Black-hole perturbation theory: The asymptotic spectrum of the prolate spin-weighted spheroidal harmonics

Abstract

Prolate spin-weighted spheroidal harmonics play a key role in black-hole perturbation theory. In particular, the highly damped quasinormal resonances of rotating Kerr black holes are closely related to the asymptotic eigenvalues of these important functions. We here present a novel and compact derivation of the asymptotic eigenvalues of the prolate spin-weighted spheroidal harmonics. Our analysis is based on a simple trick which transforms the corresponding spin-weighted spheroidal angular equation into a Schr\"odinger-like wave equation which is amenable to a standard WKB analysis. Our analytical results for the prolate asymptotic spectrum agree with previous numerical computations of the eigenvalues which appear in the literature.