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3.6 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 (t-,t+). 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 [246]. 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 (0, oo ). This result was extended to more general vacuum spacetimes with two Killing vectors in [50]. Andréasson [16] extended it in another direction to the case of collisionless matter in a spacetime with Gowdy symmetry. This development was completed in [20] where general cosmological solutions of the Einstein-Vlasov system with two commuting spacelike Killing vectors were treated. Corresponding results for spacetimes with hyperbolic symmetry were obtained in [19Jump To The Next Citation Point].

In all of these cases other than Gowdy the areal time coordinate was proved to cover the maximal globally hyperbolic development, but the range of the coordinate was only shown to be (R0, oo ) for an undetermined constant R0 > 0. It was not known whether R0 was necessarily zero except in the Gowdy case. This issue was settled in [195] for the vacuum case with two commuting Killing vectors and this was extended to include Vlasov matter in [348]. It turns out that in vacuum R0 = 0 apart from the exceptional case of the flat Kasner solution and an unconventional choice of the two Killing vectors. With Vlasov matter and a distribution function which does not vanish identically, R0 = 0 without exception. The corresponding result in cosmological models with spherical symmetry was proved in [336Jump To The Next Citation Point] where the case of a negative cosmological constant was also included. For solutions of the Einstein-Vlasov system with hyperbolic symmetry the question is still open, although the homogeneous case was treated in [336Jump To The Next Citation Point].

Another attractive time coordinate is constant mean curvature (CMC) time. For a general discussion of this see [292]. A global existence theorem in this time for spacetimes with two Killing vectors and certain matter models (collisionless matter, wave maps) was proved in [295]. That the choice of matter model is important for this result was demonstrated by a global non-existence result for dust in [294]. As shown in [194], 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 [28972].

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 [178] and global existence of PMC foliations can be proved in a way analogous to that previously done for CMC foliations [177176]. 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 [19]. One important observation there is that hypersurfaces of constant areal time in spacetimes with symmetry often have mean curvature of a definite sign. Related problems for the Einstein equations coupled to fields motivated by string theory have been studied by Narita [252253254Jump To The Next Citation Point255Jump To The Next Citation Point].

Once global existence has been proved for a preferred time coordinate, the next step is to investigate the asymptotic behaviour of the solution as t-- > t±. There are few cases in which this has been done successfully. Notable examples are Gowdy spacetimes [113190Jump To The Next Citation Point116Jump To The Next Citation Point] and solutions of the Einstein-Vlasov system with spherical and plane symmetry [272Jump To The Next Citation Point]. These last results have been extended to allow a non-zero cosmological constant in [336Jump To The Next Citation Point]. 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. Detailed information has been obtained on the late-time behaviour of a class of inhomogeneous solutions of the Einstein-Vlasov system with positive cosmological constant in [337Jump To The Next Citation Point338Jump To The Next Citation Point] (see Section 7.6).

In the case of polarized Gowdy spacetimes a description of the late-time asymptotics was given in [116]. A proof of the validity of the asymptotic expansions can be found in [199]. The central object in the analysis of these spacetimes is a function P that satisfies the equation Ptt + t-1Pt = Phh. The picture that emerges is that the leading asymptotics are given by P = A logt + B for constants A and B, this being the form taken by this function in a general Kasner model, while the next order correction consists of waves whose amplitude decays like -1/2 t, where t is the usual Gowdy time coordinate. The entire spacetime can be reconstructed from P by integration. It turns out that the generalized Kasner exponents converge to (1,0, 0) for inhomogeneous models. This shows that if it is stated that these models are approximated by Kasner models at late times it is necessary to be careful in what sense the approximation is supposed to hold.

General (non-polarized) Gowdy models, which are technically much more difficult to handle, have been analysed in [316Jump To The Next Citation Point]. Interesting and new qualitative behaviour was found. This is one of the rare examples where a rigorous mathematical approach has discovered phenomena which had not previously been suspected on the basis of heuristic and numerical work. In the general Gowdy model the function P is joined by a function Q and these two functions satisfy a coupled system of nonlinear wave equations. Assuming periodic boundary conditions the solution at a fixed time t defines a closed loop in the (P, Q) plane. (In fact it is natural to interpret it as the hyperbolic plane.) Thus the solution as a whole can be represented by a loop which moves in the hyperbolic plane. On the basis of what happens in the polarized case it might be expected that the following would happen at late times. The diameter of the loop shrinks like t-1/2 while the centre of the loop, defined in a suitable way, moves along a geodesic. In [316Jump To The Next Citation Point] Ringström shows that there are solutions which behave in the way described but there are also just as many solutions which behave in a quite different way. The shrinking of the diameter is always valid but the way the resulting small loop moves is different. There are solutions where it converges to a circle in the hyperbolic plane which is not a geodesic and it continues to move around this circle forever. A physical interpretation of this behaviour does not seem to be known.

Ringström has also obtained important new results on the structure of singularities in Gowdy spacetimes. They are discussed in Section 8.4.


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