Null Surfaces and Legendre Submanifolds

TL;DR

This paper explores the geometric structure of null surfaces in general relativity, showing that a key variable acts as a generating function for Legendre submanifolds on an energy surface, with analysis of caustic behavior.

Contribution

It demonstrates that the main variable in the Null Surface Formulation is a generating function for constrained Lagrange submanifolds on the energy surface, providing new geometric insights.

Findings

Z is the generating function of a constrained Lagrange submanifold

Level surfaces Z=const. are Legendre submanifolds on the energy surface

Analysis of Z's behavior at caustic points and its generalization

Abstract

It is shown that the main variable Z of the Null Surface Formulation of GR is the generating function of a constrained Lagrange submanifold that lives on the energy surface H=0 and that its level surfaces Z=const. are Legendre submanifolds on that energy surface. The behaviour of the variable Z at the caustic points is analysed and a genralization of this variable is discussed.