THE VACUUM EINSTEIN EQUATIONS VIA HOLONOMY AROUND CLOSED LOOPS ON CHARACTERISTIC SURFACES

We reformulate the standard local equations of general relativity for asymptotically flat space-times in terms of two non-local quantities, the holonomy H around certain closed null loops on characteristic surf aces and the light cone cut function Z, which describes the intersecti on of the future null cones from arbitrary space-time points, with fut ure null infinity. We obtain a set of differential equations for H and Z equivalent to the vacuum Einstein equations. By finding an algebrai c relation between H and Z and integrating a linear o.d.e. these equat ions are reduced to just two coupled equations: an integro-differentia l equation for Z which yields the conformal structure of the underlyin g space-time and a linear differential equation for the ''vacuum'' con formal factor. These equations, which apply to all vacuum asymptotical ly flat space-times are however lengthy and complicated. They neverthe less are amenable to an attractive perturbative scheme which has Minko wski space as a zeroth order solution.