### 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 [14]. 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 [272, 336], and a subset of general Gowdy spacetimes [317, 315]. 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.