Critical phenomena of charged Einstein-Gauss-Bonnet black holes with charged scalar hair

TL;DR

This paper explores the critical phenomena and phase structure of charged Einstein-Gauss-Bonnet black holes with charged scalar hair, revealing the conditions for their existence and establishing a no-hair theorem in higher dimensions.

Contribution

It demonstrates the critical scalar charge condition for charged scalar hair and proves a no-hair theorem, highlighting the role of Gauss-Bonnet terms in black hole solutions.

Findings

Charged scalar hair exists under specific charge conditions.

The hairy black hole branch connects to charged EGB black holes but not to boson stars.

Gauss-Bonnet term is essential for scalar hair existence.

Abstract

Einstein-Gauss-Bonnet-gravity (EGB) coupled minimally to a $U(1)$ gauged, massive scalar field possesses -- for appropriate choices of the $U(1)$ charge -- black hole solutions that carry charged scalar hair if the frequency of the harmonic time-dependence of the scalar field is equal to the upper bound on the superradiant frequency. The existence of these solutions has first been discussed in \cite{Grandi:2017zgz}. In this paper, we demonstrate that the critical value of the scalar charge results from the requirement of non-extremality of the charged black hole solutions and the fact that the scalar field should not escape to infinity. Moreover, we investigate the hairy black holes in more detail and demonstrate that the branch of these solutions joins the branch of the corresponding charged EGB black hole for vanishing scalar field, but is {\it not} connected to the branch of boson stars in the limit of vanishing horizon radius. This indicates that it is unlikely that these black holes appear from the collapse of the corresponding boson stars. Finally, we prove a No-hair theorem for charged scalar fields with harmonic time-dependence for static, spherically symmetric, asymptotically flat electro-vacuum black holes in $d$ space-time dimensions and hence demonstrate that the GB term is crucial for the existence of the hairy black holes discussed in this paper.