Warp Drives

	Notice:
	Please bear with us. Your ride's still a work in progress.

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. 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, 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!

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.)	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.

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.

"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:

8πG	T = R -	1	R g
c4		2

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.)

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 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.

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.

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, 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(r_s)

Here, r_s is the distance from the center of the warp bubble, and f(r_s) can be any function that is 1 near r_s = 0 and is 0 for r_s 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.

1. Pfenning and Ford found that if the negative energy density wall around the bubble obeys 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.
2. 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.
3. As with any method of faster than light travel, the warp drive could be used to make a time machine.
4. 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.
5. 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.

   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

6. 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, which rather makes warp drives unnecessary from the get-go.
7. 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. As a consequence, the bubble cannot be controlled, steered, nor stopped from within the bubble itself.
8. 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 (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.
9. 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.

Helpfully, Van Den Broeck suggested several solutions – or at least mitigations – for some of these problems

1. Quantum inequalities had not been shown to be true in general for highly curved space-times.
2. A proposed warp drive geometry (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.
3. Time travel is always going to be a worry with faster than light travel. There's no neat solution to this one.
4. 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.

Other mitigating research has also shown

9. 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.

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. 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. 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. 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 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: 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 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 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 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.

A representation of the motion of space in a zero-expansion Natário drive with a vanishingly thin warp shell around the bubble.

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.

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 v_s 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 + v_s. For sub-luminal travel, light coming from behind will be redshifted by a factor 1 - v_s. In general for n as the unit direction along which the light is propagating, the blueshift factor will be 1 + v_s ⋅ 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. 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 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 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 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.)

(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 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

Hyperwave makes use of this warp railroad idea.

Quantum effects

(quantum stuff here)

From

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.

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. 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