Linear perturbations of Einstein-Gauss-Bonnet black holes

TL;DR

This paper analyzes linear perturbations of non-rotating black holes in scalar-tensor theories, revealing well-behaved perturbations in some models and pathological behaviors in others, highlighting differences from general relativity.

Contribution

It provides a detailed study of perturbations in Einstein-Gauss-Bonnet black holes within scalar-tensor theories, including new methods for analyzing asymptotic behaviors.

Findings

Einstein-scalar-Gauss-Bonnet black holes have well-behaved perturbations.

4D Einstein-Gauss-Bonnet black holes exhibit pathological perturbation behavior.

Effective Schrödinger-like equations are derived for axial perturbations.

Abstract

We study linear perturbations about non rotating black hole solutions in scalar-tensor theories, more specifically Horndeski theories. We consider two particular theories that admit known hairy black hole solutions. The first one, Einstein-scalar-Gauss-Bonnet theory, contains a Gauss-Bonnet term coupled to a scalar field, and its black hole solution is given as a perturbative expansion in a small parameter that measures the deviation from general relativity. The second one, known as 4-dimensional-Einstein-Gauss-Bonnet theory, can be seen as a compactification of higher-dimensional Lovelock theories and admits an exact black hole solution. We study both axial and polar perturbations about these solutions and write their equations of motion as a first-order (radial) system of differential equations, which enables us to study the asymptotic behaviours of the perturbations at infinity and at the horizon following an algorithm we developed recently. For the axial perturbations, we also obtain effective Schr\"odinger-like equations with explicit expressions for the potentials and the propagation speeds. We see that while the Einstein-scalar-Gauss-Bonnet solution has well-behaved perturbations, the solution of the 4-dimensional-Einstein-Gauss-Bonnet theory exhibits unusual asymptotic behaviour of its perturbations near its horizon and at infinity, which makes the definition of ingoing and outgoing modes impossible. This indicates that the dynamics of these perturbations strongly differs from the general relativity case and seems pathological.