Belinfante Tensors Induced by Matter-Gravity Couplings

TL;DR

This paper demonstrates that matter fields coupled to gravity produce conserved, symmetric energy-momentum tensors, with the Belinfante tensor emerging naturally and showing equivalence to the canonical tensor in flat space, regardless of the coupling specifics.

Contribution

It provides a general algorithm linking matter-gravity couplings to conserved energy-momentum tensors, clarifying their equivalence and differences in various theories.

Findings

Belinfante tensor arises from matter-gravity couplings.

Different couplings yield conserved tensors differing by identically conserved terms.

Metric energy-momentum tensor is on-shell equivalent to the flat-space canonical tensor.

Abstract

We show that any generally covariant coupling of matter fields to gravity gives rise to a conserved, on-shell symmetric energy-momentum tensor equivalent to the canonical energy-momentum tensor of the flat-space theory. For matter fields minimally coupled to gravity our algorithm gives the conventional Belinfante tensor. We establish that different matter-gravity couplings give metric energy-momentum tensors differing by identically conserved tensors. We prove that the metric energy-momentum tensor obtained from an arbitrary gravity theory is on-shell equivalent to the canonical energy-momentum tensor of the flat-space theory.