The Dirac equation in the Kerr-de Sitter metric

TL;DR

This paper analyzes the Dirac equation in the Kerr-de Sitter spacetime, deriving new horizon formulas, separating the equation into radial and angular parts, and exploring spectral properties and solutions.

Contribution

It introduces new horizon formulas, constructs a symmetry operator via the Chandrasekhar ansatz, and characterizes the spectrum and polynomial solutions of the angular operator.

Findings

Spectrum of the angular operator is discrete with simple eigenvalues.

No bound states exist in the non-extreme Kerr-de Sitter case.

Conditions for polynomial solutions of the angular operator are derived.

Abstract

We consider a fermion in the presence of a rotating black hole immersed in a universe with positive cosmological constant. After deriving new formulae for the event, Cauchy and cosmological horizons we adopt the Carter tetrad to separate the aforementioned equation into a radial and angular equation. We show how the Chandrasekhar ansatz leads to the construction of a symmetry operator that can be interpreted as the square root of the squared total angular momentum operator. Furthermore, we prove that the the spectrum of the angular operator is discrete and consists of simple eigenvalues and by means of the functional Bethe ansatz method we also derive a set of necessary and sufficient conditions for the angular operator to have polynomial solutions. Finally, we show that there exist no bound states for the Dirac equation in the non-extreme case.