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

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.

22 pages, 6 figures. Minor corrections, to match published version

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• Late-time evolution of charged massive Dirac fields in the Kerr-Newman background
• Quantum Bound States Around Black Holes
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• Kermions: quantization of fermions on Kerr space-time
• The massive Dirac equation in the Kerr-Newman-de Sitter and Kerr-Newman black hole spacetimes
• Dirac perturbations on Schwarzschild-Anti-de Sitter spacetimes: Generic boundary conditions and new quasinormal modes
• The Massive Dirac Equation in the Kerr Geometry: Separability in Eddington-Finkelstein-type Coordinates and Asymptotics
• Dirac dynamical resonance states around Schwarzschild black holes
• Identifying Spin Properties of Evaporating Black Holes through Asymmetric Neutrino and Photon Emission
• Charged Dirac perturbations on Reissner-Nordström-Anti-de Sitter spacetimes: quasinormal modes with Robin boundary conditions
• The Dirac equation in Schwarzschild mass coupled to a Stationary Electromagnetic Field
• The effective method to calculate eigenvalues of Chandrasekhar-Page angular equations
• Stationary solutions of the second-order equation for fermions in Kerr-Newman space-time
• On the eigenvalues of the fermionic angular eigenfunctions in the Kerr metric
• Interacting Kerr-Newman Electromagnetic Fields