Nonlinear Stability of Rotating Hairy Black Holes

TL;DR

This study investigates the nonlinear stability of rotating hairy black holes with scalar fields, finding that configurations with subdominant scalar mass are stable over long timescales, while scalar-dominant ones are unstable on shorter timescales.

Contribution

The paper provides the first non-linear numerical stability analysis of rotating hairy black holes, revealing conditions under which they remain stable or become unstable.

Findings

Subdominant scalar mass configurations are stable over long timescales.

Scalar-dominant configurations develop instabilities similar to rotating boson stars.

Results imply superradiant formation of these black holes likely leads to stable configurations.

Abstract

Rotating hairy black holes (RHBHs) are axisymmetric equilibrium solutions of the Einstein--Klein--Gordon equations, consisting of a spinning black hole surrounded by a toroidal distribution of complex scalar field. Despite their potential astrophysical relevance, the stability of these configurations -- naturally expected to form through superradiant growth of light bosonic fields -- remains uncertain. In this work, we investigate the stability of RHBHs by performing fully non-linear numerical evolutions of several configurations that differ in the relative mass contribution of the scalar-field torus. We find that configurations in which the scalar field mass is subdominant compared to the black hole mass remain stable throughout the evolution, at least on timescales of order $\mu t > 1600$, where $\mu$ is the scalar field mass. These configurations might therefore be stable, with possible superradiant instabilities developing only on much longer timescales $\mu t \sim 10^{11}$, according to previous linear stability analyses. In contrast, when the scalar-field mass dominates, the system develops an instability on a much shorter timescale around $\mu t \sim \mathcal{O}(100)$, similar to the non-axisymmetric instability observed in rotating boson stars. Given the expected upper limits on scalar-field mass growth achievable through superradiance, our results suggest that rotating hairy black holes formed predominantly via this mechanism are likely to remain stable, at least up to the onset of the superradiant instability.