Resonance spectrum of near-extremal Kerr black holes in the eikonal limit

The fundamental resonances of rapidly rotating Kerr black holes in the eikonal limit are derived analytically. We show that there exists a critical value, mu(c) = root 15-root 193/2, for the dimensionless ratio mu m/l between the azimuthal harmonic index m and the spheroidal harmonic index l of the perturbation mode, above which the perturbations become long lived. In particular, it is proved that above mu(c) the imaginary parts of the quasinormal frequencies scale like the black-hole temperature: omega(1)(n: mu > mu(c)) = 2 pi T-BH(n + 1/2). This implies that for perturbations modes in the interval mu(c) < mu <= 1. the relaxation period tau similar to 1/omega(l), of the black hole becomes extremely long as the extremal limit T-BH -> 0 is approached. A generalization of the results to the case of scalar quasinormal resonances of near-extremal Kerr-Newman black holes is also provided. In particular, we prove that only black holes that rotate fast enough (with M Omega >= 2/5, where M and Omega are the black-hole mass and angular velocity, respectively) possess this family of remarkably long-lived perturbation modes. (C) 2012 Elsevier B.V. All rights reserved.