Comments on the paper 'Static solutions of the Vlasov-Einstein system' by G. Wolansky

TL;DR

This note addresses the attempted proof of the existence of static solutions to the Einstein–Vlasov system as given in Wolansky (Arch Ration Mech Anal 156:205–230, 2001), and shows that two important claims in Wolanski are incorrect and the validity of a third claim is questioned.

Abstract

In this note we address the attempted proof of the existence of static solutions to the Einstein-Vlasov system as given in \cite{Wol}. We focus on a specific and central part of the proof which concerns a variational problem with an obstacle. We show that two important claims in \cite{Wol} are incorrect and we question the validity of a third claim. We also discuss the variational problem and its difficulties with the aim to stimulate further investigations of this intriguing problem: to answer the question whether or not static solutions of the Einstein-Vlasov system can be found as local minimizers of an energy-Casimir functional.