On the energy-momentum tensor

TL;DR

This paper clarifies the relationships among different definitions of energy-momentum tensors in tensor field theories, introducing a new tensor to simplify their computation and showing their equivalence on-shell.

Contribution

It introduces a new tensor field to easily compute the Belinfante tensor and clarifies its relation to canonical and metric energy-momentum tensors.

Findings

Belinfante tensor can be computed from a new tensor field.

Belinfante tensor coincides with the metric tensor on-shell.

The Belinfante tensor naturally arises from Noether's theorem.

Abstract

We clarify the relation among canonical, metric and Belinfante's energy-momentum tensors for general tensor field theories. For any tensor field T, we define a new tensor field $\til {\bm T}$, in terms of which the Belinfante tensor is readily computed. We show that the latter is the one that arises naturally from Noether Theorem for an arbitrary spacetime and it coincides on-shell with the metric one.