The massive Dirac equation in the Kerr-Newman-de Sitter and Kerr-Newman black hole spacetimes

TL;DR

This paper derives and analyzes exact solutions of the massive Dirac equation in Kerr-Newman-(anti) de Sitter black hole spacetimes, revealing complex differential equations and their asymptotic behaviors, and showing the absence of bound states in certain conditions.

Contribution

The paper provides a detailed derivation of the Dirac equation solutions in Kerr-Newman-(anti) de Sitter backgrounds and extends the analysis to generalized Heun equations, including asymptotic behaviors and eigenvalue problems.

Findings

Radial and angular equations reduce to generalized Heun equations.

Solutions near horizons and at infinity are explicitly derived.

No bound states exist for < in non-extreme Kerr-Newman geometry.

Abstract

Exact solutions of the Dirac general relativistic equation (DE) that describe the dynamics of a massive, electrically charged particle with half-integer spin in the curved spacetime geometry of an electrically charged, rotating Kerr-Newman-(anti) de Sitter black hole (BH) are investigated. We first, derive the DE in the Kerr-Newman-de Sitter (KNdS) BH background using a generalised Kinnersley null tetrad in the Newman-Penrose formalism. In this frame, we prove the separation of the DE into ordinary differential equations for the radial and angular parts. Under specific transformations of the independent and dependent variables we prove that the transformed radial equation for a massive charged spin $\frac{1}{2}$ fermion in the background KNdS BH constitutes a highly non-trivial generalisation of Heun's equation. Using a Regge-Wheeler-like independent variable we transform the radial equation in the KNdS background into a Schr\"{o}dinger-like differential equation and investigate its asymptotic behaviour near the event and cosmological horizons. For a massive fermion (MF) in the background of a Kerr-Newman (KN) BH we first prove that the radial and angular equations that result from the separation of DE reduce to the generalised Heun differential equation (GHE). The local solutions of such GHE are derived and can be described by holomorphic functions whose power series coefficients are determined by a four-term recurrence relation. Using asymptotic analysis we derive the solutions for the MF far away from the KN BH and the solutions near the event horizon . The determination of the separation constant as an eigenvalue problem in the KN background is investigated. Using the aforementioned four-term recursion formula we prove that in the non-extreme KN geometry there are no bound states with $\omega^2<\mu^2$, where $\omega$ and $\mu$ are the energy and mass of the fermion respectively.