Black hole nonmodal linear stability: the Schwarzschild (A)dS cases

TL;DR

This paper extends the nonmodal linear stability analysis of Schwarzschild black holes to include cases with nonnegative cosmological constant, demonstrating boundedness and decay of perturbations, and exploring stability conditions in anti-de Sitter space.

Contribution

It generalizes the stability results to Schwarzschild (A)dS spacetimes, introduces gauge-invariant scalars for perturbation analysis, and discusses boundary conditions affecting stability.

Findings

Perturbations decay over time, leaving a Kerr black hole.

Boundedness of gauge-invariant scalars in the outer static region.

Instability of Schwarzschild anti-de Sitter black holes under certain boundary conditions.

Abstract

The nonmodal linear stability of the Schwarzschild black hole established in Phys. Rev. Lett. 112 (2014) 191101 is generalized to the case of a nonnegative cosmological constant $\Lambda$. Two gauge invariant combinations $G_{\pm}$ of perturbed scalars made out of the Weyl tensor and its first covariant derivative are found such that the map $[h_{\alpha \beta}] \to \left( G_- \left([h_{\alpha \beta}] \right), G_+ \left([h_{\alpha \beta}] \right) \right)$ with domain the set of equivalent classes $[h_{\alpha \beta}]$ under gauge transformations of solutions of the linearized Einstein's equation, is invertible. The way to reconstruct a representative of $[h_{\alpha \beta}]$ in terms of $(G_-,G_+)$ is given. It is proved that, for an arbitrary perturbation consistent with the background asymptote, $G_+$ and $G_-$ are bounded in the the outer static region. At large times, the perturbation decays leaving a linearized Kerr black hole around the Schwarzschild or Schwarschild de Sitter background solution. For negative cosmological constant it is shown that there is a choice of boundary conditions at the time-like boundary under which the Schwarzschild anti de Sitter black hole is unstable. The root of Chandrasekhar's duality relating odd and even modes is exhibited, and some technicalities related to this duality and omitted in the original proof of the $\Lambda=0$ case are explained in detail.