General Relativity as a Genuine Connection Theory

TL;DR

This paper develops a connection-based formulation of general relativity using the Palatini approach, highlighting the role of spin connection and tetrads in describing gravitational interactions akin to gauge theories.

Contribution

It introduces a genuine connection theory for general relativity where the gravitational field is represented by a Lorentz-valued spin connection, unifying it with gauge theory frameworks.

Findings

Coupling between spin connection and coordinate vector-field underpins curvature.

Tetrad field reduces Bianchi identities to a single identity.

Approach aligns gravitational theory with gauge theories of fundamental interactions.

Abstract

The Palatini formulation is used to develop a genuine connection theory for general relativity, in which the gravitational field is represented by a Lorentz-valued spin connection. The existence of a tetrad field, given by the Fock-Ivanenko covariant derivative of the tangent-space coordinates, implies a coupling between the spin connection and the coordinate vector-field, which turns out to be the responsible for the onset of curvature. This connection-coordinate coupling can thus be considered as the very foundation of the gravitational interaction. The peculiar form of the tetrad field is shown to reduce both Bianchi identities of general relativity to a single one, which brings this theory closer to the gauge theories describing the other fundamental interactions of Nature. Some further properties of this approach are also examined.