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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, $μ_c=\sqrt{{15-\sqrt{193}\over{2}}}$, for the dimensionless ratio $μ\equiv m/l$ 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 $μ_c$ the imaginary parts of the quasinormal frequencies scale like the black-hole temperature: $ω_I(n;μ>μ_c)=2πT_{BH}(n+{1\over 2})$. This implies that for perturbations modes in the interval $μ_c<μ\leq 1$, the relaxation period $τ\sim 1/ω_I$ of the black hole becomes extremely long as the extremal limit $T_{BH}\to 0$ 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 $MΩ\geq {2\over 5}$, where $M$ and $Ω$ are the black-hole mass and angular velocity, respectively) possess this family of remarkably long-lived perturbation modes.