On the Dirac operator for a test electron in a Reissner--Weyl--Nordstr\"om black hole spacetime

TL;DR

This paper investigates the Dirac operator for a test electron in Reissner--Weyl--Nordstr"om black hole spacetimes, revealing conditions for self-adjointness, spectral properties, and the impact of anomalous magnetic moments.

Contribution

It demonstrates that including the electron's anomalous magnetic moment ensures the Dirac operator's essential self-adjointness and characterizes its spectral properties in black hole spacetimes.

Findings

Dirac Hamiltonian is not initially self-adjoint but has infinitely many extensions.

Anomalous magnetic moment restores essential self-adjointness.

Spectrum is purely absolutely continuous, covering the entire real line.

Abstract

The present paper studies the Dirac Hamiltonian of a test electron with a domain of bi-spinor wave functions supported on the static region inside the Cauchy horizon of the subextremal RWN black hole spacetime, respectively inside the event horizon of the extremal RWN black hole spacetime. It is found that this Dirac Hamiltonian is not essentially self-adjoint, yet has infinitely many self-adjoint extensions. Including a sufficiently large anomalous magnetic moment interaction in the Dirac Hamiltonian restores essential self-adjointness; the empirical value of the electron's anomalous magnetic moment is large enough. The spectrum of the subextremal self-adjoint Dirac operator with anomalous magnetic moment is purely absolutely continuous and consists of the whole real line; in particular, there are no eigenvalues. The same is true for the spectrum of any self-adjoint extension of the Dirac operator without anomalous magnetic moment interaction, in the subextremal black hole context. In the extremal black hole sector the point spectrum, if non-empty, consists of a single eigenvalue, which is identified.