On the Einstein-Vlasov system

In this thesis we consider the Einstein-Vlasov system, which models a system of particles within the framework of general relativity, and where collisions between the particles are assumed to be sufficiently rare to be neglected. Here the particles are stars, galaxies or even clusters of galaxies, which interact by the gravitational field generated collectively by the particles. The thesis consists of three papers, and the first two are devoted to cylindrically symmetric spacetimes and the third treats the spherically symmetric case. In the first paper the time-dependent Einstein-Vlasov system with cylindrical symmetry is considered. We prove global existence in the so called polarized case under the assumption that the particles never reach a neighborhood of the axis of symmetry. In the more general case of a non-polarized metric we need the additional assumption that the derivatives of certain metric components are bounded in a vicinity of the axis of symmetry to obtain global existence. The second paper of the thesis considers static cylindrical spacetimes. In this case we prove global existence in space and also that the solutions have finite extension in two of the three spatial dimensions. It then follows that it is possible to extend the spacetime by gluing it with a Levi-Civita spacetime, i.e. the most general vacuum solution of the static cylindrically symmetric Einstein equations. In the third and last paper, which is a joint work with C. Uggla and M. Heinzle, the static spherically symmetric Einstein-Vlasov system is studied. We introduce a new method by rewriting the system as an autonomous dynamical system on a state space with compact closure. In this way we are able to improve earlier results and enlarge the class of distribution functions which give rise to steady states with finite mass and finite extension