Scalar Hairy Black Holes in Four Dimensions are Unstable

TL;DR

This paper investigates the stability of four-dimensional scalar hairy black holes, revealing their instability through numerical analysis and identifying modes similar to those of a massive scalar field on Kerr backgrounds.

Contribution

It introduces a new gauge where scalar perturbations decouple, enabling detailed analysis of quasinormal modes and stability properties of these black holes.

Findings

Unstable modes with characteristic growth rates identified

Scalar hair black holes are shown to be unstable

Perturbation spectrum similar to massive scalar fields on Kerr

Abstract

We present a numerical analysis of the stability properties of the black holes with scalar hair constructed by Herdeiro and Radu. We prove the existence of a novel gauge where the scalar field perturbations decouple from the metric perturbations, and analyse the resulting quasinormal mode spectrum. We find unstable modes with characteristic growth rates which for uniformly small hair are almost identical to those of a massive scalar field on a fixed Kerr background.