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

TL;DR

The paper develops a unified, Leaver-based framework to compute the eigenvalues and eigenfunctions of spin-weighted spheroidal harmonics in four and higher dimensions, addressing both oblate (real c) and prolate (imaginary c) regimes. It delivers a complete 4D treatment of SWSHs, including a practical angular-series solution, robust numerical algorithms, small-c and large-c asymptotics, and Kerr quasinormal mode scalar products, complemented by a comprehensive higher-dimensional generalization (HSHs) with analogous series solutions and asymptotics. The results enable accurate angular decompositions for Kerr perturbations and higher-dimensional black-hole spacetimes, with direct applications to gravitational-wave data analysis and TeV-scale gravity scenarios. Overall, the work fills important gaps by providing analytic expansions, validated numerics, and cross-checks across dimensions, paving the way for precise modeling of spin-weighted perturbations in diverse gravitational settings.

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.

Figures (10)

• Figure 1: Angular eigenvalue $_{0}A_{lm}$ as a function of $c_I$ (prolate case) or $c$ (oblate case), for selected values of $l$ and $m$.
• Figure 2: Angular eigenvalue for the lowest radiatable multipoles with $s=-1$ (left) and $s=-2$ (right) in the oblate case.
• Figure 3: Prolate angular eigenvalues for $s=-1$ (left) and $s=-2$ (right). At variance with Fig. \ref{['fig:2']}, the angular eigenvalue ${}_sA_{lm}$ is now complex. Lines limiting to ${}_sA_{lm}=l(l+1)-s(s+1)$ as $|c|\to 0$ are the real parts of the eigenvalues. Lines approaching zero as $|c|\to 0$ are the imaginary parts if $c_I<0$ (or their modulus if $c_I>0$: see property (iv) of Sec.\ref{['eqns']}).
• Figure 4: $\Re _{-2}A_{22}$ as a function of $|c|$ for different, fixed values of the phase angle $\theta$.
• Figure 5: Modulus of the eigenfunction for prolate SWSHs with $s=-2$ and different values of $m$. The top row refers to $l=2$, the bottom row to $l=3$. Left to right: $c_I=0,~-1$ and $-5$.