Differential Geometrical Methods for Deriving Dirac's Equation in Curved Spacetime to Account for the Presence of Matter

TL;DR

This paper introduces novel differential geometric methods to derive the Dirac equation in curved spacetime, explicitly incorporating matter through Einstein's field equations, extending previous approaches that only addressed matter-free scenarios.

Contribution

The paper presents a new approach using differential geometry to derive the Dirac equation in curved spacetime with matter, filling a gap in existing literature.

Findings

Derived Dirac equation including matter effects

Extended previous methods to matter-present scenarios

Provided a unified geometric framework for Dirac in curved spacetime

Abstract

Differential geomtrical methods for deriving the Dirac equation in Curved Spacetime are presented. Einstein's field equation is applied in a novel manner; in the most current standard reference, Birrell and Davies, 1994 [1], the suggestions for deriving the Dirac equation in Curved Spacetime make no mention of employing Einstein's field equation. Thus, to date, the literature on the derivation of the Dirac equation could not include an expression for the presence of matter. This lack is consistent with earlier publications, including Lichnerowicz's well-known 1964 journal article [3], which presented the first such derivation, and Dimock's 1982 article [2]. The new differential geometrical methods go beyond all previous suggestions, which only apply to cases in the absence of matter. These differential geometrical methods have resulted in derivations of the Dirac equation in Curved Spacetime that apply to either the presence or absence of matter.