The Mixmaster model

The Mixmaster model arises in the study of solutions of the vacuum Einstein equations which are spatially homogeneous. To say that a solution is spatially homogeneous means that it is invariant under a group of symmetries whose orbits are three-dimensional and spacelike. This implies that it can be described by functions depending only on time and that the Einstein equations reduce to a system of ordinary differential equations. The details of these equations depend on the group defining the symmetry. In all but one case this group can be taken to be three-dimensional. Then classifying the symmetry type essentially comes down to classifying all three-dimensional Lie algebras.This was done by Bianchi in the 1890’s. He introduced types I to IX and this terminology is used in the study of the Einstein equations to this day. Types VI and VII are actually one-parameter families of non-isomorphic Lie algebras and are therefore denoted by VI and VII, where is the parameter. The most general Bianchi types (in the sense that the solutions depend on the largest number of parameters) are types VIII, IX and VI. All Bianchi types contain solutions with initial singularities and the three types just listed exhibit complicated oscillatory behaviour in the approach to the singularity. The case which has been studied most is Bianchi IX and this was christened the Mixmaster model by Charles Misner in 1969. He named it after a kind of food mixer which was popular at the time. There has been a great deal of heuristic and numerical work on the dynamics of this system of ODE. Rigorous mathematical results have been relatively rare. The most notable results in this direction are those arising from the PhD thesis of Hans Ringström, some of which will now be described.

It is useful in the study of Bianchi models to distinguish between the Lie algebras which are unimodular (trace of the structure constants zero) and the rest. These are called Class A and Class B respectively. All models of Class A can be incorporated into a single dynamical system. The dynamical systems for types VIII and IX occupy open subsets of the state space and the other types are represented on lower dimensional submanifolds on the boundaries of these open sets. Class A consists of the types I, II, VI, VII, VIII and IX. It was shown by Ringström that the limit points of solutions of Bianchi type IX in the approach to the singularity (technically the -limit points) are of type I and II. For almost all initial data it was proved that the approach to the singularity is oscillatory in the sense that there are at least three distinct -limit points. More details can be found in this paper. The wider context of these results and aspects of the history of the problem are discussed in the paper ‘Mixmaster: Fact and Belief‘ by Mark Heinzle and Claes Uggla.

Recently there has been renewed interest in these problems. A project which I have started in cooperation with the research group of Bernold Fiedler has generated a lot of activity. The central idea is to bring together knowledge about the concrete application with sophisticated techniques from the theory of dynamical systems. What are the problems to be solved? A first question is whether the direct analogues of the Bianchi IX results hold for type VIII. This has not been proved and there are serious reasons for doubting if it is even true. The key issue, at least at our present state of understanding is whether Bianchi VIII solutions may have -limit points of type VI. For type VI almost nothing is known about the corresponding questions on a rigorous level.

Misner’s original motivation for introducing the Mixmaster model was to find an explanation for the observed isotropy of the universe. Whether this explanation can work depends on the nature of the causal structure of the geometry near the singularity. These days it is usual to explain the isotropy of the universe on the basis of inflation and so Misner’s proposal no longer has the same direct physical significance. The mathematical problem remains and has lost none of its intrinsic fascination. In this context it is desirable to understand the detailed structure of the -limit sets of generic and non-generic solutions. In particular there are three special points in the phase space, the flat Kasner solutions. Whether Misner’s mechanism is effective depends on how much time solutions spend close to these special points. Quantifying this requires a very detailed understanding of the asymptotics of solutions near the singularity. This issue is under intensive investigation. I hope to write about the results here when they are ready to come into the public domain.

These questions are of interest for the understanding of solutions far beyond the homogeneous case. This has to do with the picture of general singularities in solutions of the Einstein equations developed by Belinkii, Khalatnikov and Lifshitz, which is one of the most outstanding challenges in mathematical relativity.

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