An analytic representation for the quasi-normal modes of Kerr black holes

The gravitational quasi-normal frequencies of both stationary and rotating black holes are calculated by constructing exact eigensolutions to the radiative boundary-value problem of Chandrasekhar and Detweiler. The method is that employed by Jaffé in his determination of the electronic spectra of the hydrogen molecule ion in 1934, and analytic representations of the quasi-normal mode wavefunctions are presented here for the first time. Numerical solution of Jaffé’s characteristic equation indicates that for each l-pole there is an infinite number of damped Schwarzschild quasi-normal modes. The real parts of the corresponding frequencies are bounded, but the imaginary parts are not. Figures are presented that illustrate the trajectories the five least-damped of these frequencies trace in the complex frequency plane as the angular momentum of the black hole increases from zero to near the Kerr limit of maximum angular momentum per unit mass, a = M, where there is a coalescence of the more highly damped frequencies to the purely real value of the critical frequency for superradiant scattering.