On the Einstein-Vlasov system: Stationary Solutions and Small Data Solutions with Charged and Massless Particles

Doctoral thesis, 2020

The Vlasov matter model describes an ensemble of collisionless particles moving through space-time. These particles interact via the gravitational field which they create collectively. In the framework of General Relativity this gravitational field is described by space-time curvature. Mathematically the situation is captured by the Einstein-Vlasov system. If the particles are charged an electro-magnetic field is created as well and the Maxwell equations are coupled to the system in addition. In astrophysics Vlasov matter is widely used to describe galaxies, globular clusters or galaxy clusters. Also in cosmology or plasma physics the Vlasov matter model plays an important role.

In this thesis a collection of results on the Einstein-Vlasov system is presented. The Papers I to IV are concerned with stationary solutions and the Papers V and VI contain stability results for Minkowski space-time (the trivial solution of Einstein's field equations describing an empty, flat space-time), i.e.~global existence results for the time evolution problem with small initial data.

In Paper I, spherically symmetric, static solutions of the Einstein-Vlasov system with massless particles are constructed. These solutions constitute very thin and highly dense shells of matter with a vacuum region at the center. One can think of these shells as highly energetic, bent light which keeps itself together through the strong gravitational field created by itself. In Paper II, charged particles are considered and the existence of spherically symmetric, static solutions of the Einstein-Vlasov-Maxwell system is proven. It is possible to obtain the large variety of different spherically symmetric, static solutions that are known in the uncharged case, as for example balls, shells and multi-shells. Paper III is concerned with isotropic solutions, i.e.~solutions where the momenta are equally distributed among the particles. In this case Vlasov matter resembles a perfect fluid in many respects. It is shown that a uniqueness result for perfect fluids can be applied to Vlasov matter. This implies that every isotropic, static solution is uniquely determined by the surface potential and in particular spherically symmetric, if its overall pressure is not too high. In Paper IV solutions are constructed where the momenta are not equally distributed among the particles. These solutions have preferred axes of rotation or even an overall angular momentum. This way axially symmetric (but not spherically symmetric), stationary solutions of the Einstein-Vlasov-Maxwell system are obtained.

In Paper V, exploiting the convenient conformal invariance properties of massless Vlasov matter, this matter model is integrated into the framework of the conformal Einstein field equations. In this framework, via a conformal rescaling, the physical space-time, which might be a perturbation of Minkowski space-time or de Sitter space-time, is identified with a compact portion of the Einstein-cylinder (or perturbations thereof). This way global Cauchy problems are turned into local Cauchy problems for which methods to obtain local existence are available. A semi-global stability result for Minkowski space-time and a global stability result for de Sitter space-time is obtained this way. In Paper VI the stability of Minkowski space-time for perturbations with massless Vlasov matter is proved with a completely different method, the vector field method for relativistic transport equations. Thereby an asymptotic stability result with very weak assumptions on the initial data is obtained, in particular no compact support assumptions of any kind are necessary for the initial data.