Another attractive time coordinate is constant mean curvature
(CMC) time. For a general discussion of this see [144]. A global existence theorem in this time for spacetimes with
two Killing vectors and certain matter models (collisionless
matter, wave maps) was proved in [146]. That the choice of matter model is important for this result
was demonstrated by a global non-existence result for dust in
[145]. As shown in [100], 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 [146] and [4] 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 [131]. 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.
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Local and Global Existence Theorems for the Einstein
Equations
Alan D. Rendall http://www.livingreviews.org/lrr-2000-1 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |