Geometrische Transport-Gleichungen und der nicht-vakuum Einstein-Fluss

The recent groundbreaking discovery of gravitational waves once more demonstrated how accurately Einstein’s theory of General Relativity describes the geometry of spacetime. It is therefore reasonable to expect that further predictions of Einstein’s 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 Einstein’s 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.