A geometric framework for conservation laws along null hypersurfaces

TL;DR

This paper develops a geometric framework for understanding conservation laws specifically along null hypersurfaces in Lorentzian manifolds, focusing on the wave operator's behavior in this context.

Contribution

It introduces a novel geometric approach to conservation laws on null hypersurfaces, extending the theory to the wave operator on Lorentzian manifolds.

Findings

Framework applies to arbitrary PDEs on null hypersurfaces

Provides new insights into wave operator behavior

Enhances understanding of conservation laws in general relativity

Abstract

We use the general theory of local conservation laws for arbitrary partial differential equations to provide a geometric framework for conservation laws on characteristic null hypersurfaces. The operator of interest is the wave operator on general four-dimensional Lorentzian manifolds restricted on a null hypersurface.