On the Characteristic Initial Value Problem in Gravitational Theory

Abstract

The problem of determining a solution of the Einstein field equations for the gravitational field from data set on a pair of intersecting characteristic (that is, null) hypersurfaces and on their intersection Σ is considered. It is shown that by giving the conformal inner metric of each hypersurface, the inner geometry of Σ, the two mean extrinsic curvatures of Σ, and one additional extrinsic quantity for Σ, one completely determines a solution in a space-time region. In a suitable coordinate system giving the above data amounts to giving four functions of three variables, four functions of two variables, and one function of one variable; all these can be given without any constraints. In a given space time with given Σ the functions are almost, but not entirely, known functions of their arguments. All the derivations and discussions are applicable only in a sufficiently small four-dimensional region.