A rotating hairy AdS3 black hole with the metric having only one Killing vector field

TL;DR

This paper constructs a perturbative rotating AdS3 black hole with scalar hair, showing it can be a superradiant instability endpoint, with only one Killing vector and higher entropy than BTZ black holes.

Contribution

It introduces a new rotating hairy black hole solution with a single Killing vector, expanding understanding of superradiant instabilities in AdS3.

Findings

Rotating BTZ black holes are unstable under superradiant modes with general boundary conditions.

Constructed a perturbative rotating hairy black hole with scalar hair up to fourth order.

The hairy black hole has higher entropy than the corresponding BTZ black hole, suggesting it as an instability endpoint.

Abstract

We perturbatively construct a three-dimensional rotating AdS black hole with a real scalar hair. We choose the mass of a scalar field slightly above the Breitenlohner-Freedman bound and impose a more general boundary condition for the bulk scalar field at AdS infinity. We first show that rotating BTZ black holes are unstable against superradiant modes under our more general boundary condition. Next we construct a rotating hairy black hole perturbatively with respect to a small amplitude $\epsilon$ of the scalar field, up to $O(\epsilon^4)$. The lumps of non-linearly perturbed geometry admit only one Killing vector field and co-rotate with the black hole, and it shows no dissipation. We numerically show that the entropy of our hairy black hole is larger than that of the BTZ black hole with the same energy and the angular momentum. This indicates, at least in the perturbative level, that our rotating hairy black hole in lumpy geometry can be the endpoint of the superradiant instability.