Parametrization and Stress-Energy-Momentum Tensors in Metric Field Theories

TL;DR

This paper explores a formalism that makes classical field theories with a metric background generally covariant by introducing covariance fields, linking multimomenta to stress-energy-momentum tensors, and illustrating with electromagnetic and Klein-Gordon fields.

Contribution

It develops a parametrization method to render metric field theories generally covariant using dynamic covariance fields within a multisymplectic framework.

Findings

Multimomenta of covariance fields form Piola-Kirchhoff stress-energy-momentum tensors.

The introduced covariance fields have no additional physical content.

The formalism is illustrated with electromagnetic and Klein-Gordon fields.

Abstract

We give an exposition of the parametrization method of Kuchar [1973] in the context of the multisymplectic approach to field theory, as presented in Gotay and Marsden [2008a]. The purpose of the formalism developed herein is to make any classical field theory, containing a metric as a sole background field, generally covariant (that is, "parametrized," with the spacetime diffeomorphism group as a symmetry group) as well as fully dynamic. This is accomplished by introducing certain "covariance fields" as genuine dynamic fields. As we shall see, the multimomenta conjugate to these new fields form the Piola-Kirchhoff version of the stress-energy-momentum tensor field, and their Euler-Lagrange equations are vacuously satisfied. Thus, these fields have no additional physical content; they serve only to provide an efficient means of parametrizing the theory. Our results are illustrated with two examples, namely an electromagnetic field and a Klein-Gordon vector field, both on a background spacetime.