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It is well established that the response of a black hole to a genericperturbation is characterized by a spectrum of damped resonances, calledquasinormal modes; and that, in the limit of large angular momentum ($l \gg1$), the quasinormal mode frequency spectrum is related to the properties ofunstable null orbits. In this paper we develop an expansion method to explorethe link. We obtain new closed-form approximations for the lightly-damped partof 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 theresonance damping to the Lyapunov exponent, of the relevant null orbit. We gosomewhat further than previous studies to establish (i) a spin-dependentcorrection 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 theapproach by testing the closed-form approximations against frequencies obtainednumerically with Leaver's method.

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