An important question that has been open for a long time
concerns the mixmaster model, as discussed in [213]. This is a class of spatially homogeneous solutions of the
vacuum Einstein equations, which are invariant under the group
*SU*
(2). A special subclass of these
*SU*
(2)-invariant solutions, the (parameter-dependent) Taub-NUT
solution, is known explicitly in terms of elementary functions.
The Taub-NUT solution has a simple initial singularity which is
in fact a Cauchy horizon. All other vacuum solutions admitting a
transitive action of
*SU*
(2) on spacelike hypersurfaces (Bianchi type IX solutions)
will be called generic in the present discussion. These generic
Bianchi IX solutions (which might be said to constitute the
mixmaster solution proper) have been believed for a long time to
have singularities that are oscillatory in nature where some
curvature invariant blows up. This belief was based on a
combination of heuristic considerations and numerical
calculations. Although these together do make a persuasive case
for the accepted picture, until very recently there were no
mathematical proofs of the these features of the mixmaster model
available. This has now changed. First, a proof of curvature
blow-up and oscillatory behaviour for a simpler model (a solution
of the Einstein-Maxwell equations) which shares many qualitative
features with the mixmaster model, was obtained by Weaver [240]. In the much more difficult case of the mixmaster model itself,
corresponding results were obtained by Ringström [223]. Later he extended this in several directions in [222]. In that paper more detailed information was obtained
concerning the asymptotics and an attractor for the evolution was
identified. It was shown that generic solutions of Bianchi
type IX with a perfect fluid whose equation of state is
with
are approximated near the singularity by vacuum solutions. The
case of a stiff fluid () which has a different asymptotic behaviour was analysed
completely for all models of Bianchi class A, a class which
includes Bianchi type IX.

Ringström's analysis of the mixmaster model is potentially of great significance for the mathematical understanding of singularities of the Einstein equations in general. Thus, its significance goes far beyond the spatially homogeneous case. According to extensive investigations of Belinskii, Khalatnikov and Lifshitz (see [164, 30, 31] and references therein), the mixmaster model should provide an approximate description for the general behaviour of solutions of the Einstein equations near singularities. This should apply to many matter models as well as to the vacuum equations. The work of Belinskii, Khalatnikov, and Lifshitz (BKL) is hard to understand and it is particularly difficult to find a precise mathematical formulation of their conclusions. This has caused many people to remain sceptical about the validity of the BKL picture. Nevertheless, it seems that nothing has ever been found to indicate any significant flaws in the final version. As long as the mixmaster model itself was not understood this represented a fundamental obstacle to progress on understanding the BKL picture mathematically. The removal of this barrier opens up an avenue to progress on this issue. The BKL picture is discussed in more detail in Section 8 .

Some recent and qualitatively new results concerning the asymptotic behaviour of spatially homogeneous solutions of the Einstein-matter equations, both close to the initial singularity and in a phase of unlimited expansion, (and with various matter models) can be found in [219, 220, 200, 238, 183, 135]. These show in particular that the dynamics can depend sensitively on the form of matter chosen. (Note that these results are consistent with the BKL picture.) The dynamics of indefinitely expanding cosmological models is discussed further in Section 7 .

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