4 Newtonian theory and special 3 Global symmetric solutions3.4 Cylindrically symmetric solutions

3.5 Spatially compact solutions

  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 tex2html_wrap_inline1366 . 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 [120]. 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 tex2html_wrap_inline1368 This result was extended to more general vacuum spacetimes with two Killing vectors in [21]. Andréasson [4Jump To The Next Citation Point In The Article] extended it in another direction to the case of collisionless matter in a spacetime with Gowdy symmetry.

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  [146Jump To The Next Citation Point In The Article]. 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 tex2html_wrap_inline1370 . 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.

4 Newtonian theory and special 3 Global symmetric solutions3.4 Cylindrically symmetric solutions

image Local and Global Existence Theorems for the Einstein Equations
Alan D. Rendall
© Max-Planck-Gesellschaft. ISSN 1433-8351
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