The Eikonal equation in asymptotically flat space-times

In an arbitrary Lorentzian manifold we provide a description for the constr uction of null surfaces and their associated singularities, via solutions o f the Eikonal equation. In particular, we study the singularities of the pa st light-cones from points on null infinity, the future light-cones from ar bitrary interior points and the intersection of these with null infinity an d unifying relationships between the different singularities. The starting point for this work is the assumption of a known family of solutions to the Eikonal equation. The work is based on the standard theory of singularitie s of smooth maps by Arnold and his colleagues. Though the work is intended to stand on its own, it can be thought of as being closely related to the r ecently developed null surface reformulation of GR. (C) 1999 American Insti tute of Physics. [S0022-2488(99)01302-X].