Static cylindrically symmetric spacetimes

We prove existence of static solutions to the cylindrically symmetric Einstein-Vlasov system, and we show that the matter cylinder has finite extension. The same results are also proved for a quite general class of equations of state for perfect fluids coupled to the Einstein equations, extending the class of equations of state considered in \cite{BL}. We also obtain this result for the Vlasov-Poisson system.Comment: Added acknowledgemen

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

Verklighetsbaserad matematikinlärning ur ett lokalt perspektiv

Validerat; 20101217 (root

Self-gravitating stationary spherically symmetric systems in relativistic galactic dynamics

We study equilibrium states in relativistic galactic dynamics which are described by stationary solutions of the Einstein-Vlasov system for collisionless matter. We recast the equations into a regular three-dimensional system of autonomous first order ordinary differential equations on a bounded state space. Based on a dynamical systems analysis we derive new theorems that guarantee that the steady state solutions have finite radii and masses