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| Dirac's wave equation for a point electron in the electromagnetic Kerr--Newman spacetime is studied in the zero-gravity limit; here, ``zero-gravity'' means $G\to 0$, where $G$ is Newton's constant of universal gravitation. The zero-$G$ limit eliminates the troublesome Cauchy horizon of the Kerr--Newman spacetime and also its physically problematic acausal region of closed timelike loops. While the gravitational features of the Kerr--Newman manifold vanish as well when $G\to 0$, this limit retains the nontrivial topology associated with its ring singularity, and all its electromagnetic features. We first show that the formal Dirac Hamiltonian on a static spacelike slice of the maximal analytically extended zero-$G$ Kerr--Newman spacetime is essentially self-adjoint, and that the spectrum of its self-adjoint extension is symmetric about zero. It is next shown that the Dirac operator on the zero-$G$ Kerr--Newman spacetime has a continuous spectrum with both positive and negative energies, separated by a gap about zero that contains a pure point spectrum. The pure point spectrum is associated with time-periodic normalizable solutions, representing bound states of Dirac's point electron in the electromagnetic field of the ring singularity of the zero-$G$ Kerr--Newman spacetime. |
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