The Massive Dirac Equation in the Kerr Geometry: Separability in Eddington-Finkelstein-type Coordinates and Asymptotics

TL;DR

This paper demonstrates the separability of the massive Dirac equation in Kerr black hole spacetime using horizon-penetrating coordinates, deriving asymptotic solutions and analyzing scattering of particles.

Contribution

It introduces a new separability proof for the Dirac equation in Kerr geometry with advanced Eddington-Finkelstein coordinates and explores its implications for scattering and spectral analysis.

Findings

Separable form of the Dirac equation in Kerr spacetime established.

Asymptotic solutions at infinity and horizons derived.

Analysis of Dirac particle scattering by Kerr black holes conducted.

Abstract

The separability of the massive Dirac equation in the non-extreme Kerr geometry in horizon-penetrating advanced Eddington-Finkelstein-type coordinates is shown. To this end, the Kerr geometry is described in the Newman-Penrose formalism by a regular Carter tetrad and the Dirac spinors and matrices are defined in a chiral Newman-Penrose dyad representation. Applying Chandrasekhar's mode ansatz, the Dirac equation is separated into radial and angular systems of ordinary differential equations. Asymptotic radial solutions at infinity, the event horizon, and the Cauchy horizon are derived, and the decay of the associated errors is analyzed. Moreover, specific aspects of the angular eigenfunctions and eigenvalues are discussed. Finally, as an application, the scattering of massive Dirac particles by the gravitational field of a rotating Kerr black hole is studied. This work provides the basis for a Hamiltonian formulation of the massive Dirac equation in the non-extreme Kerr geometry in horizon-penetrating coordinates and for the construction of an integral spectral representation of the Dirac propagator that yields the dynamics of Dirac particles outside, across, and inside the event horizon, up to the Cauchy horizon.