Asymptotic stability of vacuum type II metrics

TL;DR

This paper extends stability results from Schwarzschild to Petrov type II twisting metrics, showing that under certain conditions, these metrics asymptotically approach the Kerr solution, with Bondi energy serving as a Lyapunov functional.

Contribution

It generalizes the asymptotic stability analysis to a broader class of metrics, specifically Petrov type II twisting metrics, under asymptotic flatness conditions.

Findings

The final state of perturbations is the Kerr metric.

Bondi energy acts as a Lyapunov functional indicating stability.

Stability results are extended beyond Schwarzschild to type II twisting metrics.

Abstract

We generalize the result of Lukacs {\it et al.} on asymptotic stability of the Schwarzschild metric with respect to perturbations in the Robinson-Trautman class of metrics to the case of Petrov type II twisting metrics, uder the condition of asymptotic flatness at future null infinity. The Bondi energy is used as the Lyapunov functional and we prove that the "final state" of such metrics is the Kerr metric.