Pauli equation and charged spin-1/2 particle in a weak gravitational field

TL;DR

This paper derives a nonrelativistic Pauli equation for a charged spin-1/2 particle in a weak gravitational field, comparing different approximation methods and analyzing specific gravitational backgrounds.

Contribution

It introduces a direct nonrelativistic approximation method for the curved-space Dirac equation, providing new insights into particle dynamics in weak gravitational fields.

Findings

Agreement with previous results for gravitational waves

Discrepancy with earlier methods in static gravitational fields

Potential energy proportional to mass affects approximation validity

Abstract

Using the nonrelativistic approximation in the curved-space Dirac equation, the analog of the Pauli equation is derived for a weak gravitational field with a gauge fixing condition related to the synchronous gauge, in the presence of an electromagnetic field. Different from the previous works which were employing either the exact or conventional Foldy-Wouthuysen transformations, here we perform calculations by directly performing nonrelativistic approximation which reduced in the power series expansion in the inverse mass of the spinning particle. On top of that, the equations of motion for the massive spin-$1/2$ charged particle are obtained. The two particular cases of the previously explored backgrounds, namely a) plane gravitational wave and b) homogeneous static gravitational field are considered for control. In the case a) we meet correspondence with the previous results. On the other hand, in case b), there is no correspondence with neither perturbative nor with exact Foldy-Wouthuysen transformations, which we also recalculate and agree with the previous works. The disagreement is a kind of a theoretical challenge and most likely occurs because the potential energy, in the particular case of Newtonian approximation, is proportional to the mass of the particle.