Quasinormal modes of asymptotically flat rotating black holes

TL;DR

This work advances the understanding of gravitational perturbations of higher-dimensional rotating black holes by computing the full QNM spectra for singly spinning and equal angular momenta MP BHs, including instability channels such as ultraspinning and bar modes. It employs the Kodama–Ishibashi formalism for Schwarzschild and a CP^N harmonic decomposition for cohomogeneity-one MP backgrounds to reduce perturbations to tractable equations, solved numerically. A central finding is the emergence of two classes of Schwarzschild QNMs in the large-$d$ limit—saturating modes with finite frequencies and scaling modes that diverge with $d$—and the demonstration that MP BH instabilities at large $d$ connect to saturating Schwarzschild modes as rotation vanishes. These results illuminate the stability landscape of high-dimensional black holes and offer a framework for exploring related spacetimes (e.g., Kerr–Newman) and large-$d$ gravity limits.

Abstract

We study the main properties of general linear perturbations of rotating black holes in asymptotically flat higher-dimensional spacetimes. In particular, we determine the quasinormal mode (QNM) spectrum of singly spinning and equal angular momenta Myers-Perry black holes (MP BHs). Emphasis is also given to the timescale of the ultraspinning and bar-mode instabilities in these two families of MP BHs. For the bar-mode instabilities in the singly spinning MP BH, we find excellent agreement with our linear analysis and the non-linear time evolution of Shibata and Yoshino for d=6,7 spacetime dimensions. We find that d=5 singly spinning BHs are linearly stable. In the context of studying general relativity in the large dimension limit, we obtain the QNM spectrum of Schwarzschild BHs and rotating MP BHs for large dimensions. We identify two classes of modes. For large dimensions, we find that in the limit of zero rotation, unstable modes of the MP BHs connect to a class of Schwarzschild QNMs that saturate to finite values.

Figures (25)

• Figure 1: Schwarzschild. The complex QNM frequencies for scalar ( left panel), vector ( middle panel), and tensor ( right panel) perturbations. In these plots, the dimension ranges from $d=6$ to $d=100$. For the scalar plot, $\widetilde{\ell}_S = 2,3,4,5,6$ modes are displayed, while for vectors $\widetilde{\ell}_V = 2,3,4$, and for tensors $\widetilde{\ell}_T = 1,2$.
• Figure 2: Schwarzschild. The real and imaginary parts of the saturating scalar QNM's for $\widetilde{\ell}_S = 2,3,4,5$. Higher $\widetilde{\ell}_S$ curves lie above lower $\ell_S$ curves.
• Figure 3: Schwarzschild. The saturating scalar QNM's in the complex $\omega$ plane for $\widetilde{\ell}_S = 2$ (top left), $\widetilde{\ell}_S = 3$ (top right), $\widetilde{\ell}_S = 4$ (bottom left), $\widetilde{\ell}_S = 5$ (bottom right). The curves begin at $d=6,6,9,15$ (large dots) for $\widetilde{\ell}_S = 2,3,4,5$, respectively, and $d$ increases along the curve, reaching $d=100$ at the other endpoint.
• Figure 4: Schwarzschild. Plot of non-saturating scalar modes for $\widetilde{\ell}_S = 2,3,4,5$. For both plots the curves appear in terms of increasing $\widetilde{\ell}_S$, from bottom to top.
• Figure 5: Schwarzschild. The imaginary part of the saturating vector QNM's for $\widetilde{\ell}_V = 2,3,4$. Higher $\widetilde{\ell}_V$ curves lie above lower $\widetilde{\ell}_V$ curves.