Self-similar spherically symmetric solutions of the massless Einstein-Vlasov system

TL;DR

This paper constructs and analyzes self-similar solutions to the Einstein-Vlasov system with massless particles, revealing their singularity structure and properties, and also presents static solutions for comparison.

Contribution

It provides the first general construction of spherically symmetric, self-similar solutions of the Einstein-Vlasov system with massless particles, including their regularity and support properties.

Findings

Solutions have a curvature singularity by construction.

Initial data can have compact support in momentum space.

Vlasov distribution function cannot be bounded.

Abstract

We construct the general spherically symmetric and self-similar solution of the Einstein-Vlasov system (collisionless matter coupled to general relativity) with massless particles, under certain regularity conditions. Such solutions have a curvature singularity by construction, and their initial data on a Cauchy surface to the past of the singularity can be chosen to have compact support in momentum space. They can also be truncated at large radius so that they have compact support in space, while retaining self-similarity in a central region that includes the singularity. However, the Vlasov distribution function can not be bounded. As a simpler illustration of our techniques and notation we also construct the general spherically symmetric and static solution, for both massive and massless particles.