The equations of Einstein and Cartan

Abstract

A formulation of Einstein's gravitational field equations in four space-time dimensions is presented using generalized differential forms and Cartan's equations for metric geometries. Cartan's structure equations are extended by using generalized metric connections. They are then employed to represent Einstein's field equations and their solutions. When the energy-momentum tensor is zero the generalized connections can be chosen to be flat and different solutions of Einstein's equations can be related by generalizations of the Poincar\'e group. An action for the vacuum field equations is constructed by generalizing the Nieh-Yan three-form.