Nonlinear Stability of Rotating Hairy Black Holes

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.