Symmetry properties under arbitrary field redefinitions of the metric energy–momentum tensor in classical field theories and gravity

Abstract

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