Solutions of the Einstein equations with cylindrical symmetry which are asymptotically flat in all directions allowed by the symmetry represent an interesting variation on asymptotic flatness. Since black holes are incompatible with this symmetry, one may hope to prove geodesic completeness of solutions under appropriate assumptions. This has been accomplished for the Einstein vacuum equations and for the source-free Einstein-Maxwell equations in, building on global existence theorems for wave maps[31, 30]. For a quite different point of view on this question see.
In the context of spatially compact spacetimes it is first necessary to ask what kind of global statements are to be expected. In a situation where the model expands indefinitely it is natural to pose the question whether the spacetime is causally geodesically complete towards the future. In a situation where the model develops a singularity either in the past or in the future one can ask what the qualitative nature of the singularity is. It is very difficult to prove results of this kind. As a first step one may prove a global existence theorem in a well-chosen time coordinate. In other words, a time coordinate is chosen which is geometrically defined and which, under ideal circumstances, will take all values in a certain interval . The aim is then to show that, in the maximal Cauchy development of data belonging to a certain class, a time coordinate of the given type exists and exhausts the expected interval. The first result of this kind for inhomogeneous spacetimes was proved by Moncrief in . This result concerned Gowdy spacetimes. These are vacuum spacetimes with two commuting Killing vectors acting on compact orbits. The area of the orbits defines a natural time coordinate. Moncrief showed that in the maximal Cauchy development of data given on a hypersurface of constant time, this time coordinate takes on the maximal possible range, namely . This result was extended to more general vacuum spacetimes with two Killing vectors in .
Another attractive time coordinate is constant mean curvature (CMC) time. For a general discussion of this see . A global existence theorem in this time for spacetimes with two Killing vectors and certain matter models (collisionless matter, wave maps) was proved in . That the choice of matter model is important for this result was demonstrated by a global non-existence result for dust in . Related results have been obtained for spherical and hyperbolic symmetry [70, 11].
Once global existence has been proved for a preferred time coordinate, the next step is to investigate the asymptotic behaviour of the solution as . There are few cases in which this has been done successfully. Notable examples are Gowdy spacetimes [32, 50, 35] and solutions of the Einstein-Vlasov system with spherical and plane symmetry.
|Local and global existence theorems for the Einstein
Alan D. Rendall
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