Strong Conservation Laws and Equations of Motion in Covariant Field Theories

Abstract

The relationship between the covariance of a field theory and the equations of motion is discussed in both the Lagrangian and the Hamiltonian formalism. All theories whose field equations are derivable from a variational principle and are covariant under arbitrary (curvilinear) coordinate transformations posses Bianchi identities and, hence, "strong" conservation laws. Because the strong conservation laws are ordinary divergences equal to zero, whether or not the field equations are satisfied, there exist certain skew-symmetric expressions whose divergences yield the components of the energy-momentum tensor. These superpotentials, as the skew-symmetric expressions are called, enable us to write the energy and momentum content of the field as two-dimensional surface integrals. Also, by using the superpotentials together with the field equations, one can find certain surface integrals which are independent of the surface of integration and which yield the equations of motion for the singularities enclosed by the surface. If the Einstein-Infeld approximation method is applied in the general theory of relativity, the above surface integrals reduce to integrals which are equivalent to those used by Einstein and Infeld to obtain the equations of motion for the field sources.