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8.2 Inhomogeneous solutions

Consider now inhomogeneous solutions of the Einstein equations where, according to the BKL picture, oscillations of mixmaster type are to be expected. This is for instance the case for general solutions of the vacuum Einstein equations. There is only one rigorous result to confirm the presence of these oscillations in an inhomogeneous spacetime of any type, and that concerns a family of spacetimes depending on only finitely many parameters [53]. They are obtained by applying a solution-generating technique to the mixmaster solution. Perhaps a reason for the dearth of results is that oscillations usually only occur in combination with the formation of local spatial structure discussed in Section 8.3. On the other hand, there is a rich variety of numerical and heuristic work supporting the BKL picture in the inhomogeneous case. For a review of the former see [49]. There are different heuristic procedures in the literature. For an approach using dimensionless variables and a dynamical systems picture see [14Jump To The Next Citation Point]. In this context it has been recognized that spatially self-similar solutions play an important role in understanding the structure of cosmological singularities. There is now a numerical calculation which shows mixmaster oscillations in vacuum solutions without any symmetry [150]. It uses a gauge condition proposed in [227]. Other interesting types of heuristics based on the Hamiltonian formulation of cosmology are the method of consistent potentials [52] and the refined billiard picture [129].

A situation where there is more hope of obtaining rigorous results is where the BKL picture suggests that there should be monotone behaviour near the singularity. This is the situation for which Fuchsian techniques can often be applied to prove the existence of large classes of spacetimes having the expected behaviour near the initial singularity (see Section 6.2). It would be desirable to have a stronger statement than these techniques have provided up to now. Ideally, it should be shown that a non-empty open set of solutions of the given class (by which is meant all solutions corresponding to an open set of initial data on a regular Cauchy surface) lead to a singularity of the given type. The only results of this type in the literature concern polarized Gowdy spacetimes [190], plane symmetric spacetimes with a massless scalar field [291], spacetimes with collisionless matter and spherical, plane or hyperbolic symmetry [272336], and a subset of general Gowdy spacetimes [317315]. The work of Christodoulou [98] on spherically symmetric solutions of the Einstein equations with a massless scalar field should also be mentioned in this context, although it concerns the singularity inside a black hole rather than singularities in cosmological models. Note that all these spacetimes have at least two Killing vectors so that the PDE problem to be solved reduces to an effective problem in one space dimension.


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