The Vacuum Einstein Equations via Holonomy around Closed Loops on Characteristic Surfaces

Abstract

We reformulate the standard local equations of general relativity for asymptotically flat spacetimes in terms of two non-local quantities, the holonomy $H$ around certain closed null loops on characteristic surfaces and the light cone cut function $Z$, which describes the intersection of the future null cones from arbitrary spacetime points, with future null infinity. We obtain a set of differential equations for $H$ and $Z$ equivalent to the vacuum Einstein equations. By finding an algebraic relation between $H$ and $Z$ this set of equations is reduced to just two coupled equations: an integro-differential equation for $Z$ which yields the conformal structure of the underlying spacetime and a linear differential equation for the ``vacuum'' conformal factor. These equations, which apply to all vacuum asymptotically flat spacetimes, are however lengthy and complicated and we do not yet know of any solution generating technique. They nevertheless are amenable to an attractive perturbative scheme which has Minkowski space as a zeroth order solution.