The Quasinormal Mode Spectrum of a Kerr Black Hole in the Eikonal Limit

TL;DR

This paper develops an expansion method to analytically approximate the quasinormal mode spectrum of Kerr black holes in the large angular momentum limit, linking frequencies to null orbit properties and validating with numerical results.

Contribution

It introduces new closed-form approximations for the quasinormal mode spectrum of Kerr black holes in the eikonal limit, including spin-dependent and polar mode corrections.

Findings

Derived explicit formulas relating mode frequencies to null orbit parameters.

Confirmed the approximations match numerical calculations with high accuracy.

Extended previous models to include spin effects and polar modes.

Abstract

It is well established that the response of a black hole to a generic perturbation is characterized by a spectrum of damped resonances, called quasinormal modes; and that, in the limit of large angular momentum ($l \gg 1$), the quasinormal mode frequency spectrum is related to the properties of unstable null orbits. In this paper we develop an expansion method to explore the link. We obtain new closed-form approximations for the lightly-damped part of the spectrum in the large-$l$ regime. We confirm that, at leading order in $l$, the resonance frequency is linked to the orbital frequency, and the resonance damping to the Lyapunov exponent, of the relevant null orbit. We go somewhat further than previous studies to establish (i) a spin-dependent correction to the frequency at order $1 / l$ for equatorial ($m = \pm l$) modes, and (ii) a new result for polar modes ($m = 0$). We validate the approach by testing the closed-form approximations against frequencies obtained numerically with Leaver's method.