Spatially homogeneous solutions can be classified into Bianchi and Kantowski-Sachs solutions. The Bianchi solutions in turn can be subdivided into types I to IX according to the Lie algebra of the isometry group of the spacetime. Two of the types, VI and VII are in fact one-parameter families of non-isomorphic Lie algebras labelled by h . The generality of the different symmetry types can be judged by counting the number of parameters in the initial data for each type. The result of this is that the most general types are Bianchi VIII, Bianchi IX, and Bianchi VI . The usual picture is that Bianchi VIII and Bianchi IX have more complicated dynamics than all other types and that the dynamics is similar in both these cases. This leads one to concentrate on Bianchi type IX and the mixmaster solution (see Section 3.2). Bianchi type VI was apparently never mentioned in the work of BKL and has been largely ignored in the literature. This is a gap in understanding that should be filled. Here we follow the majority and focus on Bianchi type IX.
Another aspect of the BKL picture is that most types of matter should become negligible near the singularity for suitably general solutions. In the case of perfect fluid solutions of Bianchi type IX with a linear equation of state, this has been proved by Ringström . In the case of collisionless matter it remains an open issue, since rigorous results are confined to Bianchi types I, II and III and Kantowski-Sachs, and have nothing to say about Bianchi type IX. If it is accepted that matter is usually asymptotically negligible then vacuum solutions become crucial. The vacuum solutions of Bianchi type IX (mixmaster solutions) play a central role. They exhibit complicated oscillatory behaviour, and essential aspects of this have been captured rigorously in the work of Ringström [223, 222] (compare Section 3.2).
Some matter fields can have an important effect on the dynamics near the singularity. A scalar field or stiff fluid leads to the oscillatory behaviour being replaced by monotone behaviour of the basic quantities near the singularity, and thus to a great simplification of the dynamics. An electromagnetic field can cause oscillatory behaviour that is not present in vacuum models or models with perfect fluid of the same symmetry type. For instance, models of Bianchi type I with an electromagnetic field show oscillatory, mixmaster-like behaviour . However, it seems that this does not lead to anything essentially new. It is simply that the effects of spatial curvature in the more complicated Bianchi types can be replaced by electromagnetic fields in simpler Bianchi types.
A useful heuristic picture that systematizes much of what is known about the qualitative dynamical behaviour of spatially homogeneous solutions of the Einstein equations is the idea developed by Misner  of representing the dynamics as the motion of a particle in a time-dependent potential. In the approach to the singularity the potential develops steep walls where the particle is reflected. The mixmaster evolution consists of an infinite sequence of bounces of this kind.
|Theorems on Existence and Global Dynamics for the
Alan D. Rendall
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