Nonlinear Stability of Rotating Hairy Black Holes

TL;DR

This work investigates the nonlinear stability of rotating hairy black holes in the Einstein–Klein–Gordon system by performing fully nonlinear evolutions of equilibria with varying scalar-hair content. It demonstrates a stability threshold near $M_{\Phi}/M \approx 0.5$ for moderate spin: configurations with subdominant scalar hair remain axisymmetric and long-lived (up to $\mu t \approx 1.6\times 10^3$), while higher hair fractions trigger a non-axisymmetric instability (NAI) that drives the BH to drift and disrupt the scalar torus. The instability resembles the NAI seen in rotating boson stars, and the Noether charge remains conserved throughout, suggesting that RHBHs formed via superradiant growth are likely stable on astrophysical timescales. Overall, the findings support a stable branch of RHBHs for physically relevant hair content, with implications for gravitational-wave signals and black-hole mimicker scenarios, and point to future work to precisely map the stability boundary and to complement nonlinear results with linear perturbation analyses.

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.

Figures (7)

• Figure 1: Global properties of the RHBH configurations. Relative contribution of the horizon and of the scalar field to the mass (top) and to the angular momentum (bottom). The leftmost solution represents Kerr background with no scalar field.
• Figure 2: Dynamics of the configuration RHBH34. Snapshots of the scalar field density $|\Phi|^2$, along the meridional and the equatorial plane, for some illustrative times. At early times there is transient in the torus induced by the initial strong perturbation, which propagates back and forth until that it relaxes to a final solution slightly different from the initial one. This new solution remains axisymmetric and stable for long timescales, at least up to $\mu t \approx 1600$.
• Figure 3: Dynamics of the configuration RHBH68. Snapshots of the quantity $|\Phi|^2$ along the meridional and equatorial planes at selected illustrative times. There is an initial transient in the torus induced by the initial perturbation, which excites non axisymmetric modes. More specifically, the instability manifests as an exponential growth for azimuthal modes $m>0$ (see section \ref{['sec:stability_analysis']}) within the scalar field torus, visible only at late times (bottom row). The black hole also experiences a slight displacement from the origin, though this is not discernible at the plot’s scale.
• Figure 4: Dynamics of the RHBH configurations. (Top) Maximum of $|\Phi|^2$ for all the configurations studied here. All cases exhibit an initial transient phase induced by the strong perturbation in the initial data, followed by a slight increase in $\max \left(|\Phi|^2\right)$. For configurations with $M_{\Phi}/M \lesssim 0.5$, the system subsequently relaxes into a stationary, oscillatory state. For larger mass fractions $\max \left(|\Phi|^2\right)$ grows more significantly, and the initial transient becomes more violent. A subsequent decay followed by a rebound is then observed, signaling a strong perturbation of the scalar field torus by the black hole. (Bottom) In all cases, the global Noether charge remains constant with high-accuracy, confirming the reliability of the simulations and indicating that the scalar field is neither escaping to infinity nor being accreted by the black hole.
• Figure 5: Dynamics of the RHBH configurations. Fourier transforms of two quantities are shown: the real part of the scalar field, $\Phi_R$ (solid lines), and of its density, $|\Phi|^2$ (dashed lines). The spectrum of $\Phi_R$ exhibits a dominant peak at $\omega/\mu \approx 1$, as expected from the synchronization condition $\omega = \Omega_{\rm BH}$ (recall that $\mu \approx \Omega_{\rm BH}$). In contrast, the oscillations in $|\Phi|^2$ exhibit a much richer spectral structure. The dominant peak appears at a higher frequency, approximately ${\bar{\omega}} \simeq 2 \Omega_{\rm BH}$, which could induce analogous perturbations in the spacetime metric.