horizon the escape velocity is faster than light and nothing can go faster than light. A warp drive can go faster than light.

The more detailed reason from general relativity is that at the event horizon space and time are rotated sufficiently that "inward" becomes "forward in time." You can no more go farther away from the singularity in a black hole once you are inside the event horizon than you can go backward in time. Any super-luminal travel can go backward in time, and warp drives can engage in super-luminal travel.

The Hawking area theorem that says that black holes can only grow, never shrink relies on the condition that the energy is everywhere positive. Warp drives have negative energy density.

So in principle, there's nothing that prevents a warp drive from taking a dip into a black hole and coming back out again.

Garattini and Zatrimaylov looked into the problem of a warp drive near (and in) a black hole<ref name="Garattini and Zatrimaylov">R. Garattini and K. Zatrimaylov, "Black holes, warp drives, and energy conditions", Physics Letters B 856 138910 (2024) https://doi.org/10.1016/j.physletb.2024.138910</ref>. The found that the black hole reduces the negative energy requirement for the warp bubble to move inward toward the black hole singularity, but increases the negative energy needed to move away from the center of the black hole. In addition, a warp drive parked at the event horizon would allow light from inside the horizon to pass through the bubble and escape back outside.

== Credit ==
Author: Luke Campbell

== References ==

[[Category:Physics]][[Category:Infrastructure]][[Category:Physics & Engineering‏‎]][[Category:Physics & Math & Engineering]][[Category:Transportation & Infrastructure‏‎]][[Category:Metric Engineering]]