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

We give an exposition of the parametrization method of Kuchaˇr [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. 1 1 Introduction