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 . As shown in , this leads to the examples of spacetimes which are not covered by a CMC slicing. Related results have been obtained for spherical and hyperbolic symmetry [143, 31]. The results of  and  have many analogous features and it would be desirable to establish connections between them, since this might lead to results stronger than those obtained by either of the techniques individually.
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 [58, 98, 61] and solutions of the Einstein-Vlasov system with spherical and plane symmetry . Progress in constructing spacetimes with prescribed singularities will be described in section 6 . In the future this could lead in some cases to the determination of the asymptotic behaviour of large classes of spacetimes as the singularity is approached.
|Local and Global Existence Theorems for the Einstein
Alan D. Rendall
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