Gravitational stability of simply rotating Myers-Perry black holes: tensorial perturbations

TL;DR

This work analyzes the stability of higher-dimensional simply rotating Myers-Perry black holes against tensor-type gravitational perturbations in asymptotically flat spacetimes (D≥7). It solves the angular and radial perturbation equations using an angular continued-fraction method for $\mu(\omega)$ and Leaver/Frobenius plus JWKB techniques for the radial spectrum, scanning a wide range of dimensions, multipoles, and rotation. The authors find no growing quasinormal modes (Im $\omega$ < 0) across the explored parameter space, with negative-m $\omega$ modes becoming increasingly damped as $|m|$ grows, supporting stability. They also provide extensive QNM data and discuss implications for higher-dimensional gravity and potential bulk graviton emission, along with notes on the AdS limit and future work.

Abstract

We study the stability of $D \geq 7$ asymptotically flat black holes rotating in a single two-plane against tensor-type gravitational perturbations. The extensive search of quasinormal modes for these black holes did not indicate any presence of growing modes, implying the stability of simply rotating Myers-Perry black holes against tensor-type perturbations.

Figures (10)

• Figure 1: Quasinormal modes as a function of $a$ obtained by the Frobenius method for $D=7$, $l=2$, $j=0$: $m=4$ (blue), $m=3$ (red), $m=2$ (yellow), $m=1$ (green), $m=0$ (light blue). Higher values of $m$ correspond to larger real part of $\omega$.
• Figure 2: Quasinormal modes obtained by the Frobenius method for $D=7$, $l=2$, $j=0$: $m=-1$ (blue), $m=-2$, $m=-3$, $m=-4$, $m=-5$ (light blue), $m=-10$ (green), $m=-15$ (red), $m=-20$ (magenta), $m=-25$ (brown). Higher negative values of $m$ correspond to the larger real and imaginary part of $\omega$. The imaginary part of the QN frequency stays negative, implying stability against perturbations with the high negative azimuthal number $m$.
• Figure 3: Maximum value of the imaginary part of the quasi-normal mode for negative values of $m$ together with the found fit $max(Im(\omega)\propto (-m)^{-\alpha}~(\alpha>0)$ for $D=7$, $l=2$, $j=0$ (left figure, log-log scale) and corresponding values of the rotation parameter $a_{max}$ as a function of $-m$ (right figure).
• Figure 4: Quasinormal modes obtained by the Frobenius method for $D=10$, $l=2$, $j=0$: $m=5$ (blue), $m=4$ (red), $m=3$ (yellow), $m=2$ (green), $m=1$ (light blue). Higher values of $l$ correspond to larger real part of $\omega$.
• Figure 5: Quasinormal modes obtained by the Frobenius method for $D=7$: ($l=2$, $m=0$, $j=0$ - black), ($l=2$, $m=0$, $j=1$ - yellow), ($l=2$, $m=0$, $j=2$ - orange), ($l=2$, $m=1$, $j=0$ - purple), ($l=2$, $m=1$, $j=1$ - blue), ($l=2$, $m=1$, $j=2$ - magenta), ($l=3$, $m=1$, $j=0$ - green), ($l=4$, $m=1$, $j=0$ - cyan), ($l=5$, $m=1$, $j=0$ - red). The larger values of $j$ correspond to the smaller values of the imaginary part of $\omega$. The real part increases with $l+m+2j$ for small rotation. When $a$ is large, the real part for the multipole index $j\neq0$ approaches zero, while that for $j=0$ approaches a constant that increases with $l$.