Stationary solutions of the axially symmetric Einstein-Vlasov system: present status and open problems

TL;DR

This paper reviews the current status and open problems related to stationary solutions of the axially symmetric Einstein-Vlasov system, highlighting analytical and numerical challenges, including existence proofs, properties of relativistic solutions, and potential astrophysical applications.

Contribution

It provides a comprehensive overview of the open problems in analytical and numerical studies of stationary solutions, including recent open-source computational tools and future research directions.

Findings

Open source code for constructing solutions is available.

Open problems include proving existence of non-spherical solutions.

Highly relativistic solutions may approach black holes or cosmic string limits.

Abstract

The purpose of this work is to review the status about stationary solutions of the axially symmetric Einstein-Vlasov system with a focus on open problems of both analytical and numerical nature. For the latter we emphasize that the code used to construct stationary solutions in \cite{Ames2016,Ames2019} is open source, see \cite{Ames2023joss}. In the analytical setting the open problems include establishing methods for proving existence of axisymmetric stationary solutions which are far from spherically symmetric, both in the general case and for certain special classes of solutions pointed out in the text. In the numerical setting there are intriguing properties of highly relativistic solutions that demand further attention, such as whether a sequence of such stationary solutions can approach a Kerr black hole, or if they necessarily approach the thin ring limit reminiscent of cosmic strings. The question of whether stationary solutions include states with thin-disk like morphologies as seen in many galaxies is also open. Finally, there are opportunities to extend this research to new settings such as the case of massless particles and coupled black hole-matter systems. We believe that some of the open problems highlighted here are of central importance for the understanding of nature.