Covariant Transformation Law for the Field Equations

Abstract

Ordinarily, the existence of Bianchi identities is proven on the strength of the transformation properties of the lagrangian of the theory. In this paper, nothing is assumed concerning the lagrangian, except that the field equations themselves are covariant with respect to general coordinate transformations. It is then shown that at least the coefficients of the second-order derivatives in the field equations satisfy the usual relationships. Furthermore, a very weak restriction on the transformation law of the field equations is sufficient to derive conservation laws that hold even in the presence of matter.