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 rules out by physics – yet!

Almost as soon as Alcubierre proposed his warp drive, others began picking it apart. Van Den Broeck identified several issues.

1. 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; which is starting to get so small as to make thickness a meaningless concept.
2. 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.

Helpfully, Van Den Broeck then went on to suggest 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. 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.

(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 adiffuse 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. In their analysis, the bubble surface is only infinitesimally thick, giving a step function jump in the "shape function" value defining the warp bubble from 0 (outside) to 1 (inside). 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. The space behind the bubble is swept clear of forward-moving light.

Van Den Broeck warp drive

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Natário warp drive

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Lentz warp drive

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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