Not a solution of the EFE!
[EDIT: The following comment occurred in the context of a thread on science-fiction plot devices; Wallace has stated--- if I understand him correctly--- that he would have clarified his remarks in another context. Still, this being PF, I thought it important to enter a clarification.]
Wallace said:
What I'm referring to is the 'Alcubierre drive'. This is a formal (i.e. fully scientific and mathemical and stuff) solution to the equations of General Relativity that allows this kind of warp motion.
No! The Alcubierre spacetime is a
Lorentzian manifold, and it does have a very intriguing interpretation which corresponds quite well to many of the features of the fictional Star Trek warp bubbles, but it is
not a solution of the EFE in any physically meaningful sense.
I have had to say this so often that I am rather tired of repeating myself, but to repeat myself: the EFE is G^{ab} = 8 \pi \, T^{ab}. Einstein's intention was that the problem of solving this be understood as finding a spacetime together with say the tensor describing an "empty space" solution of (the curved spacetime version of) the Maxwell field equations, such that when you compute the stress-tensor of the EM field (using the given EM field tensor and the given metric tensor) according to (the curved spacetime version of) Maxwell's theory, and plug the result into the RHS of the EFE, this matches the Einstein tensor computed from the given metric tensor. This would then be an
exact electrovacuum solution of the EFE. Similarly one can define a notion of exact perfect fluid solutions, and then one can combine (by adding contributions to the RHS of the EFE) to define more general notions of an exact solution. We can say for short that we "reason from right to left" in solving the EFE, although this is potentially misleading because in all cases we are in effect finding
simultaneous solutions of the EFE (for the purely gravitational part of the model) plus other relevant field equations (or the hydrodynamical equations). Indeed, electrovacuums are also known as "free space Einstein-Maxwell solutions".
Alcubierre OTH wrote down a metric ingeniously constructed using bump functions specifically in order to satisfy the requirements of the Star Trek "warp bubble". One can then "reason from left to right" in the EFE to
infer what the putative stress-energy tensor would have to be, namely \frac{1}{8 \pi} \, G^{ab}. Unfortunately, the result turns out not to correspond with anything we can imagine making using physically realistic stress-energy tensors T^{ab}. This takes quite a bit of back and forth to explain, since effective field theories derived from considerations from QFTs briefly suggested otherwise, but the consensus has been for some time that warp bubbles appear to be unrealizable for a variety of reasons.
This review paper touches most of the technical points, although when Lobo says
All these solutions [Kerr vacuum, FRW dusts, etc.] have been obtained by first considering a plausible distribution of matter, and through the Einstein field equation, the spacetime metric of the geometry is determined. However, one may solve the Einstein field equation in the reverse direction, namely, one first considers an interesting and exotic spacetime metric, then finds the matter source responsible for the respective geometry.
one shouldn't read this as implying that it is always possible to find such a source! Indeed, it we could do that, metric theories of gravitation would be utterly vacuous! In fact, in gtr--- as you will soon discover if you try to solve the EFE "reasoning from right to left"--- the vast majority of Lorentzian manifolds are
ruled out as candidates for spacetime models providing the geometrical arena for some physical scenario (e.g. "a spherically collapsing dust cloud interacts with a passing gravitational wave),
simply because the Einstein tensor has the wrong form to be matched to any sum of contributions from physically reasonable stress-momentum-energy tensors.
IMO it only makes sense to "reason from left to right" when one is trying to figure out whether certain geometrical properties of putative spacetime models appear to be physically realizable
eventwise (level of "jet spaces" in the sense of differential geometry) according to known physics, or not terribly wildly speculative physics. As I said, in the case of warp bubbles, the mainstream answer is currently "apparently not", even eventwise (at the level of jet spaces), and there are independent "local neighborhood" and "global" level arguments.
It is unfortunate that Lobo failed to stress these crucial points in his review. I can assure you that they are very widely appreciated by researchers in gravitation physics!
