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 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.

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.
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?)

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.

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

Lentz warp drive

(stuff goes here)

Conservation laws

(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".)

Credit

Author: Luke Campbell