Eikonal quasinormal modes of highly-spinning black holes in higher-curvature gravity: a window into extremality

TL;DR

This work computes gravitational QNMs of highly spinning black holes in a quartic-curvature EFT that preserves eikonal isospectrality and connects to string theory. By deriving a master equation for large-momentum perturbations and employing WKB methods, the authors obtain analytic corrections to Kerr QNMs for arbitrary rotation and angular-momentum ratio $\mu=m/(\ell+1/2)$, revealing dramatic near-extremal amplification, especially near the critical $\mu_{\rm cr}\approx0.744$. They develop a geometric-optics (graviton-sphere) perspective linking equatorial orbits to $\ell=m$ QNMs and show a modified relation between the Lyapunov exponent and the imaginary part. An alternative approach directly solves the effective scalar equation and yields a corrected Teukolsky potential, enabling a WKB treatment for arbitrary rotation. The findings imply strong, observable deviations from GR in near-extremal, highly-rotating black holes, offering a new window to test high-energy corrections via black-hole spectroscopy within the EFT’s regime of validity.

Abstract

We carry out the first computation of gravitational quasinormal modes of black holes with arbitrary rotation in a theory with higher-derivative corrections. Our analysis focuses on a recently identified quartic-curvature theory that preserves the isospectrality of quasinormal modes in the eikonal limit and that is connected to string theory. We find a master equation that governs large-momentum gravitational perturbations in this theory. By solving this equation with WKB methods, we provide complete results for the corrections to the Kerr quasinormal mode frequencies for arbitrary spin and arbitrary $μ=m/(\ell+1/2)$, where $\ell$ and $m$ are the harmonic numbers. Our results show that the corrections become orders of magnitude larger when the spin is close to extremality, with the modes close to the critical value of $μ$ that separates damped and zero-damped modes being particularly sensitive. We also perform a geometric-optics analysis of gravitational-wave propagation around black holes and relate the equatorial ``graviton-sphere'' orbits to quasinormal mode frequencies with $\ell=m$. We find that the usual correspondence between the Lyapunov exponent of those orbits and the imaginary part of the frequency is modified.

Figures (7)

• Figure 1: Kerr QNM frequencies with $\ell=m$ (top row) and their corrections (bottom row) as a function of the spin parameter $\chi$ ranging from zero rotation up to extremality. In the case of the imaginary part we show the fundamental mode $n=0$.
• Figure 2: Potential as a function of the tortoise coordinate. Left: full potential for $\hat{\alpha}=0$ (GR) and for $\hat{\alpha}=0.005$. We plot $\omega^2-U$, which vanishes at infinity and represents the intuitive definition of a potential. The corrections introduce in this case a local minimum, but it disappears for smaller values of $\hat{\alpha}$. Right: correction to the potential, which exhibits a minimum quite close to the horizon. The plots correspond to $\chi=0.99$, $\mu=1/4$, and $\omega=\omega_{R}^{\rm Kerr}$.
• Figure 3: Corrections to the Kerr QNM frequencies for $(\ell,m,n)=(4,0,0), (4,2,0)$. The colored dotted and dashed lines are the prediction from the modified Teukolsky equation, and they show a mild isospectrality breaking of the two families of modes $\delta\omega^{+}$ and $\delta\omega^{-}$. The solid black lines are the prediction from our eikonal computation.
• Figure 4: Convergence towards the eikonal regime for $\ell=m$ modes. The colored dotted and dashed lines are the prediction from the modified Teukolsky equation for $(\ell,m,n)=(2,2,0), (3,3,0), (4,4,0), (5,5,0)$. The solid black lines are the prediction from our eikonal computation.
• Figure 5: Kerr QNM frequencies (top row) and their corrections (middle and bottom row) as a function of $\mu=m/L$ for different values of the black hole rotation $\chi$. In the bottom row we show the corrections at extremality. We normalize the frequencies by their scaling with $L$, and the imaginary part corresponds to the fundamental mode $n=0$.