Black hole perturbations in Maxwell-Horndeski theories

TL;DR

This paper analyzes the linear stability of black holes in Maxwell-Horndeski theories, deriving conditions for stability of various perturbation modes and applying these to specific gravity models, revealing stability or instability of known solutions.

Contribution

It provides a comprehensive stability analysis for black holes in Maxwell-Horndeski theories, including new conditions for ghost and Laplacian stability across different perturbation sectors.

Findings

Hairy black holes in certain theories are stable under linear perturbations.

Charged black holes in regularized Einstein-Gauss-Bonnet gravity exhibit instabilities.

Conditions for absence of ghost and Laplacian instabilities are derived for all perturbation modes.

Abstract

We study the linear stability of black holes in Maxwell-Horndeski theories where a $U(1)$ gauge-invariant vector field is coupled to a scalar field with the Lagrangian of full Horndeski theories. The perturbations on a static and spherically symmetric background can be decomposed into odd- and even-parity modes under the expansion of spherical harmonics with multipoles $l$. For $l \geq 2$, the odd-parity sector contains two propagating degrees of freedom associated with the gravitational and vector field perturbations. In the even-parity sector, there are three dynamical perturbations arising from the scalar field besides the gravitational and vector field perturbations. For these five propagating degrees of freedom, we derive conditions for the absence of ghost/Laplacian stabilities along the radial and angular directions. We also discuss the stability of black holes for $l=0$ and $l=1$, in which case no additional conditions are imposed to those obtained for $l \geq 2$. We apply our general results to Einstein-Maxwell-dilaton-Gauss-Bonnet theory and Einstein-Born-Infeld-dilaton gravity and show that hairy black hole solutions present in these theories can be consistent with all the linear stability conditions. In regularized four-dimensional Einstein-Gauss-Bonnet gravity with a Maxwell field, however, exact charged black hole solutions known in the literature are prone to instabilities of even-parity perturbations besides a strong coupling problem with a vanishing kinetic term of the radion mode.