$D=5$ static, charged black holes, strings and rings with resonant, scalar $Q$-hair

TL;DR

This paper demonstrates a mechanism to endow various five-dimensional black objects with scalar hair by satisfying a resonance condition, extending previous four-dimensional results and producing singularity-free solutions with potential balancing effects.

Contribution

It generalizes the scalar hair mechanism to five-dimensional black objects, including black holes, rings, and strings, showing the universality of the resonance condition for scalar hair.

Findings

Scalar hair can be added to 5D black objects via a resonance condition.

Charged scalar hair can balance black rings, avoiding singularities.

The solutions exhibit features similar to 4D cases, with a Q-ball potential.

Abstract

A mechanism for circumventing the Mayo-Bekenstein no-hair theorem allows endowing four dimensional $(D=4)$ asymptotically flat, spherical, electro-vacuum black holes with a minimally coupled $U(1)$-gauged scalar field profile: $Q$-$hair$. The scalar field must be massive, self-interacting and obey a {\it resonance condition} at the threshold of (charged) superradiance. We establish generality for this mechanism by endowing three different types of static black objects with scalar hair, within a $D=5$ Einstein-Maxwell-gauged scalar field model: asymptotically flat black holes and black rings; and black strings which asymptote to a Kaluza-Klein vacuum. These $D=5$ $Q$-hairy black objects share many of the features of their $D=4$ counterparts. In particular, the scalar field is subject to a resonance condition and possesses a $Q$-ball type potential. For the static black ring, the charged scalar hair can balance it, yielding solutions that are singularity free on and outside the horizon.