  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.
