The vacuum and Bach equations in terms of light cone cuts

Abstract

Light cone cuts are the intersection of the light cones of space–time events with an initial data surface which is usually taken to be null infinity. These are determined by a ‘‘cut function,’’ a function of the space–time coordinates and the sphere of generators of null infinity. The Bach equations and the conformal vacuum equations are shown to lead to a single scalar ‘‘main’’ equation on the cut function. In simplified cases, i.e., when the Weyl curvature is either small or self-dual, this equation can be integrated to give a simple scalar conformally invariant elliptic equation on the sphere of null directions. It has the remarkable property that the general solution of the Einstein vacuum equations in these cases can be derived from the solution to this auxiliary equation in two dimensions. In general, it will not be possible to reduce it to an auxilliary equation on the sphere. Nevertheless, the main equation is still a necessary condition and it is conjectured that it is also sufficient to imply the Bach equations in general and supporting arguments are given. It is also shown how to extend the Kozameh–Newman framework to the case of light cone cuts of a spacelike Cauchy hypersurface. The analogous procedures for Yang–Mills are also discussed. A conformally invariant formalism for calculations on null infinity is described in an Appendix. This is used for all the calculations which are only described in brief form in the main text.