Bound State Solutions of the Dirac Equation in the Extreme Kerr Geometry

TL;DR

This paper investigates bound state solutions of the Dirac equation around extreme Kerr black holes, establishing conditions for their existence, explicitly solving for eigenfunctions, and revealing the energy spectrum's dependence on black hole parameters.

Contribution

It provides the first explicit solutions for Dirac bound states in extreme Kerr geometry and characterizes the conditions for their existence.

Findings

Bound states exist for specific angular momentum and black hole parameters.

Energy levels are explicitly determined by black hole and quantum numbers.

Radial eigenfunctions are expressed in terms of Laguerre polynomials.

Abstract

In this paper we consider bound state solutions, i.e., normalizable time-periodic solutions of the Dirac equation in the exterior region of an extreme Kerr black hole with mass $M$ and angular momentum $J$. It is shown that for each azimuthal quantum number $k$ and for particular values of $J$ the Dirac equation has a bound state solution, and that the energy of this Dirac particle is uniquely determined by $\omega = -\frac{kM}{2J}$. Moreover, we prove a necessary and sufficient condition for the existence of bound states in the extreme Kerr-Newman geometry, and we give an explicit expression for the radial eigenfunctions in terms of Laguerre polynomials.