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

Guido Magnano·Leszek M. Sokolowski
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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.

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Guido Magnano, Leszek M. Sokolowski. 2002-01-23. Symmetry properties of the metric energy-momentum tensor in classical field theories and gravity. https://doi.org/10.1088/0264-9381%2F19%2F2%2F304

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