Misplaced Pages

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This article is seriously misleading in one important respect.

The Alcubierre spacetimes are a certain family of Lorentzian spacetimes, but they are not solutions of the Einstein field equation.

Every Lorentzian manifold has, mathematically speaking, a well-defined Einstein tensor, which you can compute from its Riemann curvature tensor. To be a solution of the EFE, this must agree (up to a certain constant factor) with a physically reasonable matter tensor. Consider two easy examples:

1. vacuum solutions are easy to recognize, since the matter tensor vanishes identically in a vacuum region, so vacuum solutions have purely a mathematical characterization: they are Lorentzian manifolds whose Einstein tensor vanishes (equivalently, whose Ricci tensor vanishes).

2. electrovacuum solutions (no mass-energy other than the field energy of an electromagnetic field) must have Einstein tensors which agree with matter tensors having the form appropriate for an electromagnetic field (as described in Maxwell's theory). Furthermore, their definition must include specifying an antisymmetric second order tensor field ${\displaystyle F_{ab}}$, and this must not only satisfy (a curved spacetime version of) Maxwell's equations but must rise to a stress-energy tensor (according to Maxwell's theory, and following some general principles for getting from a field equation to the corresponding contribution to the matter tensor), and of course this stress-energy tensor must match the Einstein tensor computed directly from the Riemann tensor.

There are other kinds of classical fields which one can propose besides classical electromagnetism, and there are other types of solutions which arise from the notion of a perfect fluid or from a nonzero cosmological constant, but AFAIK (and few years ago I read the ENTIRE literature on warp drives) there is no well-defined classical field theory which gives contributions to the matter tensor which could match the Einstein tensor of a nontrivial Alcubierre spacetime.

Put in other words: in principle, we could take any Lorentzian spacetime, compute its Einstein tensor, and declare this to be the 'matter tensor' of some funky field. But this would be an absurd procedure, since it would imply that every Lorentzian spacetime is a 'solution' of the EFE! If we allowed this, gtr would be unfalsifiable-- hence useless! Of course, it is not useless, because in fact physicists have stringent expectations about what can stand as a legal matter tensor.

Einstein himself had a rather stringent notion of 'solution' in mind, similar to what I described above in the special case of 'electrovacuum'. Others with a more speculative turn of mind have proposed to allow matter tensors which merely satisfy one or more of a list of energy conditions which may (or may not) characterize some properties shared by all (or some) known matter tensors which everyone would agree are acceptable, such as electromagnetic field energy/momentum/stress, or a perfect fluid.

Accordingly, I propose to rewrite this article to correct the mistaken impression.