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

TL;DR

This paper critically examines a previous proof claiming static solutions to the Einstein-Vlasov system, identifying errors in key claims and discussing the challenges in establishing such solutions as energy-Casimir functional minimizers.

Contribution

It provides a detailed critique of the proof's variational approach, highlighting errors and uncertainties to guide future research on static Einstein-Vlasov solutions.

Findings

Identifies errors in the proof's key claims.

Questions the validity of the variational approach used.

Stimulates further investigation into the existence of static solutions.

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.