Cartan's equivalence method and null coframes in General Relativity

TL;DR

This paper applies Cartan's equivalence method to derive conformal geometries from differential equations, explicitly constructs null tetrads and conformal groups in Lorentzian metrics, and links these to the Null Surface Formulation of General Relativity.

Contribution

It introduces a first-principles approach to conformal geometry in GR using Cartan's method, revealing the role of the Wünschmann invariant in the structure of null connections.

Findings

Constructed null tetrads for Lorentzian metrics.

Derived the conformal group in 3 and 4 dimensions.

Connected the geometric construction to the Null Surface Formulation of GR.

Abstract

Using Cartan's equivalence method for point transformations we obtain from first principles the conformal geometry associated with third order ODEs and a special class of PDEs in two dimensions. We explicitly construct the null tetrads of a family of Lorentzian metrics, the conformal group in three and four dimensions and the so called normal metric connection. A special feature of this connection is that the non vanishing components of its torsion depend on one relative invariant, the (generalized) W\"unschmann Invariant. We show that the above mentioned construction naturally contains the Null Surface Formulation of General Relativity.