Static Solutions to the Einstein--Vlasov System with a Nonvanishing Cosmological Constant

Source abstract

We construct spherically symmetric static solutions to the Einstein--Vlasov system with nonvanishing cosmological constant $\Lambda$. The results are divided as follows. For small $\Lambda>0$ we show the existence of globally regular solutions which coincide with the Schwarzschild--deSitter solution in the exterior of the matter regions. 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 for obtaining a large class of global, nonvacuum spacetimes with topologies $\mathbb R\times S^3$ and $\mathbb R\times S^2\times \mathbb R$ which arise from our solutions as a result of using the periodicity of the Schwarzschild--deSitter solution. A subclass of these solutions contains black holes of different masses.