Spectral Analysis of Radial Dirac Operators in the Kerr-Newman Metric and its Applications to Time-periodic Solutions

TL;DR

This paper analyzes the existence of time-periodic solutions to the Dirac equation in Kerr-Newman black hole backgrounds, proving non-existence in non-extreme cases and demonstrating existence for specific particle masses in the extreme case.

Contribution

It provides a spectral analysis of the Dirac equation in Kerr-Newman spacetime and identifies conditions under which time-periodic solutions exist or do not exist.

Findings

No time-periodic solutions in non-extreme Kerr-Newman black holes.

Existence of time-periodic solutions for specific particle masses in the extreme case.

Solution period depends only on black hole parameters.

Abstract

We investigate the existence of time-periodic solutions of the Dirac equation in the Kerr-Newman background metric. To this end, the solutions are expanded in a Fourier series with respect to the time variable $t$ and the Chandrasekhar separation ansatz is applied so that the question of existence of a time-periodic solution is reduced to the solvability of a certain coupled system of ordinary differential equations. First, we prove the already known result that there are no time-periodic solutions in the non-extreme case. Then it is shown that in the extreme case for fixed black hole data there is a sequence of particle masses $(m_N)_{N\in\mathbb N}$ for which a time-periodic solution of the Dirac equation does exist. The period of the solution depends only on the data of the black hole described by the Kerr-Newman metric.