A spinor reformulation of the null surface treatment of GR

We begin with a four-dimensional manifold, M, that possesses a two paramete r family of (local) foliations by three-surfaces, with the two parameters b eing the coordinates on the sphere of directions at each point of the manif old expressed via homogeneous coordinates pi(A) and pi(A'). By then requiri ng that each foliation (of the two parameter set of foliations) be a one pa rameter family of null surfaces for some (as yet unkown) conformal Lorentzi an metric-we derive an explicit expression, in terms of the foliation descr iption, 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 ver y much connected to the null surface formulation (NSF) of GR developed earl ier. The advantages of the present formulation are that one can much more e asily see the logical structure of the NSF, one can calculate with much gre ater ease and finally it allows [because of the use of (spinor) index calcu lus] generalizations of the NSF so that the study of the evolutionary devel opment of the null surface singularities (caustics, etc.) can be developed. (C) 2000 American Institute of Physics. [S0022-2488(00)04609-0].