On the notions of energy tensors in tetrad-affine gravity

TL;DR

This paper clarifies how energy tensors are defined and behave in tetrad-affine gravity, showing their equivalence and extension to include gravity, and analyzing their divergence properties related to energy conservation with torsion.

Contribution

It demonstrates the exact coincidence of canonical and stress-energy tensors in tetrad-affine theories and extends these notions to gravitational fields.

Findings

Canonical and stress-energy tensors coincide without adjustments.

Energy tensors can be extended to include gravitational fields.

On-shell divergences relate to local energy conservation with torsion.

Abstract

We are concerned with the precise modalities by which mathematical constructions related to energy-tensors can be adapted to a tetrad-affine setting. We show that, for fairly general gauge field theories formulated in that setting, two notions of energy tensor--the canonical tensor and the stress-energy tensor--exactly coincide with no need for tweaking. Moreover we show how both notions of energy-tensor can be naturally extended to include the gravitational field itself, represented by a couple constituted by the tetrad and a spinor connection. Then we examine the on-shell divergences of these tensors in relation to the issue of local energy-conservation in the presence of torsion.