Compact Relativistic Matter Shells with Massless and Charged Particles

Licentiatavhandling, 2017

The Vlasov model is a matter model that is widely used in physics. In the context of astrophysics and cosmology it describes an ensemble of self gravitating, collisionless particles. These particles can be interpreted as for example ions, stars, or galaxies. If the gravity that the particles create collectively is described by General Relativity the Vlasov equation couples to Einstein’s field equations and the Einstein-Vlasov system is obtained. If it is furthermore assumed that the particles carry a charge, then an electromagnetic field is created collectively which in addition to the space-time curvature governs the particles’ trajectories and which satisfies Maxwell’s equations. This thesis focuses on static solutions, in spherical symmetry. In the introduction the Einstein-Vlasov-Maxwell system is introduced formally and the equations in spherical symmetry are derived. Further, the results of this thesis are set into relation to the current sate of research. Finally, a numerical section characterizing the solutions constructed analytically in the papers concludes the introduction. In Paper I spherically symmetric, static shell solutions for massless particles are constructed. The difficulty lies in showing that the matter quantities of these solutions are of compact support. A different approach than in the massive case is necessary. Furthermore the obtained solutions are compared to the concept of geons, introduced by John Wheeler in 1955. Geons are solutions of the Einstein-Maxwell system that are spherically symmetric on a time average and have long lifetimes. They were originally intended to serve as models for individual particles. In paper II charged particles are considered. In the first part local existence of spherically symmetric, static solutions of the Einstein-Vlasov-Maxwell system around the center of symmetry is established. Then, based on that, the existence of compactly supported, static solutions with finite mass is shown for small particle charges by a perturbation argument. In the last part the existence of thin shell solutions is proven for arbitrary values of the particle charge. The proof yields sequences of shell solutions approaching an infinitesimally thin shell. Some properties of this limit are discussed.