On the Mathematics of Coframe Formalism and Einstein-Cartan Theory -- A Brief Review

TL;DR

This paper reviews the fundamental mathematical structures underlying Einstein-Cartan theory, including principal bundles, connections, curvature, torsion, and associated field equations, providing a comprehensive mathematical overview.

Contribution

It offers a concise synthesis of the core mathematical formalism essential for understanding Einstein-Cartan theory, highlighting its geometric and gauge-theoretic foundations.

Findings

Clarifies the role of torsion in Einstein-Cartan theory

Derives Einstein-Cartan field equations from geometric principles

Summarizes conservation laws within the formalism

Abstract

This article is a review of what could be considered the basic mathematics of Einstein-Cartan theory. We discuss the formalism of principal bundles, principal connections, curvature forms, gauge fields, torsion form, and Bianchi identities, and eventually, we will end up with Einstein-Cartan-Sciama-Kibble field equations and conservation laws in their implicit formulation.