A spinor reformulation of the null surface treatment of GR

Abstract

We begin with a four-dimensional manifold, M, that possesses a two parameter family of (local) foliations by three-surfaces, with the two parameters being the coordinates on the sphere of directions at each point of the manifold expressed via homogeneous coordinates πA and πA′. By then requiring that each foliation (of the two parameter set of foliations) be a one parameter family of null surfaces for some (as yet unkown) conformal Lorentzian metric—we derive an explicit expression, in terms of the foliation description, for this conformal metric. We then show (1) how a conformal factor can be chosen to convert the conformal metric into a metric and (2) how to impose on the foliation and conformal factor conditions so that the metric satisfies the vacuum Einstein equations. The material described here is very much connected to the null surface formulation (NSF) of GR developed earlier. The advantages of the present formulation are that one can much more easily see the logical structure of the NSF, one can calculate with much greater ease and finally it allows [because of the use of (spinor) index calculus] generalizations of the NSF so that the study of the evolutionary development of the null surface singularities (caustics, etc.) can be developed.