Stability of the Toroidal AdS Schwarzschild Solution in the Einstein--Klein-Gordon System

TL;DR

This paper proves the nonlinear stability of toroidal AdS-Schwarzschild black holes within the Einstein--Klein-Gordon system under symmetric perturbations, considering different boundary conditions for the scalar field.

Contribution

It establishes the orbital and asymptotic stability of these black holes for a range of Klein-Gordon masses, including the analysis of boundary conditions and energy divergence.

Findings

Stability holds for Dirichlet and Neumann boundary conditions.

Neumann boundary conditions lead to infinite energy solutions.

Stability results are valid for a range of scalar field masses.

Abstract

We consider the stability of the toroidal AdS-Schwarzshild black holes as solutions of the Einstein--Klein-Gordon system, with Dirichlet or Neumann boundary conditions for the scalar field. Restricting to perturbations that respect the toroidal symmetry we show both orbital and asymptotic stability for the full nonlinear problem, for a range of choices of the Klein-Gordon mass. The solutions we construct with Neumann boundary conditions have a Hawking mass which diverges towards infinity, reflecting the infinite energy of the Klein-Gordon field for perturbations satisfying these boundary conditions.