Numerical calculation of the relativistic acceleration of an electron in curved spacetime using the Dirac equation

TL;DR

This paper numerically and analytically investigates the relativistic acceleration of an electron in curved spacetime, revealing a velocity-dependent factor consistent with classical geodesic predictions, and introduces methods to improve approximation accuracy.

Contribution

It presents a novel numerical and analytical approach to calculating electron acceleration in curved spacetime without relying on WKB approximation.

Findings

Acceleration proportional to (1-3v^2/c^2) in Schwarzschild coordinates

Results align with classical geodesic predictions

Introduces alternative iterative analytical methods

Abstract

The relativistic acceleration of an electron in a uniform gravitational field is calculated numerically using the generalization of the Dirac equation to curved spacetime. Equivalent results are also obtained analytically using an iterative approach that is not based on the WKB approximation, which has been used extensively in the past. The acceleration is found to be proportional to a factor of $(1-3v^2/c^2)$ using Schwarzschild coordinates, which is consistent with the classical geodesic of a particle. These techniques may be useful in resolving the differences between commonly-used approximations.