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

Abstract

The fundamental resonances of rapidly rotating Kerr black holes in the eikonal limit are derived analytically. We show that there exists a critical value, μc=15−1932, for the dimensionless ratio μ≡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 μc the imaginary parts of the quasinormal frequencies scale like the black-hole temperature: ωI(n;μ>μc)=2πTBH(n+12). This implies that for perturbations modes in the interval μc<μ⩽1, the relaxation period τ∼1/ωI of the black hole becomes extremely long as the extremal limit TBH→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 MΩ⩾25, where M and Ω are the black-hole mass and angular velocity, respectively) possess this family of remarkably long-lived perturbation modes.