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

Abstract

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.

Caustic echoes from a Schwarzschild black hole

“…There is no photon sphere in Kerr spacetime; instead, there are spherical photon orbits bounded by the location of proand retrograde circular photon orbits [37] which may participate in the Cauchy evolution in a similar way as does the photon sphere in Schwarzschild spacetime. The large limit of the quasinormal mode spectrum and its relation to spherical photon orbits in Kerr spacetime have recently been analyzed in [38]. Such analytic knowledge can be combined with numerical experiments to reveal the structure of the Green function in Kerr spacetimes and to provide good approximations to it.…”

Post-Kerr black hole spectroscopy

“…This connection between null geodesics and QNMs has been explored in depth for Kerr black holes [22][23][24][25][26]. Our goal here is to extend this connection beyond the Kerr spacetime, and to turn it into a practical scheme to test experimentally whether a set of QNM frequencies (such as those potentially observable by LIGO) is consistent with the dynamics of the Kerr spacetime.…”

Magnetosphere of a Kerr black hole immersed in magnetized plasma and its perturbative mode structure

“…Similar WKB treatments for uncoupled waves can be found in Refs. [43,[66][67][68], and readers interested in further details can consult these references. Note that the phase matching between α m and β m follows from the assumption that we are solving for a single eigen-mode, in which case phase coherence should be preserved during propagation.…”