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 .