The Einstein-Vlasov system in spherical symmetry II: spherical perturbations of static solutions

TL;DR

This paper simplifies the analysis of spherical perturbations in the Einstein-Vlasov system to a wave equation, identifies conditions for stability, and finds an unstable mode matching previous nonlinear results.

Contribution

It reduces the complex perturbation equations to a single wave equation and numerically identifies an unstable mode in static solutions.

Findings

Identified a symmetric operator H for perturbations

Numerically approximated eigenvalues using the Ritz method

Found an unstable mode with growth rate matching previous nonlinear studies

Abstract

We reduce the equations governing the spherically symmetric perturbations of static spherically symmetric solutions of the Einstein-Vlasov system (with either massive or massless particles) to a single stratified wave equation $-\psi_{,tt}=H\psi$, with $H$ containing second derivatives in radius, and integrals over energy and angular momentum. We identify an inner product with respect to which $H$ is symmetric, and use the Ritz method to approximate the lowest eigenvalues of $H$ numerically. For two representative background solutions with massless particles we find a single unstable mode with a growth rate consistent with the universal one found by Akbarian and Choptuik in nonlinear numerical time evolutions.