Static solutions to the Einstein-Vlasov system with non-vanishing cosmological constant

TL;DR

This paper constructs and analyzes static, spherically symmetric solutions to the Einstein-Vlasov system with a non-zero cosmological constant, revealing conditions for regularity, singularities, and topologies, including black hole configurations.

Contribution

It provides new existence results for globally regular and singular solutions with non-zero cosmological constant, extending the understanding of Einstein-Vlasov systems with diverse topologies.

Findings

Existence of globally regular solutions for small positive and negative cosmological constants.

Construction of solutions with Schwarzschild singularity at the center.

Identification of solutions with non-vacuum topologies and black holes.

Abstract

We construct spherically symmetric, static solutions to the Einstein-Vlasov system with non-vanishing cosmological constant $\Lambda$. The results are divided as follows. For small $\Lambda>0$ we show existence of globally regular solutions which coincide with the Schwarzschild-deSitter solution in the exterior of the matter sources. For $\Lambda<0$ we show via an energy estimate the existence of globally regular solutions which coincide with the Schwarzschild-Anti-deSitter solution in the exterior vacuum region. We also construct solutions with a Schwarzschild singularity at the center regardless of the sign of $\Lambda$. For all solutions considered, the energy density and the pressure components have bounded support. Finally, we point out a straightforward method to obtain a large class of globally non-vacuum spacetimes with topologies $\mathbb R\times S^3$ and $\mathbb R\times S^2\times \mathbb R$ which arise from our solutions using the periodicity of the Schwarzschild-deSitter solution. A subclass of these solutions contains black holes of different masses.