The Eikonal equation in asymptotically flat space–times

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Abstract
In an arbitrary Lorentzian manifold we provide a description for the construction of null surfaces and their associated singularities, via solutions of the Eikonal equa- tion. In particular, we study the singularities of the past light-cones from points on null infinity, the future light-cones from arbitrary interior points and the intersection of these with null infinity and unifying relationships between the different singu- larities. 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 singularities 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 recently developed null surface reformulation of GR. © 1999 American Institute of Physics. @S0022-2488~99!01302-X#

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