Semi-analytical Solution of Dirac equation in Schwarzschild Geometry

TL;DR

This paper provides a semi-analytical WKB solution to the Dirac equation in Schwarzschild spacetime, deriving reflection and transmission coefficients, and compares it with a quantum mechanical step potential approach.

Contribution

It offers a novel semi-analytical WKB method for solving the Dirac equation in Schwarzschild geometry, including explicit expressions for wave functions and scattering coefficients.

Findings

WKB and quantum mechanical solutions agree across parameter space

Reflection and transmission coefficients are analytically derived

Wave scattering properties depend on potential well height and energy

Abstract

Separation of the Dirac equation in the spacetime around a Kerr black hole into radial and angular coordinates was done by Chandrasekhar in 1976. In the present paper, we solve the radial equations in a Schwarzschild geometry semi-analytically using Wentzel-Kramers-Brillouin approximation (in short WKB) method. Among other things, we present analytical expression of the instantaneous reflection and transmission coefficients and the radial wave functions of the Dirac particles. Complete physical parameter space was divided into two parts depending on the height of the potential well and energy of the incoming waves. We show the general solution for these two regions. We also solve the equations by a Quantum Mechanical approach, in which the potential is approximated by a series of steps and found that these two solutions agree. We compare solutions of different initial parameters and show how the properties of the scattered wave depend on these parameters.