8.3 Formation of localized structure8 Structure of General Singularities8.1 Lessons from homogeneous solutions

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 [35]. 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 [32].

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 [139], plane symmetric spacetimes with a massless scalar field [208], spacetimes with collisionless matter and spherical, plane or hyperbolic symmetry [190], and a subset of general Gowdy spacetimes [85]. The work of Christodoulou [69] 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.



8.3 Formation of localized structure8 Structure of General Singularities8.1 Lessons from homogeneous solutions

image Theorems on Existence and Global Dynamics for the Einstein Equations
Alan D. Rendall
http://www.livingreviews.org/lrr-2002-6
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