Non-Existence of Time-Periodic Solutions of the Dirac Equation in an Axisymmetric Black Hole Geometry

TL;DR

This paper proves that in non-extreme Kerr-Newman black hole geometries, the Dirac equation admits no normalizable, time-periodic solutions, implying quantum particles cannot remain in stable orbits but must either fall into the black hole or escape.

Contribution

It establishes a non-existence theorem for time-periodic solutions of the Dirac equation in axisymmetric black hole geometries, extending previous results to a broader class of metrics.

Findings

No normalizable, time-periodic Dirac solutions in Kerr-Newman geometry.

Quantum Dirac particles cannot form stable orbits around black holes.

Particles must either fall into the black hole or escape to infinity.

Abstract

We prove that, in the non-extreme Kerr-Newman black hole geometry, the Dirac equation has no normalizable, time-periodic solutions. A key tool is Chandrasekhar's separation of the Dirac equation in this geometry. A similar non-existence theorem is established in a more general class of stationary, axisymmetric metrics in which the Dirac equation is known to be separable. These results indicate that, in contrast with the classical situation of massive particle orbits, a quantum mechanical Dirac particle must either disappear into the black hole or escape to infinity.