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Quasinormal mode frequencies of Kerr black holes from Regge trajectories

A large portion of the studies concerning the quasinormal mode frequencies of a Kerr black hole have focused only on achieving higher numerical accuracy with limited emphasis on providing their physical interpretation. In this article, we partially address this issue by computing the quasinormal mode frequency spectrum of a Kerr black hole using the theory of Regge poles. By considering the retarded Green's function of the Teukolsky equation, we establish for scalar, electromagnetic and gravitational perturbations an equation linking the Regge poles to the quasinormal frequencies and we solve it in the high-frequency regime to get "semiclassical" relations permitting us to obtain the complex frequencies of the weakly damped quasinormal modes from the Regge trajectories. Numerical results concerning gravitational perturbations ($s=-2$) are displayed. They are in excellent agreement with the "exact" ones in the eikonal regime $(\ell \gg 1)$ and in very good agreement even for lower values of $\ell$. Moreover, the splitting of each Regge pole of the Schwarzschild black hole into an infinite number of Kerr Regge poles explains the breaking of the azimuthal degeneracy of the quasinormal frequencies of the Schwarzschild black hole due to rotation. Our work is a first step to extend to Kerr black holes the approach developed for static spherically symmetric black holes which allowed, from a geometrical interpretation of the Regge poles in terms of the properties of the unstable circular null geodesics lying on the photon sphere, to derive accurate analytical formulas for the Regge trajectories and, as a by-product, for the complex frequencies of the weakly damped quasinormal modes.