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

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 Elie 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.

💡 Research Summary

The paper presents a comprehensive review of how differential forms, Pfaffian systems, and Cartan connections provide a modern geometric framework for General Relativity (GR) and, in particular, for the Null Surface Formulation (NSF). It begins by recalling the first‑order (tetrad) formulation of GR, where the orthonormal co‑frame θ⁽ᵃ⁾ and the spin‑connection ω⁽ᵃ⁾₍ᵇ₎ are treated as 1‑forms. The Cartan structure equations, dθ⁽ᵃ⁾+ω⁽ᵃ⁾₍ᵇ₎∧θ⁽ᵇ⁾=T⁽ᵃ⁾ and dω⁽ᵃ⁾₍ᵇ₎+ω⁽ᵃ⁾₍ᶜ₎∧ω⁽ᶜ⁾₍ᵇ₎=Ω⁽ᵃ⁾₍ᵇ₎, encode torsion and curvature as 2‑forms, turning the Einstein equations into algebraic relations among differential forms. This exterior‑calculus language dramatically simplifies tensor calculations and makes the underlying geometric structures manifest.

The discussion then moves to conformal geometry, emphasizing that many physical problems (e.g., propagation of massless fields, asymptotic structure at null infinity) depend only on the conformal class