Nonlinear Stability of Rotating Hairy Black Holes

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. In contrast, when the scalar-field mass dominates, the system develops an instability akin to the non-axisymmetric instability observed in rotating boson stars. Given the expected limits on the scalar-field mass growth achievable through superradiance, our results suggest that rotating hairy black holes formed predominantly by this process are expected to be stable.