Dirac Equation and Its Solution Around Schwarzschild–de Sitter Black Hole

Abstract

This paper aims to solve the radial parts of Dirac equation between the inner and the outer horizon in Schwarzschild–de Sitter geometry numerically. A “logarithm” approximation which almost has the same nature with the modified “tortoise” coordinate r* even close to the two horizons is found. It is used to replace the modified “tortoise” coordinate r*, this leads to a simple analytically invertible relation between r* and the radius r. Then, the potential V (r*) is replaced by a collection of step functions. By a quantum mechanical method, the solution of the wave equation as well as the reflection and transmission coefficients are computed. The resulting wave turns out to be not close to that of a harmonic wave globally.