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[7], building on global existence theorems for wave maps[31, 30]. For a quite different point of view on this question see[83].
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 [58]. 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 [6].
Another attractive time coordinate is constant mean curvature (CMC) time. For a general discussion of this see [71]. A global existence theorem in this time for spacetimes with two Killing vectors and certain matter models (collisionless matter, wave maps) was proved in [72]. That the choice of matter model is important for this result was demonstrated by a global non-existence result for dust in [66]. 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[64].
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Local and global existence theorems for the Einstein
equations
Alan D. Rendall http://www.livingreviews.org/lrr-1998-4 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |