Gravitational Perturbations of Higher Dimensional Rotating Black Holes: Tensor Perturbations

TL;DR

The study demonstrates that odd-dimensional Myers-Perry black holes with equal angular momenta are perturbatively stable in asymptotically flat spacetimes, while their AdS counterparts exhibit a superradiant instability once the horizon angular velocity surpasses the speed of light at the boundary. A decoupled tensor perturbation sector reduces to a single radial equation, allowing both scalar-like and gravitational perturbations to be treated within a unified framework using CP^N harmonics. Through analytic (WKB) and numerical analyses, the authors map the onset of instability in AdS black holes, derive scaling relations for large angular quantum numbers, and discuss potential end-states involving stationary nonaxisymmetric black holes. The work advances understanding of stability in higher-dimensional rotating black holes and clarifies conditions under which AdS instabilities arise and how they might terminate. It also lays groundwork for exploring endpoint solutions and bifurcations in the landscape of higher-dimensional gravity.

Abstract

Assessing the stability of higher-dimensional rotating black holes requires a study of linearized gravitational perturbations around such backgrounds. We study perturbations of Myers-Perry black holes with equal angular momenta in an odd number of dimensions (greater than five), allowing for a cosmological constant. We find a class of perturbations for which the equations of motion reduce to a single radial equation. In the asymptotically flat case we find no evidence of any instability. In the asymptotically anti-de Sitter case, we demonstrate the existence of a superradiant instability that sets in precisely when the angular velocity of the black hole exceeds the speed of light from the point of view of the conformal boundary. We suggest that the endpoint of the instability may be a stationary, nonaxisymmetric black hole.

Figures (5)

• Figure 1: Plot of solution $\Psi(r)$ against $r/\ell$ for doubly transverse tensor perturbations with $N=2$, $l=6$, $m=8$, $r_+/\ell=0.7$, $\Omega_H\ell = 1.46242$.
• Figure 2: Plot of $\Omega_H\ell$ against $r_+/\ell$ for doubly transverse tensor perturbations with $N=2$, $\epsilon=1$. No black holes exist in the empty region bounded by the curve in the top right of the diagram. Curves $\Omega_H(r_+,l,m)$ corresponding to existence of a normal mode with frequency $\omega = m \Omega_H$ are displayed for different values of $l,m$. From top to bottom, $(l,m) = (5,1),(4,2),(3,3),(2,4),(3,5),(4,6),(5,7),(6,8)$.
• Figure 3: Plots of critical value of $\Omega_H$ for doubly transverse tensor perturbations for $N=2$, $\epsilon=1$, $(l,m)=(38,40)$ (top) and $(78,80)$. No black holes exist in the empty region bounded by the curve in the top right of the diagram. The fact that the $\Omega_H$ curves do not quite meet this curve is due to the limitations of our numerical method. Note the scale of the vertical axis.
• Figure 4: Plots of $Z$ againt $\omega/(m\Omega_H)$ for (from bottom to top) $\Omega_H/\Omega_{\rm max} = 0.5,0.7,0.9,0.99,0.999$ with $\epsilon=1,l=2,m=4$. Note that $Z=1$ for $\omega=m\Omega_H$ but this point has been deleted from the topmost two curves to make the figure clearer.
• Figure 5: Plot of $Z$ against $l=2,3,\ldots$ for $m=m_{\rm max} = l+2$, $\epsilon=1$, $\Omega_H/\Omega_{\rm max}=0.99$, $\omega/(m\Omega_H) = 0.99$.