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 that is geometrically defined and that, under ideal circumstances, will take all values in a certain interval tex2html_wrap_inline1959 . 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 [176]. This result concerned Gowdy spacetimes. These are vacuum spacetimes with a two-dimensional Abelian group of isometries acting on compact orbits. The area of the orbits defines a natural time coordinate (areal 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_inline1961 This result was extended to more general vacuum spacetimes with two Killing vectors in [33]. Andréasson [8] 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 [209]. A global existence theorem in this time for spacetimes with two Killing vectors and certain matter models (collisionless matter, wave maps) was proved in [212]. That the choice of matter model is important for this result was demonstrated by a global non-existence result for dust in [211]. As shown in [141], this leads to examples of spacetimes that are not covered by a CMC slicing. Results on global existence of CMC foliations have also been obtained for spherical and hyperbolic symmetry [206, 51].

A drawback of the results on the existence of CMC foliations just cited is that they require as a hypothesis the existence of one CMC Cauchy surface in the given spacetime. More recently, this restriction has been removed in certain cases by Henkel using a generalization of CMC foliations called prescribed mean curvature (PMC) foliations. A PMC foliation can be built that includes any given Cauchy surface [130] and global existence of PMC foliations can be proved in a way analogous to that previously done for CMC foliations [131, 132]. These global foliations provide barriers that imply the existence of a CMC hypersurface. Thus, in the end it turns out that the unwanted condition in the previous theorems on CMC foliations is in fact automatically satisfied. Connections between areal, CMC, and PMC time coordinates were further explored in [9]. One important observation there is that hypersurfaces of constant areal time in spacetimes with symmetry often have mean curvature of a definite sign.

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_inline1963 . There are few cases in which this has been done successfully. Notable examples are Gowdy spacetimes [84, 139Jump To The Next Citation Point In The Article, 87Jump To The Next Citation Point In The Article] and solutions of the Einstein-Vlasov system with spherical and plane symmetry [190Jump To The Next Citation Point In The Article]. 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 Theorems on Existence and Global Dynamics for the Einstein Equations
Alan D. Rendall
© Max-Planck-Gesellschaft. ISSN 1433-8351
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