Geometry of Curved Spacetimes Equipped with Torsionful Affinities and Einstein-Cartan's Theory in Two-Component Spinor Form

TL;DR

This paper reviews the geometry of spacetimes with torsion, introduces curvature spinors in this context, and provides a spinor formulation of Einstein-Cartan theory, clarifying its relation to general relativity.

Contribution

It offers a detailed two-component spinor formulation of Einstein-Cartan theory and explores the symmetry properties of curvature spinors in torsionful spacetimes.

Findings

Definition and analysis of Witten curvature spinors for torsionful spacetimes

Spinor transcription of Einstein-Cartan's theory

Clarification of the correspondence between Einstein-Cartan and general relativity

Abstract

The classical world structures borne by spacetimes endowed with torsionful affinities are reviewed. Subsequently, the definition and symmetry properties of a typical pair of Witten curvature spinors for such spacetimes are exhibited along with a comprehensive two-component spinor transcription of Einstein-Cartan's theory. A full description of the correspondence principle that interrelates Einstein-Cartan's theory and general relativity is likewise presented.