The scalar perturbation of the higher-dimensional rotating black holes

TL;DR

This work analyzes massless scalar perturbations of higher-dimensional rotating black holes (Myers-Perry with a single rotation axis). It demonstrates separability of the scalar wave equation in arbitrary dimensions and computes the five-dimensional quasi-normal-mode spectrum using Leaver's continued fraction method, probing potential instabilities. Detweiler–Ipser-type bounds are derived, and numerical searches up to $|m|,j\le5$ reveal no unstable modes, supporting stability of the scalar field in these spacetimes. In the rapidly rotating limit, the perturbation time scale scales as $t \sim M^{1/2}M_{\rm Pl}^{-3/2}$ rather than the horizon light-crossing time, highlighting distinctive dynamical behavior of higher-dimensional black holes and informing their phenomenology.

Abstract

The massless scalar field in the higher-dimensional Kerr black hole (Myers- Perry solution with a single rotation axis) has been investigated. It has been shown that the field equation is separable in arbitrary dimensions. The quasi-normal modes of the scalar field have been searched in five dimensions using the continued fraction method. The numerical result shows the evidence for the stability of the scalar perturbation of the five-dimensional Kerr black holes. The time scale of the resonant oscillation in the rapidly rotating black hole, in which case the horizon radius becomes small, is characterized by (black hole mass)^{1/2}(Planck mass)^{-3/2} rather than the light-crossing time of the horizon.

Figures (7)

• Figure 1: For non-rotating black hole, first eight quasi-normal frequencies for $2\ell+j+|m|=0,\cdots,10$ are plotted in the complex $\omega_*$ plane.
• Figure 2: The first six quasi-normal modes of Kerr black hole for $j=0$, $m=\pm1$, $\ell=1$ are plotted in the $\sqrt{1+a_*^2}\omega_*$ plane. The bullets are the quasi-normal modes of Schwarzschild black hole ($a_*=0$). The right part starting from the Schwarzschild mode is the branch of the $m=+1$ modes. The left part is the branch of the $m=-1$ modes. The crosses are the modes with integer values of $a_*$.
• Figure 3: The first six quasi-normal modes of Kerr black hole for $j=1$, $m=\pm1$, $\ell=1$ are plotted in the $\sqrt{1+a_*^2}\omega_*$ plane. The bullets are the quasi-normal modes of Schwarzschild black hole ($a_*=0$). The right part starting from the Schwarzschild mode is the branch of the $m=+1$ modes. The left part is the branch of the $m=-1$ modes. The crosses are the modes with integer values of $a_*$.
• Figure 4: The first six quasi-normal modes of Kerr black hole for $j=2$, $m=\pm1$, $\ell=1$ are plotted in the $\sqrt{1+a_*^2}\omega_*$ plane. The bullets are the quasi-normal modes of Schwarzschild black hole ($a_*=0$). The right part starting from the Schwarzschild mode is the branch of the $m=+1$ modes. The left part is the branch of the $m=-1$ modes. The crosses are the modes with integer values of $a_*$.
• Figure 5: The first six quasi-normal modes of Kerr black hole for $j=0$, $m=\pm2$, $\ell=1$ are plotted in the $\sqrt{1+a_*^2}\omega_*$ plane. The bullets are the quasi-normal modes of Schwarzschild black hole ($a_*=0$). The right part starting from the Schwarzschild mode is the branch of the $m=+2$ modes. The left part is the branch of the $m=-2$ modes. The crosses are the modes with integer values of $a_*$.