Eigenvalues and eigenfunctions of spin-weighted spheroidal harmonics in four and higher dimensions

TL;DR

This paper reviews and extends the analysis of spin-weighted spheroidal harmonics, including eigenvalues, eigenfunctions, and their applications in black hole physics and higher-dimensional theories, through analytic and numerical methods.

Contribution

It provides a comprehensive review and new calculations of eigenvalues and eigenfunctions in four and higher dimensions, including for Kerr black hole modes.

Findings

Eigenvalues and eigenfunctions in four dimensions are computed and tabulated.

Angular dependence of harmonics for Kerr black hole quasinormal modes is characterized.

Scalar spheroidal harmonics in higher dimensions are thoroughly analyzed both analytically and numerically.

Abstract

Spin-weighted spheroidal harmonics are useful in a variety of physical situations, including light scattering, nuclear modeling, signal processing, electromagnetic wave propagation, black hole perturbation theory in four and higher dimensions, quantum field theory in curved space-time and studies of D-branes. We first review analytic and numerical calculations of their eigenvalues and eigenfunctions in four dimensions, filling gaps in the existing literature when necessary. Then we compute the angular dependence of the spin-weighted spheroidal harmonics corresponding to slowly-damped quasinormal mode frequencies of the Kerr black hole, providing numerical tables and approximate formulas for their scalar products. Finally we present an exhaustive analytic and numerical study of scalar spheroidal harmonics in (n+4) dimensions.