Static solutions to the spherically symmetric Einstein-Vlasov system: a particle-number-Casimir approach

TL;DR

This paper constructs static solutions to the spherically symmetric Einstein-Vlasov system using a variational approach based on particle number and Casimir functionals, addressing open problems and correcting previous flawed attempts.

Contribution

It rigorously constructs static solutions via a variational method, connecting particle number-Casimir functionals to the Einstein-Vlasov system, and clarifies previous unresolved issues.

Findings

Constructed static solutions as fixed points of an Euler-Lagrange equation.

Linked energy density to phase space density function.

Resolved open problem on existence of static solutions.

Abstract

Existence of spherically symmetric solutions to the Einstein-Vlasov system is well-known. However, it is an open problem whether or not static solutions arise as minimizers of a variational problem. Apart from being of interest in its own right, it is the connection to non-linear stability that gives this topic its importance. This problem was considered in \cite{Wol}, but as has been pointed out in \cite{AK}, the paper \cite{Wol} contained serious flaws. In this work we construct static solutions by solving the Euler-Lagrange equation for the energy density $\rho$ as a fixed point problem. The Euler-Lagrange equation originates from the particle number-Casimir functional introduced in \cite{Wol}. We then define a density function $f$ on phase space which induces the energy density $\rho$ and we show that it constitutes a static solution of the Einstein-Vlasov system. Hence we settle rigorously parts of what the author of \cite{Wol} attempted to prove.