Hairy Black Hole Stability in AdS, Quantum Mechanics on the Half-Line and Holography

TL;DR

This paper analyzes the linear stability of hairy black holes in AdS space with mixed boundary conditions, linking mathematical spectral theory to holographic dual field theories and establishing stability restrictions.

Contribution

It demonstrates how self-adjoint extensions of the Schrödinger operator determine stability conditions for hairy black holes in AdS, connecting boundary conditions to dual field theory potentials.

Findings

Stability depends on boundary conditions via self-adjoint extensions.

Negative modes are excluded by integral restrictions on the potential.

Results have implications for holographic dual field theories.

Abstract

We consider the linear stability of $4$-dimensional hairy black holes with mixed boundary conditions in Anti-de Sitter spacetime. We focus on the mass of scalar fields around the maximally supersymmetric vacuum of the gauged $\mathcal{N}=8$ supergravity in four dimensions, $m^{2}=-2l^{-2}$. It is shown that the Schr\"{o}dinger operator on the half-line, governing the $S^{2}$, $H^{2}$ or $\mathbb{R}^{2}$ invariant mode around the hairy black hole, allows for non-trivial self-adjoint extensions and each of them correspons to a class of mixed boundary conditions in the gravitational theory. Discarding the self-adjoint extensions with a negative mode impose a restriction on these boundary conditions. The restriction is given in terms of an integral of the potential in the Schr\"{o}dinger operator resembling the estimate of Simon for Schr\"{o}dinger operators on the real line. In the context of AdS/CFT duality, our result has a natural interpretation in terms of the field theory dual effective potential.