**Manuel Tecchiolli**

Institute for Theoretical Physics, ETH Zürich, Wolfgang-Pauli-Str. 27, 8093 Zürich, Switzerland; manuelte@student.ethz.ch

Received: 3 September 2019; Accepted: 24 September 2019; Published: 27 September 2019

**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.

**Keywords:** general relativity; torsion-gravity; mathematical physics

### **1. Introduction**

The formulation of torsion gravity and the consequent coupling with spin rely on a different formulation compared to the one of original works on General Relativity. This formulation regards geometrical objects called principal bundles. In this context, we can formulate General Relativity (or Einstein–Cartan–Sciama–Kibble (ECSK) theory in the presence of torsion) with a principal connection, which can be pulled back to the base manifold in a canonical way and further restricted to the only antisymmetric component, giving birth to the well-known spin connection. This process shows the possibility of formulating General Relativity as a proper gauge theory rather than using the affine formulation and Christoffel symbols Γ. What permits the equivalence of the two formulations is a bundle isomorphism called *tetrads* or *vierbein*, which is supposed to respect certain compatibility conditions. Then, we can define the associated torsion form and postulate the Palatini–Cartan action as a functional of such tetrads and spin connection. This leads to ECSK field equations.

We will first set up all the abstract tools of principal bundles, tetrads, and principal connection; secondly, we will derive the Einstein–Cartan–Sciama–Kibble theory in its implicit version; and finally, we will discuss conservation laws coming from local SO(3, 1) and diffeomorphism invariance of ECSK theory.

Throughout the article, we will give theorems and definitions. However, we would like to stress that hypotheses for such theorems will often be slightly redundant: we will take spaces and functions to be differentiable manifolds and smooth, even though weaker statements would suffice. This is because we prefer displaying the setup for formalizing the theory rather than presenting theorems and definitions with weaker hypotheses that we will never use for the theory. Nontheless, we will specify where such hypotheses are strengthened. In spite of this, the discussion will be rather general, probably more general than what is usually required in formulating ECSK (Einstein–Cartan–Sciama–Kibble) theory.
