Homogeneous cosmological models

The most drastic symmetry assumption which still allows some dynamics is that of spatially homogeneity. In that case the partial differential equations reduce to ordinary differential equations since all variables depend on time alone. In general relativity the gravitational field has interesting and complicated dynamics even in the absence of matter ("vacuum case"). The homogeneous vacuum Einstein equations lead to a system of four ordinary differential equations (the Wainwright-Hsu system). When the simplest type of matter, a perfect fluid, is included the number of equations is increased to five. The dynamics of general solutions of this system is still far from understood, despite major advances by Ringström in the late 1990's. I have a project with the group of Bernold Fiedler at the Free University in Berlin to bring sophisticated techniques from the theory of dynamical systems to bear on this subject. This is project B7 of the special research area SFB 647, Space-Time-Matter. It has been hypothesized that the dynamics is controlled by an object called the Mixmaster attractor. The aim of this project is to find out in what sense and to what extent this is true. It seems a priori difficult to obtain information about the dynamics of general solutions of a specific five-dimensional dynamical system. That it is possible for the WH system is due to some very special properties. It has some lower dimensional invariant submanifolds and there are Liapunov functions which drive the solution towards these submanifolds during the evolution. Thus much of the evolution of general solutions can be encoded in terms of the dynamics on the submanifolds. It is nevertheless the case that these lower dimensional dynamical systems, which themselves have relatively simple behaviour, combine to produce very subtle effects. This is why it has taken the research community so long to make progress on this problem. An interesting type of matter source for the gravitational field, both for physics and for mathematics, is collisionless matter described by kinetic theory. For this matter model the basic unknown depends on time, position and velocity. The spatial dependence is eliminated by spatial homogeneity but the velocity dependence remains. Thus in this case a partial differential equation is retained even in the homogeneous case. There are special cases where the problem can be reduced to a system of ordinary differential equations and their dynamics have been analysed by Paul Tod, Claes Uggla and myself, cf. gr-qc/9811051 , gr-qc/0005116 , and gr-qc/0112040 . To go beyond this restricted situation is the goal of the PhD project of Ernesto Nungesser.

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Alan Rendall
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