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There is a well-known, intuitive geometric correspondence betweenhigh-frequency QNMs of Schwarzschild black holes and null geodesics that resideon the light-ring : the real part of the mode's frequency relates to thegeodesic's orbital frequency, and the imaginary part of the frequencycorresponds to the Lyapunov exponent of the orbit. For slowly rotating blackholes, the QNM real frequency is a linear combination of a the orbit'sprecessional and orbital frequencies, but the correspondence is otherwiseunchanged. In this paper, we find a relationship between the QNM frequencies ofKerr black holes of arbitrary (astrophysical) spins and general sphericalphoton orbits, which is analogous to the relationship for slowly rotatingholes. To derive this result, we first use the WKB approximation to computeaccurate algebraic expressions for large-l QNM frequencies. Comparing our WKBcalculation to the leading-order, geometric-optics approximation to scalar-wavepropagation in the Kerr spacetime, we then draw a correspondence between thereal parts of the parameters of a QNM and the conserved quantities of sphericalphoton orbits. At next-to-leading order in this comparison, we relate theimaginary parts of the QNM parameters to coefficients that modify the amplitudeof the scalar wave. With this correspondence, we find a geometricinterpretation to two features of the QNM spectrum of Kerr black holes: First,for Kerr holes rotating near the maximal rate, a large number of modes havenearly zero damping; we connect this characteristic to the fact that a largenumber of spherical photon orbits approach the horizon in this limit. Second,for black holes of any spins, the frequencies of specific sets of modes aredegenerate; we find that this feature arises when the spherical photon orbitscorresponding to these modes form closed (as opposed to ergodically winding)curves.

Quasinormal-mode spectrum of Kerr black holes and its geometric interpretation