Symmetry properties of the metric energy-momentum tensor in classical field theories and gravity

TL;DR

This paper derives a general identity relating symmetries of the Lagrangian to the symmetry properties of the energy-momentum tensor in covariant field theories, revealing limitations in defining gravitational energy density.

Contribution

It provides a new, general identity linking Lagrangian symmetries to energy-momentum tensor symmetries and applies it to show the impossibility of gauge-invariant energy-momentum tensors for linear spin-2 fields.

Findings

Symmetry of the Lagrangian implies symmetry of the energy-momentum tensor in curved spacetime.

A gauge-invariant spin-2 field cannot have a gauge-invariant metric energy-momentum tensor.

Field-theoretic approaches to gravitational energy density face fundamental limitations.

Abstract

We derive a generic identity which holds for the metric (i.e. variational) energy-momentum tensor under any field transformation in any generally covariant classical Lagrangian field theory. The identity determines the conditions under which a symmetry of the Lagrangian is also a symmetry of the energy-momentum tensor. It turns out that the stress tensor acquires the symmetry if the Lagrangian has the symmetry in a generic curved spacetime. In this sense a field theory in flat spacetime is not self-contained. When the identity is applied to the gauge invariant spin-two field in Minkowski space, we obtain an alternative and direct derivation of a known no-go theorem: a linear gauge invariant spin-2 field, which is dynamically equivalent to linearized General Relativity, cannot have a gauge invariant metric energy-momentum tensor. This implies that attempts to define the notion of gravitational energy density in terms of the metric energy--momentum tensor in a field-theoretical formulation of gravity must fail.