Instability of hairy black holes in shift-symmetric Horndeski theories

TL;DR

This paper investigates the stability of black holes with scalar hair in shift-symmetric Horndeski theories, revealing that known solutions exhibit instabilities or strong coupling issues in perturbations.

Contribution

It develops perturbation theory for black holes with time-dependent scalar hair and demonstrates their perturbative instability or strong coupling.

Findings

All known solutions with scalar hair are unstable or strongly coupled.

Perturbation analysis is developed for odd-parity modes.

The results challenge the viability of such hairy black hole solutions.

Abstract

Recently it was pointed out that in shift-symmetric scalar-tensor theories a black hole can have nontrivial scalar hair which depends linearly on time. We develop black hole perturbation theory for such solutions and compute the quadratic action of odd-parity perturbations. We show that around all the solutions known so far with such time-dependent scalar hair the perturbations trigger instabilities or are presumably strongly coupled.