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

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.