Quasinormal modes of Kerr-Newman black holes: Revisiting the Dudley-Finley approximation

We present a comprehensive study of the Kerr-Newman quasinormal mode spectrum in the Dudley-Finley approximation, where the linear gravitoelectromagnetic perturbations are decoupled by "freezing" either one of the fields to its background value. First, we reassess the accuracy of this approximation by comparing it to calculations that solve the coupled system of gravitoelectromagnetic perturbation equations across the subextremal spin-charge parameter space. We find that for the $(\ell,m,n) = (2,2,0)$, $(2,2,1)$, and $(3,3,0)$ modes, the agreement is typically within $10\%$ and $1\%$ for the real and imaginary parts of the frequencies, respectively. Next, we investigate the spectrum in the near-extremal limit, and study the family of long-lived ("zero-damped") gravitational modes. We find that the near-extremal parameter space consists of subregions containing either only zero-damped modes, or zero-damped modes alongside modes that retain nonzero damping. We derive analytic expressions for the boundaries between these regions. Moreover, we discuss the connection between the zero-damped and damped modes in the Dudley-Finley approximation and the "near-horizon/photon-sphere" modes of the full Kerr-Newman spectrum. Finally, we analyze the behavior of the quadrupolar gravitational modes with large overtone numbers $n$, and study their trajectories in the complex plane.