The massive Dirac field on a rotating black hole spacetime: Angular solutions

TL;DR

This paper introduces a new numerical and analytical method for solving the angular part of the massive Dirac equation in Kerr-Newman black hole spacetimes, improving accuracy and correcting previous errors.

Contribution

A novel approach using a new representation of spin-half spherical harmonics and a three-term recurrence relation to compute eigenvalues of the Chandrasekhar-Page equation.

Findings

Derived eigenvalues and eigenfunctions in closed form for specific parameter cases

Provided comprehensive tables and plots of eigenvalues and eigenfunctions

Corrected errors in previous literature on Dirac equation solutions in black hole backgrounds

Abstract

The massive Dirac equation on a Kerr-Newman background may be solved by the method of separation of variables. The radial and angular equations are coupled via an angular eigenvalue, which is determined from the Chandrasekhar-Page (CP) equation. Obtaining accurate angular eigenvalues is a key step in studying scattering, absorption and emission of the fermionic field. Here we introduce a new method for finding solutions of the CP equation. First, we introduce a novel representation for the spin-half spherical harmonics. Next, we decompose the angular solutions of the CP equation (the mass-dependent spin-half spheroidal harmonics) in the spherical basis. The method yields a three-term recurrence relation which may be solved numerically via continued-fraction methods, or perturbatively to obtain a series expansion for the eigenvalues. In the case $\mu = \pm \omega$ (where $\omega$ and $\mu$ are the frequency and mass of the fermion) we obtain eigenvalues and eigenfunctions in closed form. We study the eigenvalue spectrum, and the zeros of the maximally co-rotating mode. We compare our results with previous studies, and uncover and correct some errors in the literature. We provide series expansions, tables of eigenvalues and numerical fits across a wide parameter range, and present plots of a selection of eigenfunctions. It is hoped this study will be a useful resource for all researchers interested in the Dirac equation on a rotating black hole background.