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Published November 15, 2012 | Published
Journal Article Open

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


There is a well-known, intuitive geometric correspondence between high-frequency quasinormal modes of Schwarzschild black holes and null geodesics that reside on the light ring (often called spherical photon orbits): the real part of the mode's frequency relates to the geodesic's orbital frequency, and the imaginary part of the frequency corresponds to the Lyapunov exponent of the orbit. For slowly rotating black holes, the quasinormal mode's real frequency is a linear combination of the orbit's precessional and orbital frequencies, but the correspondence is otherwise unchanged. In this paper, we find a relationship between the quasinormal-mode frequencies of Kerr black holes of arbitrary (astrophysical) spins and general spherical photon orbits, which is analogous to the relationship for slowly rotating holes. To derive this result, we first use the Wentzel-Kramers-Brillouin approximation to compute accurate algebraic expressions for large-l quasinormal-mode frequencies. Comparing our Wentzel-Kramers-Brillouin calculation to the leading-order, geometric-optics approximation to scalar-wave propagation in the Kerr spacetime, we then draw a correspondence between the real parts of the parameters of a quasinormal mode and the conserved quantities of spherical photon orbits. At next-to-leading order in this comparison, we relate the imaginary parts of the quasinormal-mode parameters to coefficients that modify the amplitude of the scalar wave. With this correspondence, we find a geometric interpretation of two features of the quasinormal-mode spectrum of Kerr black holes: First, for Kerr holes rotating near the maximal rate, a large number of modes have nearly zero damping; we connect this characteristic to the fact that a large number of spherical photon orbits approach the horizon in this limit. Second, for black holes of any spins, the frequencies of specific sets of modes are degenerate; we find that this feature arises when the spherical photon orbits corresponding to these modes form closed (as opposed to ergodically winding) curves.

Additional Information

© 2012 American Physical Society. Received 17 July 2012; published 1 November 2012. We thank Emanuele Berti for discussing this work with us and pointing out several references to us. We also thank Jeandrew Brink for insightful discussions about spherical photon orbits in the Kerr spacetime. We would also like to thank the anonymous referee for carefully reviewing our manuscript and offering many helpful suggestions. We base our numerical calculation of the QNM frequencies on the Mathematica notebook provided by Emanuele Berti and Vitor Cardoso [58]. This research is funded by NSF Grants No. PHY-1068881, No. PHY-1005655, CAREER Grant No. PHY-0956189; NASA Grant No. NNX09AF97G; the Sherman Fairchild Foundation, the Brinson Foundation, and the David and Barabara Groce Startup Fund at Caltech.

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