On the bound states of the Dirac equation in the extreme Kerr metric

TL;DR

This paper analyzes the eigenvalues of the Dirac equation in the extreme Kerr black hole metric, deriving a differential equation for these eigenvalues and proving the absence of bound states in this spacetime.

Contribution

It introduces a novel approach using a self-adjoint holomorphic operator family to study eigenvalues and proves no bound states exist in the extreme Kerr metric.

Findings

Eigenvalues satisfy a nonlinear differential equation

Explicit solution of the eigenvalue differential equation

No bound states for the Dirac equation in the extreme Kerr metric

Abstract

We study the eigenvalues of the angular equation arising after the separation of the Dirac equation in the extreme Kerr metric. To this purpose a self-adjoint holomorphic operator family associated to this eigenvalue problem is considered. We show that the eigenvalues satisfy a first order nonlinear differential equation with respect to the black hole mass and we solve it. Finally, we prove that there exist no bound states for the Dirac equation in the aforementioned metric.