Non-degeneracy of Riemannian Schwarzschild-anti de Sitter metrics: Birkhoff-type results in linearized gravity

TL;DR

This paper proves that certain linearized solutions around Riemannian Schwarzschild-anti de Sitter metrics are pure gauge unless controlled by specific functions, establishing non-degeneracy of the linearized Einstein operator in these settings.

Contribution

It extends Birkhoff-type results to higher dimensions and arbitrary horizon topologies, showing non-degeneracy of the linearized Einstein operator for these metrics.

Findings

Solutions not controlled by master functions are pure gauge

The linearized Einstein operator is non-degenerate for open mass ranges

Results apply to arbitrary dimension and horizon topology

Abstract

We prove Birkhoff-type results showing that $L^2$ solutions of the linearized Einstein equations around Riemannian Kottler ("Schwarzschild-anti de Sitter") metrics in arbitrary dimension and horizon topology, which are not controlled by "master functions" are pure gauge. Together with earlier results this implies that the $TT$-gauge-fixed linearized Einstein operator for these metrics is non-degenerate for open ranges of the mass parameter.