Instability of rotating black holes: large D analysis

TL;DR

This work analyzes the stability of odd-dimensional Myers–Perry black holes with equal angular momenta using a $1/D$ expansion, revealing a bar-mode instability at finite rotation and an axisymmetric instability at higher rotation. The authors develop a decoupled perturbation framework based on a near-horizon boost that maps to Schwarzschild modes at leading order, then compute quasinormal frequencies to next-to-next-to-leading order, confirming and extending numerical results. The analysis also extends to rotating AdS black holes, showing the same qualitative instability structure with rotation bounds modified by the AdS scale and uncovering a regime of universal high-frequency modes obtained via boosted Schwarzschild spectra. Overall, the results provide analytic control over dynamical instabilities in high dimensions and connect rotating black-hole spectra across flat and AdS spaces, with good agreement to existing numerical studies.

Abstract

We study the stability of odd-dimensional rotating black holes with equal angular momenta by performing an expansion in the inverse of the number of dimensions D. Universality at large $D$ allows us to calculate analytically the complex frequency of quasinormal modes to next-to-leading order in the expansion. We identify the onset of non-axisymmetric, bar-mode instabilities at a specific finite rotation, and axisymmetric instabilities at larger rotation. The former occur at the threshold where the modes become superradiant, and before the ultraspinning regime is reached. Our results fully confirm the picture found in numerical studies, with very good quantitative agreement. We extend the analysis to the same class of black holes in Anti-deSitter space, and find the same qualitative features. We also discuss the appearance at high frequencies of the universal set of (stable) quasinormal modes.

Figures (7)

• Figure 1: Comparison between analytical and numerical calculations of the real and imaginary frequencies of the dominant unstable quasinormal mode with $\ell=m=2$ for $N=6$ ($D=15$). Black lines: analytical result to next-to-leading order in $1/N$. Gray lines: numerical results of Hartnett:2013fba for $N=6$. Dashed lines: analytical result to leading order.
• Figure 2: Real and imaginary frequencies of quasinormal modes $(\ell,m)=(2,2)$ (in units $r_+=1$). Here and in the next plots the thick black line is the mode $\omega_{\ell,m}^{(3)}$ that becomes unstable, the thin gray line is $\omega_{\ell,m}^{(1)}$ in \ref{['qnm1']}, and the discontinuous lines are the other quasinormal modes from \ref{['QNMeq']}. For $\ell=2$ the instability occurs at $a_c=1/\sqrt{2}$. Also at that value the black line $\text{Re}\,\omega$ cuts the gray line $\omega=am$ that marks the superradiant bound. Higher $\ell=m$ modes show the same qualitative behavior. Note that $\text{Re}\,\omega_{m,m}^{(3)}\neq 0$ at $a=0$. The large $N$ expansion breaks down near the extremal limit, so the results very close to $a=1$ become less reliable.
• Figure 3: Real and imaginary frequencies of quasinormal modes $(\ell,m)=(4,2)$. The instability sets in at $a_c=\sqrt{3}/2$, where the unstable mode is also at the superradiant threshold $\text{Re}\,\omega=a m$. At $a=0$ this mode becomes purely imaginary.
• Figure 4: Real and imaginary frequencies of quasinormal modes $(\ell,m)=(4,0)$. The thick solid line is the purely imaginary mode $\omega^{(0)}_{\ell,0}$ that becomes unstable at $a_c=\sqrt{3}/2$. The gray-line mode $\omega_{\ell,0}^{(1)}$ is also purely imaginary, but stable. The dashed line is $\omega_{\ell,0}^{(+)}$, and we omit $\omega_{\ell,0}^{(-)}=-\left(\omega_{\ell,0}^{(+)}\right)^*$.
• Figure 5: Same as fig. \ref{['fig:lm2']}, now including the first $1/N$ corrections with $N=6$.