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978-3-8439-2389-7, Reihe Mathematik

Lukas Taegert Oppenheimer-Snyder type collapse with Vlasov matter

123 Seiten, Dissertation Universität Bayreuth (2015), Softcover, B5

In a paper published in 1939, Oppenheimer and Snyder proved that in the framework of general relativity, it is possible for a homogeneous ball of pressureless fluid—often referred to as dust—to ultimately contract beyond its own Schwarzschild radius solely by the influence of its own gravitational field; furthermore, they showed that any observer co-moving with the matter will cross this radius in finite proper time.

This thesis—apart from expanding upon the original investigation by using Eddington-Finkelstein instead of the originally employed Schwarzschild coordinates, which allows for the study of the formation of a spacetime singularity at the center of symmetry—aims to establish a similar result using the matter model of a collisionless gas. This choice of a matter model distinguishes itself by the fact that it is no longer pressure-free; also, in contrast to dust and other fluid models, there is strong indication from the non-relativistic case that this model should not be prone to the development of naked and non-spacetime singularities.

First we show that in the non-relativistic Vlasov-Poisson case, a ball of matter that initially sufficiently closely approximates a resting homogeneous ball of dust will contract under the influence of its own gravitational field until most of its constituting particles are concentrated in an arbitrarily small volume of space.

In order to establish a similar result in the general relativistic case, we then use a class of cosmological Friedman-Lemaître-Robertson-Walker (FLRW) type solutions of the Vlasov-Einstein system to provide initial data with a homogeneous center.

We prove that any spherically symmetric initial data with bounded particle momentum that are identical to an FLRW solution on a ball about the origin launch a solution of the Vlasov-Einstein system that exists as long as there remains a homogeneous ball at the center, the particle momentum remains bounded, and there is no singularity at the origin.

Building on techniques implemented in the non-relativistic case, we are then able to prove that there exists a class of solutions of the above type that initially do not contain any trapped surfaces but that develop trapped surfaces later in their evolution, which by Penrose's singularity theorem is a sufficient condition for the development of a spacetime singularity.