Black holes with only one Killing field

TL;DR

The paper demonstrates the existence of five-dimensional AdS black holes with scalar hair that preserve only a single Killing field, linking these objects to rotating boson stars and the endpoint of scalar and gravitational superradiant instabilities. Using a cohomogeneity-1 ansatz and matched asymptotic expansions, the authors construct both perturbative and numerical solutions, revealing a rich phase structure with nonuniqueness in the energy–angular momentum plane. They show that hairy black holes merge with MP-AdS black holes along merger lines, asymptote to boson stars as the horizon shrinks, and form a two-parameter family with intricate thermodynamics and tidal properties near extremality. The results illuminate new endpoints for superradiant instabilities in AdS and suggest broader implications for higher-dimensional gravity and holographic interpretations, while also outlining avenues for future dynamical studies.

Abstract

We present the first examples of black holes with only one Killing field. The solutions describe five dimensional AdS black holes with scalar hair. The black holes are neither stationary nor axisymmetric, but are invariant under a single Killing field which is tangent to the null generators of the horizon. Some of these solutions can be viewed as putting black holes into rotating boson stars. Others are related to the endpoint of a superradiant instability. For given mass and angular momentum (within a certain range) several black hole solutions exist.

Figures (7)

• Figure 1: Plot of angular velocity versus horizon size. MP-AdS black holes exist in the shaded regions and its boundary corresponds to extremal black holes. Curves of $\Omega_H(r_+,m)$ corresponding to the merger lines are displayed for different values of $m$. Above each line, black holes are unstable to perturbations with that $m$. Below $\Omega_H \ell =1$, black holes are stable. The inset plot zooms in the region where the $m = 1$ merger exists.
• Figure 2: The real (left panel) and imaginary (right panel) parts of the perturbation frequency as a function of angular velocity, for a constant value of the MP-AdS horizon size, $r_+/\ell = 0.1$. Different curves correspond to different values of $m$. Note that the plot on the right uses a logarithmic scale.
• Figure 3: The real (left panel) and imaginary (right panel) parts of the perturbation frequency as a function of angular velocity, for the purely gravitational superradiance instability, at fixed MP-AdS horizon size $r_+/\ell = 0.1$. The mode shown has $\ell_1 = m = 3$.
• Figure 4: Some properties of rotating boson stars. The solid curves represent the exact solution, whereas the dashed lines show the analytical estimates of section \ref{['sec:PertBstar']}.
• Figure 5: Some properties of rotating hairy black holes. Each line corresponds to a different value of $r_2/\ell\equiv r_+/\ell$. From left to right in each panel, one has $r_2/\ell = 0.05,\,0.07,\,0.09,\,0.11,\,0.13$.