Stability of Schwarzschild-AdS for the spherically symmetric Einstein-Klein-Gordon system

TL;DR

This paper proves the asymptotic stability of Schwarzschild-AdS black holes under small perturbations in the spherically symmetric Einstein-Klein-Gordon system with negative cosmological constant, overcoming key analytical challenges.

Contribution

It establishes the nonlinear stability of Schwarzschild-AdS solutions, addressing difficulties from non-monotonic mass and boundary conditions using a bootstrap approach.

Findings

Proved asymptotic stability of Schwarzschild-AdS black holes.

Derived decay estimates for perturbations.

Resolved analytical challenges with Hardy inequalities and redshift effects.

Abstract

In this paper, we study the global behavior of solutions to the spherically symmetric coupled Einstein-Klein-Gordon (EKG) system in the presence of a negative cosmological constant. We prove that the Schwarzschild-AdS spacetimes (the trivial black hole solutions of the EKG system for which $\phi=0$ identically) are asymptotically stable: Small perturbations of Schwarzschild-AdS initial data again lead to regular black holes, with the metric on the black hole exterior approaching a Schwarzschild-AdS spacetime. The main difficulties in the proof arise from the lack of monotonicity for the Hawking mass and the asymptotically AdS boundary conditions, which render even (part of) the orbital stability intricate. These issues are resolved in a bootstrap argument on the black hole exterior, with the redshift effect and weighted Hardy inequalities playing the fundamental role in the analysis. Both integrated decay and pointwise decay estimates are obtained.