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

Emanuel Gallo, Carlos N. Kozameh

TL;DR

The paper surveys the null surface formulation (NSF) of general relativity, recasting gravitational dynamics in terms of differential forms, Pfaffian systems, and Cartan connections. It shows how the conformal structure and light-cone data, captured by a scalar function $Z$ on a six-dimensional bundle, determine the spacetime metric via a normal conformal Cartan framework and metricity conditions linked to torsion. Central to the discussion are the Weyl curvature, Bach tensor, and the broader role of the $SO(4,2)$ Cartan connection in encoding conformal gravity, with explicit treatment of how Einstein equations emerge from these conformal structures. The work also reviews the geometric machinery of jet spaces, exterior differential systems, and principal bundles, and outlines extensions to higher dimensions and constrained generating families to address caustics, offering a holistic, holographic-like view of gravitational degrees of freedom beyond the metric formalism.

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 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.

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

TL;DR

The paper surveys the null surface formulation (NSF) of general relativity, recasting gravitational dynamics in terms of differential forms, Pfaffian systems, and Cartan connections. It shows how the conformal structure and light-cone data, captured by a scalar function on a six-dimensional bundle, determine the spacetime metric via a normal conformal Cartan framework and metricity conditions linked to torsion. Central to the discussion are the Weyl curvature, Bach tensor, and the broader role of the Cartan connection in encoding conformal gravity, with explicit treatment of how Einstein equations emerge from these conformal structures. The work also reviews the geometric machinery of jet spaces, exterior differential systems, and principal bundles, and outlines extensions to higher dimensions and constrained generating families to address caustics, offering a holistic, holographic-like view of gravitational degrees of freedom beyond the metric formalism.

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 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, , whose level surfaces are null.

Paper Structure

This paper contains 16 sections, 5 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Historical Perspective
  3. A Modern Perspective on NSF and Its Geometrization
  4. Differential Systems and Pfaffian Systems
  5. Geometrization of Differential Equations
  6. Motivation
  7. Jet Spaces and Contact Structures
  8. Fiber Bundles and Ehresmann and Cartan Connections
  9. Principal Fiber Bundles
  10. The Ehresmann Connection
  11. Parallel Transport, Curvature, and Torsion
  12. Parallel Transport
  13. Torsion
  14. The Cartan Connection
  15. The Normal Conformal Cartan Connection SO(4,2) and Conformal Gravity
  16. ...and 1 more sections

Key Result

Theorem 1

A submanifold $N \subset M$ is an integral submanifold of the differential system defined by the ideal $\mathcal{I}$ if and only if $\mathcal{I}$ vanishes on $N$; i.e., for every $\omega \in \mathcal{I}$, we have $\omega|_N = 0$.

Theorems & Definitions (23)

  • Definition 1
  • Theorem 1
  • Definition 2
  • Definition 3
  • Theorem 2
  • Definition 4
  • Definition 5
  • Remark 1
  • Theorem 3
  • Definition 6
  • ...and 13 more