On the steady states of the spherically symmetric Einstein-Vlasov system

TL;DR

This paper investigates static, spherically symmetric solutions of the Einstein-Vlasov system, revealing multi-peaked mass-energy densities, supporting the Buchdahl inequality, and demonstrating spiral structures in the mass-radius relation through numerical and analytical methods.

Contribution

It provides new numerical evidence and partial proofs for the Buchdahl inequality and the spiral structure in the Einstein-Vlasov system's static solutions.

Findings

Mass-energy density can be multi-peaked.

Buchdahl inequality holds for all steady states.

Static solutions form spirals in the radius-mass diagram.

Abstract

Using both numerical and analytical tools we study various features of static, spherically symmetric solutions of the Einstein-Vlasov system. In particular, we investigate the possible shapes of their mass-energy density and find that they can be multi-peaked, we give numerical evidence and a partial proof for the conjecture that the Buchdahl inequality $\sup_{r > 0} 2 m(r)/r < 8/9$, $m(r)$ the quasi-local mass, holds for all such steady states--both isotropic {\em and} anisotropic--, and we give numerical evidence and a partial proof for the conjecture that for any given microscopic equation of state--both isotropic {\em and} anisotropic--the resulting one-parameter family of static solutions generates a spiral in the radius-mass diagram.