A class of dust-like self-similar solutions of the massless Einstein-Vlasov system

TL;DR

This paper proves the existence of a new class of self-similar, dust-like solutions to the Einstein-Vlasov system, which could shed light on potential violations of cosmic censorship.

Contribution

It introduces a novel class of non-smooth, self-similar solutions bridging smooth solutions and dust, using a reduction to a four-dimensional ODE system.

Findings

Existence of dust-like self-similar solutions proved.

Solutions connect specific initial and stationary points.

Reduction to a four-dimensional ODE system via a shooting argument.

Abstract

In this paper the existence of a class of self-similar solutions of the Einstein-Vlasov system is proved. The initial data for these solutions are not smooth, with their particle density being supported in a submanifold of codimension one. They can be thought of as intermediate between smooth solutions of the Einstein-Vlasov system and dust. The motivation for studying them is to obtain insights into possible violation of weak cosmic censorship by solutions of the Einstein-Vlasov system. By assuming a suitable form of the unknowns it is shown that the existence question can be reduced to that of the existence of a certain type of solution of a four-dimensional system of ordinary differential equations depending on two parameters. This solution starts at a particular point $P_0$ and converges to a stationary solution $P_1$ as the independent variable tends to infinity. The existence proof is based on a shooting argument and involves relating the dynamics of solutions of the four-dimensional system to that of solutions of certain two- and three-dimensional systems obtained from it by limiting processes.