Stability of Gauss-Bonnet black holes in Anti-de-Sitter space-time against scalar field condensation

TL;DR

This paper investigates the stability of hyperbolic Gauss-Bonnet black holes in AdS space against scalar field condensation, revealing how Gauss-Bonnet coupling influences the emergence of scalar hair and the existence of hairy black hole solutions.

Contribution

It provides a numerical analysis of scalar hair formation on Gauss-Bonnet black holes and constructs explicit solutions with different scalar field node numbers, extending understanding in Einstein-Gauss-Bonnet gravity.

Findings

Scalar hair forms near extremality within a specific mass range.

Solutions with different scalar field nodes exist and persist in Einstein gravity.

The mass interval for scalar condensation decreases with higher Gauss-Bonnet coupling and node number.

Abstract

We study the stability of static, hyperbolic Gauss-Bonnet black holes in (4+1)-dimensional Anti-de-Sitter (AdS) space-time against the formation of scalar hair. Close to extremality the black holes possess a near-horizon topology of AdS_2 x H^3 such that within a certain range of the scalar field mass one would expect that they become unstable to the condensation of an uncharged scalar field. We confirm this numerically and observe that there exists a family of hairy black hole solutions labelled by the number of nodes of the scalar field function. We construct explicit examples of solutions with a scalar field that possesses zero nodes, one node and two nodes, respectively, and show that the solutions with nodes persist in the limit of Einstein gravity, i.e. for vanishing Gauss-Bonnet coupling. We observe that the interval of the mass for which scalar field condensation appears decreases with increasing Gauss-Bonnet coupling and/or with increasing node number.