Exact Solutions in General Relativity - Difficulties With The Definition

Difficulties With The Definition

Take any Lorentzian manifold, compute its Einstein tensor, which is a purely mathematical operation, divide by, and declare the resulting symmetric second rank tensor field to be the stress-energy tensor . Thus any Lorentzian manifold is a solution of the Einstein field equation with some right hand side. Which of course doesn't make general relativity useless, but only shows that there are two complementary ways to use it. One can fix the form of the stress-energy tensor (from some physical reasons, say) and study the solutions of the Einstein equations with such right hand side (for example, if the stress-energy tensor is chosen to be that of the perfect fluid, a spherically symmetric solution can serve as a stellar model). Alternatively, one can fix some geometrical properties of a spacetime and look for a matter source that could provide these properties. This is what cosmologists have done for the last 5–10 years: they assume that the Universe is homogenous, isotropic, and accelerating and try to realize what matter (called dark energy) can support such a structure.

Within the first approach the alleged stress-energy tensor must arise in the standard way from a "reasonable" matter distribution or nongravitational field. In practice, this notion is pretty clear, especially if you restrict the admissible nongravitational fields to the only one known in 1916, the electromagnetic field. But ideally we would like to have some mathematical characterization that states some purely mathematical test which we can apply to any putative "stress-energy tensor", which passes everything which might arise from a "reasonable" physical scenario, and rejects everything else. Unfortunately, no such characterization is known. Instead, we have crude tests known as the energy conditions, which are similar to placing restrictions on the eigenvalues and eigenvectors of a linear operator. But these conditions, it seems, can satisfy no-one. On the one hand, they are far too permissive: they would admit "solutions" which almost no-one believes are physically reasonable. On the other, they may be far too restrictive: the most popular energy conditions are apparently violated by the Casimir effect.

Einstein also recognized another element of the definition of an exact solution: it should be a Lorentzian manifold (meeting additional criteria), i.e. a smooth manifold. But in working with general relativity, it turns out to be very useful to admit solutions which are not everywhere smooth; examples include many solutions created by matching a perfect fluid interior solution to a vacuum exterior solution, and impulsive plane waves. Once again, the creative tension between elegance and convenience, respectively, has proven difficult to resolve satisfactorily.

In addition to such local objections, we have the far more challenging problem that there are very many exact solutions which are locally unobjectionable, but globally exhibit causally suspect features such as closed timelike curves or structures with points of separation ("trouser worlds"). Some of the best known exact solutions, in fact, have globally a strange character.