Can the local stress-energy conservation laws be derived solely from field equations?

TL;DR

The paper examines whether local stress-energy conservation laws can be derived solely from field equations, revealing that such derivations are generally ambiguous and more complex than traditional variational methods.

Contribution

It demonstrates that the observed property linking matter equations to energy-momentum tensors is due to a general covariance identity, but this approach has limitations and potential inaccuracies.

Findings

The identity arises from covariance of the matter Lagrangian.

In specific cases, the identity can determine the stress-energy tensor.

The method often leads to ambiguities and is more complex than variational techniques.

Abstract

According to a recent suggestion [1], the energy--momentum tensor for gravitating fields can be computed through a suitable rearrangement of the matter field equations, without relying on the variational definition. We show that the property observed by Accioly et al. in [1] is the consequence of a general identity, which follows from the covariance of the matter Lagrangian in much the same way as (generalized) Bianchi identities follow from the covariance of the purely gravitational Lagrangian. However, we also show that only in particular cases can this identity be used to obtain the actual form of the stress-energy tensor, while in general the method leads to ambiguities and possibly to wrong results. Moreover, in nontrivial cases the computations turn out to be more difficult than the standard variational technique.