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, ${\mu }_c=\sqrt{\frac{15-\sqrt{193}}{2}}$, for the dimensionless ratio $\mu \equiv m\mathrm{/}l$ 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: ${\omega }_I(n;\mu >{\mu }_c)=2\pi T_{BH}(n+\frac{1}{2})$. This implies that for perturbations modes in the interval ${\mu }_c<\mu \le 1$, the relaxation period $\tau \sim 1\mathrm{/}{\omega }_I$ 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 $M\mathrm{\Omega }\ge \frac{2}{5}$, where $M$ and $\mathrm{\Omega }$ are the black-hole mass and angular velocity, respectively) possess this family of remarkably long-lived perturbation modes.