Static solutions of the spherically symmetric Vlasov-Einstein system

TL;DR

This paper proves the existence of static, spherically symmetric solutions to the Vlasov-Einstein system, including both regular and singularity-containing configurations, contributing to understanding gravitational equilibrium states in general relativity.

Contribution

It extends previous work by establishing the existence of a broader class of static solutions, including those with singularities and anisotropic pressures, in the Vlasov-Einstein system.

Findings

Existence of asymptotically flat static solutions with finite mass

Construction of smooth, singularity-free solutions with regular centers

Identification of solutions with Schwarzschild singularities at the center

Abstract

The Vlasov-Einstein system describes the evolution of an ensemble of particles (such as stars in a galaxy, galaxies in a galaxy cluster etc.) interacting only by the gravitational field which they create collectively and which obeys Einstein's field equations. The matter distribution is described by the Vlasov or Liouville equation for a collisionless gas. Recent investigations seem to indicate that such a matter model is particularly suited in a general relativistic setting and may avoid the formation of naked singularities, as opposed to other matter models. In the present note we consider the Vlasov-Einstein system in a spherically symmetric setting and prove the existence of static solutions which are asymptotically flat and have finite total mass and finite extension of the matter. Among these there are smooth, singularity-free solutions, which have a regular center and may have isotropic or anisotropic pressure, and solutions, which have a Schwarzschild-singularity at the center. The paper extends previous work, where smooth, globally defined solutions with regular center and isotropic pressure were considered.