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 continuedfraction methods, or perturbatively to obtain a series expansion for the eigenvalues. In the case µ = ±ω (where ω and µ 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. PACS numbers: 04.70.-s, 04.62.+v
I. INTRODUCTIONThe interaction of a fermionic (Dirac) field with a rotating charged (Kerr-Newman) black hole is of deep theoretical interest. The interaction depends on three couplings: coupling between quantum-mechanical spin and black hole (BH) rotation; coupling between field mass and BH mass; and coupling between field charge and BH charge. Desire for a deeper understanding of these couplings has motivated a range of studies over the last three decades.Over thirty years ago, Chandrasekhar [1] and Page [2] showed that separation of variables is possible for the massive Dirac field on the Kerr-Newman spacetime. In other words, the partial differential equations (PDEs) governing the evolution of the Dirac field may be reduced to a set of coupled ordinary differential equations (ODEs), and the solution can be expressed as a sum over modes and integral over frequency. The radial and angular ODEs are coupled through an angular eigenvalue λ. The eigenvalue is found by solving the so-called Chandrasekhar-Page (CP) equation (see Eq. (4) below). The eigenfunctions of the CP equation -the so-called mass dependent spin-half spheroidal harmonics (MDSHs) -are required for a full reconstruction of the field. In this paper we present a new method for computing both the eigenvalues and eigenfunctions. Our aim is to show that finding MDSHs is no more difficult than finding the spin-weighted spheroidal harmonics for massless fields [3]. Along the way, we review alternative methods and highlight some inaccuracies in the literature.Motivation for this work arose from two s...