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3.2 Spatially homogeneous solutions

A solution of the Einstein equations is called spatially homogeneous if there exists a group of symmetries with three-dimensional spacelike orbits. In this case there are at least three linearly independent spacelike Killing vector fields. For most matter models the field equations reduce to ordinary differential equations. (Kinetic matter leads to an integro-differential equation.) The most important results in this area have been reviewed in the book [344Jump To The Next Citation Point] edited by Wainwright and Ellis (see, in particular, Part Two of the book). There remain a host of interesting and accessible open questions. The spatially homogeneous solutions have the advantage that it is not necessary to stop at just existence theorems; information on the global qualitative behaviour of solutions can also be obtained.

An important question that was open for a long time concerns the mixmaster model, as discussed in [296]. 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 recently there were no mathematical proofs of 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 [347]. In the much more difficult case of the mixmaster model itself, corresponding results were obtained by Ringström [312Jump To The Next Citation Point]. Later he extended this in several directions in [311Jump To The Next Citation Point]. 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 p = (g- 1)r with 1 < g < 2 are approximated near the singularity by vacuum solutions. The case of a stiff fluid (g = 2) 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 [2264748] 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 [308309302Jump To The Next Citation Point345259184180]. 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.


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