Small Hairy Black Holes in Global AdS Spacetime

TL;DR

This paper investigates small charged black holes in global AdS spacetime, demonstrating that superradiant instabilities lead to stable hairy black holes with scalar condensates, and maps their phase transitions.

Contribution

It constructs analytically the end state of superradiant instability as hairy black holes in AdS, revealing a new phase structure and phase transition in the system.

Findings

Hairy black holes form as end points of superradiant instability.

Existence of a phase transition between RNAdS and hairy black holes.

Bounded mass-charge relationship for hairy black holes.

Abstract

We study small charged black holes in global AdS spacetime in the presence of a charged massless minimally coupled scalar field. In a certain parameter range these black holes suffer from well known superradiant instabilities. We demonstrate that the end point of the resultant tachyon condensation process is a hairy black hole which we construct analytically in a perturbative expansion in the black hole radius. At leading order our solution is a small undeformed RNAdS black hole immersed into a charged scalar condensate that fills the AdS `box'. These hairy black hole solutions appear in a two parameter family labelled by their mass and charge. Their mass is bounded from below by a function of their charge; at the lower bound a hairy black hole reduces to a regular horizon free soliton which can also be thought of as a nonlinear Bose condensate. We compute the microcanonical phase diagram of our system at small mass, and demonstrate that it exhibits a second order `phase transition' between the RNAdS black hole and the hairy black hole phases.