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, $\mu_c=\sqrt{{{15-\sqrt{193}}\over{2}}}$, for the dimensionless ratio $\mu\equiv m/l$ 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 $\mu_c$ the imaginary parts of the quasinormal frequencies scale like the black-hole temperature: $\omega_I(n;\mu>\mu_c)=2\pi T_{BH}(n+{1\over 2})$. This implies that for perturbations modes in the interval $\mu_c<\mu\leq 1$, the relaxation period $\tau\sim 1/\omega_I$ 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 $M\Omega\geq {2\over 5}$, where $M$ and $\Omega$ are the black-hole mass and angular velocity, respectively) possess this family of remarkably long-lived perturbation modes.