A numerical stability analysis for the Einstein-Vlasov system

TL;DR

This paper numerically analyzes the stability of steady states in the Einstein-Vlasov system across different coordinate systems, confirming that the first binding energy maximum indicates the onset of instability and exploring the nature of perturbed solutions.

Contribution

It provides a comprehensive numerical stability analysis of the Einstein-Vlasov system in multiple coordinate systems, verifying the binding energy maximum as a stability indicator and challenging previous assumptions about negative binding energy.

Findings

Binding energy maximum signals instability onset.

Perturbed solutions can collapse, disperse, or form heteroclinic orbits.

Negative binding energy does not always mean dispersion.

Abstract

We investigate stability issues for steady states of the spherically symmetric Einstein-Vlasov system numerically in Schwarzschild, maximal areal, and Eddington-Finkelstein coordinates. Across all coordinate systems we confirm the conjecture that the first binding energy maximum along a one-parameter family of steady states signals the onset of instability. Beyond this maximum perturbed solutions either collapse to a black hole, form heteroclinic orbits, or eventually fully disperse. Contrary to earlier research, we find that a negative binding energy does not necessarily correspond to fully dispersing solutions. We also comment on the so-called turning point principle from the viewpoint of our numerical results. The physical reliability of the latter is strengthened by obtaining consistent results in the three different coordinate systems and by the systematic use of dynamically accessible perturbations.