Gravitation and the Local Symmetry Group of Spacetime

TL;DR

This paper argues that in general relativity, the spin connection should be regarded as the fundamental gravitational field, with the Lorentz group as the relevant local symmetry group, instead of the tetrad.

Contribution

It demonstrates that the spin connection, not the tetrad, is the fundamental field for gravitation, and clarifies the local symmetry group as the Lorentz group.

Findings

The spin connection and tetrad are not independent fields.

The Lorentz group underpins the local symmetry in general relativity.

Minimal coupling using the Lorentz covariant derivative matches standard prescriptions.

Abstract

According to general relativity, the interaction of a matter field with gravitation requires the simultaneous introduction of a tetrad field, which is a field related to translations, and a spin connection, which is a field assuming values in the Lie algebra of the Lorentz group. These two fields, however, are not independent. By analyzing the constraint between them, it is concluded that the relevant local symmetry group behind general relativity is provided by the Lorentz group. Furthermore, it is shown that the minimal coupling prescription obtained from the Lorentz covariant derivative coincides exactly with the usual coupling prescription of general relativity. Instead of the tetrad, therefore, the spin connection is to be considered as the fundamental field representing gravitation.