Dirac Hamiltonian and Reissner-Nordstrom Metric: Coulomb Interaction in Curved Space-Time

TL;DR

This paper derives the Dirac Hamiltonian in Reissner-Nordstrom space-time, revealing electro-gravitational spin-orbit coupling and symmetry breaking due to electrostatic interactions, with implications for bound state spectra.

Contribution

It provides the first derivation of the relativistic Dirac Hamiltonian in charged black hole space-time, including electro-gravitational effects and symmetry analysis.

Findings

Identification of gravitational and electro-gravitational spin-orbit coupling terms.

Explicit breaking of particle-antiparticle symmetry due to electrostatic coupling.

Evaluation of bound state spectra with gravitational corrections.

Abstract

We investigate the spin-1/2 relativistic quantum dynamics in the curved space-time generated by a central massive charged object (black hole). This necessitates a study of the coupling of a Dirac particle to the Reissner-Nordstrom space-time geometry and the simultaneous covariant coupling to the central electrostatic field. The relativistic Dirac Hamiltonian for the Reissner-Nordstrom geometry is derived. A Foldy-Wouthuysen transformation reveals the presence of gravitational, and electro-gravitational spin-orbit coupling terms which generalize the Fokker precession terms found for the Dirac-Schwarzschild Hamiltonian, and other electro-gravitational correction terms to the potential proportional to alpha^n G, where alpha is the fine-structure constant, and G is the gravitational coupling constant. The particle-antiparticle symmetry found for the Dirac-Schwarzschild geometry (and for other geometries which do not include electromagnetic interactions) is shown to be explicitly broken due to the electrostatic coupling. The resulting spectrum of radially symmetric, electrostatically bound systems (with gravitational corrections) is evaluated for example cases.