Geometric Transport equations and the non-vacuum Einstein-flow

The recent groundbreaking discovery of gravitational waves once more demonstrated how accurately Einsteins theory of General Relativity describes the geometry of spacetime. It is therefore reasonable to expect that further predictions of Einsteins equations will be verified in similarly impressive experiments in the future. This, once more, underlines the meaning of the mathematical study of the theory of General Relativity and related areas. The proposed project is dedicated to the mathematical study of a system consisting of Einstein equations coupled with the so-called Vlasov equation the Einstein-Vlasov system. This system describes Ensembles of particles, which move on idealized paths without collisions only mutually interacting via gravitation. These assumptions of the model are compatible with observations of our universe on large scales: Galaxies and Galaxy Clusters indeed move almost collisionless through space only subject to their mutual gravitational interaction. It is expected that the study of the Einstein-Vlasov system reveals insight on the behavior of our universe on large scales. The project is dedicated to certain related questions, which we describe in the following. Eternal expansion or recollapse? It is known that our universe is expanding. Mathematical models, however, imply that this expansion may be limited and may reach its maximum at a certain time followed by a recollapse, i.e. a contraction ending in a big crunch the opposite of a big bang. In the alternative scenario, the expansion never ends and the universe expands eternally. There are interesting conjectures about the relation between the topology of our universe and the aforementioned behavior. One aim of the envisioned project is to investigate this connection for the Einstein-Vlasov System and to prove related rigorous results. Nature of Singularities. It is known that our universe emanates from a so-called big bang. In the sense of Einsteins equations this is a singularity out of which the universe has started to expand. A fundamental questions on singularities is whether space-time ends at those points or if it is possible to extend it beyond them. A method to show that space-time indeed ends in a singularity would be to show that the curvature of spacetimes becomes infinite at this specific at this specific point. Another aim of the project is to show this effect for a class of models. Stability of Models. An important feature of models of the universe is their stability. This means that small perturbations of the parameters of the model lead to another model which only deviates mildly from the original one. To establish features of this kind for the Einstein-Vlasov system is another aim of the envisioned project.

The Einstein equations describe the dynamical behaviour of the gravitational field over time. Generalized system, such as the Einstein-Vlasov system, model the gravitational field in the presence of matter, which generates the gravitational field. The Einstein-Vlasov system is a system of this kind, which describes models of collisionless particles, which interact via gravity The rigorous study of solutions of these equations describe various spacetimes such as isolated gravitating systems as models for galaxies and galaxy clusters as well as cosmological models to describe the universe as a whole. In this research project various theorems about the long-time behaviour of the Einstein-Vlasov System were proved. In particular, the stability of two fundamental models in general relativity were established: of Minkowski space and of the Milne model. Stability in this context means that small initial perturbations close to those models decay over time and the system asymptotes to the initial unperturbed configuration Those results in particular imply that the corresponding models are suitable to describe realistic physical phenomena, as they are robust against small perturbations. To obtain these results fundamental mathematical methods to analyse the corresponding equations were developed and refined and eventually applied in combination to the problems at hand. Beside these main results comparable results for other matter models were obtained such as for Klein-Gordon field, fluids and Kaluza-Klein fields from String theory. In the context of homogeneous cosmology particular matter-dominated solutions were constructed and analyzed, were the presence of matter generates strong deviations from the spacetime geometry in the vacuum.