Cauchy problems for the conformal vacuum field equations in general relativity

Abstract

Cauchy problems for Einstein's conformal vacuum field equations are reduced to Cauchy problems for first order quasilinear symmetric hyperbolic systems. The “hyperboloidal initial value” problem, where Cauchy data are given on a spacelike hypersurface which intersects past null infinity at a spacelike two-surface, is discussed and translated into the conformally related picture. It is shown that for conformal hyperboloidal initial data of classH S,s≧4, there is a unique (up to questions of extensibility) development which is a solution of the conformal vacuum field equations of classH S. It provides a solution of Einstein's vacuum field equations which has a smooth structure at past null infinity.