Cartan rediscovered

Élie Cartan's invariant integral formalism is extended to gauge field theory, including general relativity. This constitutes an alternative procedure that is equivalent to the Rosenfeld, Bergmann, Dirac algorithm. Also a new derivation of the generator of diffeomorphism-induced canonical transformations is given that proceed's from the Poincaré-Cartan form. In addition, a Hamilton-Jacobi formalism is developed for constructing explicit phase space functions in general relativity that are invariant under the full four-dimensional diffeomorphism group. These identify equivalence classes of classical solutions of Einstein's equations. Each member is dependent on intrinsic spatial coordinates and also undergoes non-trivial evolution in intrinsic time. The intrinsic coordinates are determined by the spacetime geometry in terms of Weyl scalars. The implications of this analysis for an eventual quantum theory of gravity are profound.