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 for an undetermined constant . It was not known whether 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 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, without exception. The corresponding result in cosmological models with spherical symmetry was proved in [336] 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 [336].

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 [289, 72].

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 [177, 176]. 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 [252, 253, 254, 255].

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 [113, 190, 116] and solutions of the Einstein-Vlasov system with spherical and plane symmetry [272]. These last results have been extended to allow a non-zero cosmological constant in [336]. 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 [337, 338] (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 that satisfies the equation . The picture that emerges is that the leading asymptotics are given by for constants and , 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 , where is the usual Gowdy time coordinate. The entire spacetime can be reconstructed from by integration. It turns out that the generalized Kasner exponents converge to 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 [316]. 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 is joined by a function and these two functions satisfy a coupled system of nonlinear wave equations. Assuming periodic boundary conditions the solution at a fixed time defines a closed loop in the 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 while the centre of the loop, defined in a suitable way, moves along a geodesic. In [316] 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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