Stable singularities of wave-fronts in general relativity

TL;DR

This paper investigates the stable singularities of wave-fronts in general relativity using the contact manifold structure of null geodesics, providing insights into their behavior and evolution.

Contribution

It introduces a novel approach by analyzing the space of null geodesics as a contact manifold to study wave-front singularities in space-time.

Findings

Null geodesics form a contact manifold structure.

This framework effectively describes stable wave-front singularities.

Provides a geometric method for analyzing wave-front evolution.

Abstract

A wave-front in a space-time $\cal M$ is a family of null geodesics orthogonal to a smooth spacelike two-surface in $\cal M$; it is of some interest to know how a wave-front can fail to be a smoothly immersed surface in $\cal M$. In this paper we see that the space of null geodesics $\cal N$ of $\cal M$, considered as a contact manifold, provides a natural setting for an efficient study of the stable singularities arising in the time evolution of wave-fronts.