Boundedness and growth for the massive wave equation on asymptotically anti-de Sitter black holes

TL;DR

This paper investigates the stability and growth of massive scalar fields around asymptotically AdS black holes, revealing how boundary conditions influence stability and the existence of scalar hair.

Contribution

It establishes a stability criterion for scalar fields on AdS black holes and shows boundary conditions affect stability and allow scalar hair formation.

Findings

Stability depends on boundary conditions.

Existence of scalar hair under Robin boundary conditions.

Criteria for linear stability of scalar fields.

Abstract

We study the global dynamics of free massive scalar fields on general, globally stationary, asymptotically AdS black hole backgrounds with Dirichlet-, Neumann- or Robin- boundary conditions imposed on $\psi$ at infinity. This class includes the regular Kerr-AdS black holes satisfying the Hawking Reall bound $r_+^2 > |a|l$. We establish a suitable criterion for linear stability (in the sense of uniform boundedness) of $\psi$ and demonstrate how the issue of stability can depend on the boundary condition prescribed. In particular, in the slowly rotating Kerr-AdS case, we obtain the existence of linear scalar hair (i.e. non-trivial stationary solutions) for suitably chosen Robin boundary conditions.