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In this video, Michio Kaku says

"...general theory of relativity, where it is possible, we think, that you might be able to go faster than the speed of light..."

Does GR really allow superluminial movement?

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GR does not explicitly "forbid" superluminal travel. It just does not provide a mechanism for achieving it. Furthermore, one can construct solutions to the Einstein equations that allow for superluminal travel; it just means you have to allow for things like negative mass matter or other weird and/or fantastic phenomena – Jim Apr 17 '14 at 22:07
@Jim: what you say applies to round-trip superluminal travel. But you can certainly have superluminal "relative velocities" invoking only ordinary matter. Consider our relative motion with matter outside of the visible universe, for example. – Jerry Schirmer Apr 17 '14 at 22:56
@JerrySchirmer Those relative velocities are not actually "superluminal" (meaning faster than light). They result from expansion of space and while they are faster than $c$, light rays outside the visible universe would be still travelling away from us faster than non-relativistic objects at the same distance. Point being, those velocities do not allow an object to reach a distant destination faster than a straight-path beam of light. – Jim Apr 17 '14 at 23:05
@Jim: what is relative velocity and what is expansion of space is coordinate-dependent. – Jerry Schirmer Apr 18 '14 at 0:38
up vote 8 down vote accepted

There is a subtle difference between "local" and "global" (or apparent) superluminal travel. Kaku is only correct in the global sense.

Local superluminal travel, in the sense of increasing one's speed to exceed the speed of light, is strictly prohibited as special relativity holds in any sufficiently local frame.

However, general relativity (with the most vague energy conditions), does allow constructs like wormholes, bridges between areas of space-time. Think of going from NYC to London; normally this takes 8 hours, but lets say you were able to stretch the earth placing london right above NYC and you hoped from NYC to London. To a mere earth dwelling creature, who doesn't see the ``global" stretching, it appears that you travelled instantaneously from NYC to London, travelling faster than the speed of light. However, to those who see this new bridge (new topology), they will realize that you didn't actually travel faster than the speed of light, you just crossed a bridge. Construction of such bridges require exotic matter (i.e. negative mass energy).

Note that according to any realistic energy conditions in the universe (such as the averaged null energy condition), topological censorship conjecture holds. This conjecture states that we cannot actively probe any bridges in space-time; i.e. all topologies are causally disconnected from us outside the topology, so you cannot fly from earth through a wormhole, and then come back to earth. One can passively probe these topologies either by observing them from the past (i.e. seeing light from a white hole), or by entering a wormhole in such a way that one cannot return back (being "causally disconnected").

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Note that global superluminal travel is allowed for paths that cross apparent horizons. These restrictions are on superluminal travel that allows for global round trips. – Jerry Schirmer Apr 17 '14 at 22:57
Yes, good thing to add, I will edit my answer to clarify that such "passive" observations are allowed. – Benjamin Horowitz Apr 17 '14 at 23:32

The speed of light can be slower than c.

wikipedia: Propagation_of_light_in_non-inertial_reference_frames

So there can be things going faster in one frame than the slow light of another frame.

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When one speaks of "cannot go faster than the speed of light" it always implies "in a vacuum". – Floris Apr 18 '14 at 2:42

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