The Eikonal equation in flat space: Null surfaces and their singularities.I

The level surfaces of solutions to the eikonal equation define null or char acteristic surfaces. In this paper we study, in Minkowski space, properties of these surfaces. In particular, we are interested both in the singularit ies of these "surfaces'' (which can, in general, self-intersect and be only piecewise smooth) and in the decomposition of the null surfaces into a one -parameter family of two-dimensional wavefronts which can also have self-in tersections and singularities. We first review a beautiful method for const ructing the general solution to the flat-space eikonal equation; it allows for solutions either from arbitrary Cauchy data or for time-independent (st ationary) solutions of the form S = t - S-0(x, y, z). We then apply this me thod to obtain global, asymptotically spherical, null surfaces that are ass ociated with shearing ("bad'') two-dimensional cuts of null infinity; the s urfaces 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 thei r associated wavefronts. We then treat the same set of issues from an alter native point of view, namely from Arnold's theory of generating families. T his treatment allows one to deal (parametrically) with the regions of self- intersection and nonsmoothness of the null surfaces, regions which are diff icult to treat otherwise. Finally, we generalize the analysis of the singul arities to the case of families of characteristic surfaces. (C) 1999 Americ an Institute of Physics. [S0022-2488(99)00801-4].