The Classical World and Spinor Formalisms of General Relativity

TL;DR

This paper reviews classical spacetime geometry and explores the spinor formalisms of Infeld and van der Waerden, focusing on gauge properties, curvature algebra, and wave equations for gravitons and photons within these frameworks.

Contribution

It provides a detailed algebraic and geometric analysis of the spinor formalisms, including gauge behaviors, curvature decompositions, and derivation of wave equations for fundamental particles.

Findings

Spinor formalisms exhibit gauge-invariant wave equations for gravitons and photons.

Curvature splittings are algebraically characterized using spinor commutators.

Couplings between Dirac fields and electromagnetic curvatures are analyzed.

Abstract

A review of some facts concerning classical spacetime geometry is presented together with a description of the most elementary aspects of the two-component spinor formalisms of Infeld and van der Waerden. Special attention is concentrated upon the gauge characterization of the basic geometric objects borne by the formalisms. It is pointed out that spin-affine configurations are most naively defined by carrying out parallel displacements of null world vectors within the framework of the {\gamma}-formalism. The standard result that assigns a covariant gauge behaviour to the symmetric parts of any admissible spin connexions is deduced out of building up a generalized version of spin transformation laws. A fairly complete algebraic description of curvature splittings is carried out on the basis of the construction of a set of spinor commutators for each formalism. The pertinent computations take up the utilization of some covariant differential prescriptions which facilitate specifying the action of the commutators on arbitrary spin tensors and densities. It turns out that the implementation of such commutators under certain circumstances gives rise to a system of wave equations for gravitons and Infeld-van der Waerden photons which possess in either formalism a gauge-invariance property associated with appropriate spinor-index configurations. The situation regarding the accomplishment of the couplings between Dirac fields and electromagnetic curvatures is entertained to a considerable extent.