Probing the universality of synchronised hair around rotating black holes with Q-clouds

TL;DR

This paper demonstrates that Q-clouds are a versatile and efficient tool for establishing the existence of black holes with synchronised hair across various dimensions and topologies, without solving complex non-linear equations.

Contribution

It introduces a perturbative argument for the generality of synchronisation and applies Q-clouds to prove the existence of hairy black holes in multiple spacetime dimensions and topologies.

Findings

Q-clouds confirm the existence of synchronised hair around diverse black holes.

Synchronisation condition is shown to be broadly applicable.

Black rings can support synchronised hair, indicating topological robustness.

Abstract

Recently, various families of black holes (BHs) with synchronised hair have been constructed. These are rotating BHs surrounded, as fully non-linear solutions of the appropriate Einstein-matter model, by a non-trivial bosonic field in synchronised rotation with the BH horizon. Some families bifurcate globally from a bald BH ($e.g.$ the Kerr BH), whereas others bifurcate only locally from a bald BH ($e.g.$ the $D=5$ Myers-Perry BH). It would be desirable to understand how generically synchronisation allows hairy BHs to bifurcate from bald ones. However, the construction and scanning of the domain of existence of the former families of BHs can be a difficult and time consuming (numerical) task. Here, we first provide a simple perturbative argument to understand the generality of the synchronisation condition. Then, we observe that the study of Q-clouds is a generic tool to establish the existence of BHs with synchronised hair bifurcating (globally or locally) from a given bald BH without having to solve the fully non-linear coupled system of Einstein-matter equations. As examples, we apply this tool to establish the existence of synchronised hair around $D=6$ Myers-Perry BHs, $D=5$ black rings and $D=4$ Kerr-AdS BHs, where $D$ is the spacetime dimension. The black rings case provides an example of BHs with synchronised hair beyond spherical horizon topology, further establishing the generality of the mechanism.