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

Abstract

There is a well-known, intuitive geometric correspondence between high-frequency QNMs of Schwarzschild black holes and null geodesics that reside on the light-ring : 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 QNM real frequency is a linear combination of a the orbit's precessional and orbital frequencies, but the correspondence is otherwise unchanged. In this paper, we find a relationship between the QNM 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 WKB approximation to compute accurate algebraic expressions for large-l QNM frequencies. Comparing our WKB 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 QNM and the conserved quantities of spherical photon orbits. At next-to-leading order in this comparison, we relate the imaginary parts of the QNM parameters to coefficients that modify the amplitude of the scalar wave. With this correspondence, we find a geometric interpretation 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 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.

22 pages, 16 figures

References in corpus (5)

• Inspiral, merger and ring-down of equal-mass black-hole binaries
• Inspiral, merger and ringdown of unequal mass black hole binaries: a multipolar analysis
• Quasi-Normal Modes and Gravitational Wave Astronomy
• Slow relaxation of rapidly rotating black holes
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• Rotating black holes can have short bristles
• The Quasinormal Modes of Weakly Charged Kerr-Newman Spacetimes
• Precession during merger 1: Strong polarization changes are observationally accessible features of strong-field gravity during binary black hole merger
• Stability of Force-Free Magnetospheres
• Magnetosphere of a Kerr black hole immersed in magnetized plasma and its perturbative mode structure
• Wavefront twisting by rotating black holes: orbital angular momentum generation and phase coherent detection