Note about the spin connection in general relativity

TL;DR

This paper derives the spin connection in general relativity for fermions solely from covariance, Lorentz symmetry, and fermion bilinear properties, confirming its equivalence to the tetrad formalism approach.

Contribution

It provides a derivation of the spin connection based only on fundamental principles, clarifying its geometric origin without relying on the tetrad formalism.

Findings

Derived the spin connection from covariance and Lorentz symmetry.

Confirmed the derived spin connection matches the tetrad formalism result.

Clarified the geometric basis of the spin connection in curved spacetime.

Abstract

In general relativity the fermions are treated from the perspective of the gauged Lorentz group and by introducing the corresponding gauge fields the spin connection. This procedure is intimately related to the so-called "vielbein" formalism and stems from the general structure that can be associated to a manifold, the affine connection. In this work we derive the correct spin connection based only on the general covariance of the theory, on the gauged Lorentz symmetry and on the known space-time properties of fermion bilinears generalized to the curved space. Our result coincides exactly with the spin connection obtain through the tetrad formalism.