The equations of Einstein and Cartan

TL;DR

The paper develops a generalized Cartan formulation of four-dimensional Lorentzian gravity using type $N=2$ generalized forms, showing Einstein's field equations can be encoded in extended Cartan structure equations with generalized connections. A central result is that, in vacuum, the generalized curvature can vanish, yielding flat generalized Poincaré connections that relate all solutions by generalized gauge transformations; this structure is extended to cases with nonzero $oldsymbol{T}_{ab}$ and $oldsymbol{ abla}$ via modified connections and torsion. The author constructs a first-order action for vacuum gravity by generalizing the Nieh-Yan three-form with a flat generalized connection, and discusses extensions to other signatures, matter content, and potential links to higher gauge theories and teleparallelism. Overall, the work provides a gauge-theoretic, generalized-form framework that unifies solution spaces of Einstein's equations and offers new action principles for gravity through generalized Nieh-Yan constructs.

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 Poincaré group. An action for the vacuum field equations is constructed by generalizing the Nieh-Yan three-form.