Pfaffian Systems, Cartan Connections, and the Null Surface Formulation of General Relativity

Abstract

This review examines the role of differential forms, Pfaffian systems, and hypersurfaces in general relativity. These mathematical constructions provide the essential tools for general relativity, in which the curvature of spacetime—described by the Einstein field equations—is most elegantly formulated using the Cartan calculus of differential forms. Another important subject in this discussion is the notion of conformal geometry, where the relevant invariants of a metric are characterized by Élie Cartan’s normal conformal connection. The previous analysis is then used to develop the null surface formulation (NSF) of general relativity, a radical framework that postulates the structure of light cones rather than the metric itself as the fundamental gravitational variable. Defined by a central Pfaffian system, this formulation allows the entire spacetime geometry to be reconstructed from a single scalar function, Z, whose level surfaces are null.