Scalar hair around charged black holes in Einstein-Gauss-Bonnet gravity

Abstract

We explore charged black hole solutions in Einstein-Gauss-Bonnet gravity in five dimensions, with a charged scalar hair. We interpret such hairy black holes as the final state of the superradiant instability previously reported for this system. We explore the relation of the hairy black hole solutions with the non-backreacting quasibound states and scalar clouds, as well as with the boson star solutions. *

Diving inside a hairy black hole

“…Even if this does not seem to be the case for a minimally coupled scalar in Einstein-Maxwell theory [18], there are simple enough extensions that do present hairy solutions. That is precisely the case for charged Einstein-Gauss-Bonnet black holes in 4 + 1 dimensions [19,20], which represent a good playground to test these ideas. This is what this paper is about.…”

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

“…For the latter case, we show in the Appendix that these solutions cannot carry scalar hair. On the other hand, this is not true for the case α > 0 [63] : for sufficiently large charge q of the scalar field, the charged EGB black holes become unstable to scalar hair formation.…”

“…The equations for the functions b(r), V (r) and φ(r) are of second order, while the equation for f (r) is of first order. For the explicit form of these equations -albeit in a slightly different parametrization of the metric -we refer the reader to [63].…”

“…For these solutions, the real-valued frequency of the time-dependence has to be chosen equal to the upper bound on the superradiant frequency in analogy to the construction of hairy, rotating Kerr black holes in 4 dimensions [34,35]. The authors of [63] also studied the corresponding boson star solutions, however, did not discuss the critical phenomena of the solutions and if and how the hairy black holes are connected to the boson star solutions and/or the branches of non-hairy, charged black holes that exist in this model. The aim of this paper is to answer these questions and also provide a simple argument for the existence of an upper bound on the U (1) charge of the scalar field.…”

“…The aim of this paper is to answer these questions and also provide a simple argument for the existence of an upper bound on the U (1) charge of the scalar field. While the focus of the analysis in [63] was the determination of the domain of existence in relation to the charge of the scalar field q and the GB coupling α, we study the pattern mainly in function of the frequency of the scalar field's time-dependence, ω, and the charge Q of the black holes.…”