The Eikonal Equation in Flat Space: Null Surfaces and Their Singularities I

TL;DR

This paper investigates the properties, singularities, and decomposition of null surfaces defined by solutions to the eikonal equation in Minkowski space, with applications to null infinity and wavefront analysis.

Contribution

It introduces a method for constructing solutions to the flat-space eikonal equation and analyzes their singularities using Arnold's theory, including global null surfaces and their caustics.

Findings

Constructed global asymptotically spherical null surfaces from shearing cuts of null infinity.

Analyzed caustics and singularities of null surfaces and wavefronts.

Applied Arnold's generating families to handle self-intersections and non-smooth regions.

Abstract

The level surfaces of solutions to the eikonal equation define null or characteristic surfaces. In this note we study, in Minkowski space, properties of these surfaces. In particular we are interested both in the singularities of these ``surfaces'' (which can in general self-intersect and be only piece-wise smooth) and in the decomposition of the null surfaces into a one parameter family of two-dimensional wavefronts which can also have self-intersections and singularities. We first review a beautiful method for constructing the general solution to the flat-space eikonal equation; it allows for solutions either from arbitrary Cauchy data or for time independent (stationary) solutions of the form S=t-S_{0}(x,y,z). We then apply this method to obtain global, asymptotically spherical, null surfaces that are associated with shearing ("bad") two-dimensional cuts of null infinity; the surfaces are defined from the normal rays to the cut. This is followed by a study of the caustics and singularities of these surfaces and those of their associated wavefronts. We then treat the same set of issues from an alternative point of view, namely from Arnold's theory of generating families. This treatment allows one to deal (parametrically) with the regions of self-intersection and non-smoothness of the null surfaces, regions which are difficult to treat otherwise. Finally we generalize the analysis of the singularities to families of solutions of the eikonal equation.