Hairy black holes: stability under odd-parity perturbations and existence of slowly rotating solutions

TL;DR

This paper proves the mode stability of static, spherically symmetric or planar black holes and solitons with scalar fields under odd-parity perturbations, and explores the existence of slowly rotating solutions.

Contribution

It generalizes the Regge-Wheeler equation for scalar fields and demonstrates stability regardless of potential or asymptotic conditions, also analyzing slowly rotating configurations.

Findings

Mode stability under odd-parity perturbations for a broad class of scalar black holes.

Generalization of the Regge-Wheeler equation for scalar fields.

Existence of slowly rotating solutions with frame dragging effects.

Abstract

We show that, independently of the scalar field potential and of specific asymptotic properties of the spacetime (asymptotically flat, de Sitter or anti-de Sitter), any static, spherically symmetric or planar, black hole or soliton solution of the Einstein theory minimally coupled to a real scalar field with a general potential is mode stable under linear odd-parity perturbations. To this end, we generalize the Regge-Wheeler equation for a generic self-interacting scalar field, and show that the potential of the relevant Schr\"odinger operator can be mapped, by the so-called S-deformation, to a semi-positively defined potential. With these results at hand we study the existence of slowly rotating configurations. The frame dragging effect is compared with the Kerr black hole.