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

TL;DR

This paper derives the asymptotic eigenvalues of prolate spin-weighted spheroidal harmonics using a novel WKB-based method, providing analytical results that match previous numerical computations, and elucidating their role in black-hole perturbation theory.

Contribution

A new, compact analytical derivation of the asymptotic eigenvalues of prolate spin-weighted spheroidal harmonics using a Schrödinger-like reformulation and WKB analysis.

Findings

Analytical asymptotic eigenvalues agree with numerical results.

Method transforms angular equation into Schrödinger-like form.

Provides insights into black-hole quasinormal modes.

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.