Warp Drives

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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 are 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.

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.

Helpfully, Van Den Broeck suggested several solutions – or at least mitigations – for 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. 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. Van Den Broeck then went on to suggest that 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. The "railroad track" of devices set up along the bubble path still works, though. At least for bubbles on a predictable schedule and route.

(Van Den Broeck deals with the negative energy in front being unable to keep up with the superluminal motion and being swept back to the shock - but material in the back on the outside of the shell will also be unable to keep up ... and it will just be left behind! Gives rise to general question: Alcubierre's metric has a specified stress-energy, but can that stress-energy meet the continuity equations over time to maintain the warp bubble geometry?)

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

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. When 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 bobble 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.

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.

Natário warp drive

The Alcubierre drive is not the only way to construct a warp drive. Natário showed that it was just one example of an entire class of warp drives. He 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.

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

The horizons of the generalized Natário class of warp 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.

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. 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.

Lentz warp drive

(stuff goes here)

Quick notes (will expand on later): 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.

Santiago, Schuster, and Visser also claim the Lentz drive is a subset of the Fell-Heisenberg drive.

Fell-Heisenberg warp drives

(stuff goes here)

Quick notes: vanishing momentum 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; exact time evolution is not solved but likely leads to collapse of warp bubble.

First example I've seen yet with a non-zero ADM mass.

Natário zero expansion drive is divergenceless; the Fell-Heisenberg drive is irrotational. Opposite choices of the typical decomposition of a vector field here!

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 claims 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.

The energy needed to form a Fell-Heisenberg drive is about 10,000 times less than the mass-energy of our sun. Or only about half the mass-energy of Jupiter. A significant improvement over other proposed drives.

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

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".)

Quantum effects

(quantum stuff here)

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