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

TL;DR

This paper reformulates Einstein's vacuum equations for asymptotically flat spacetimes using holonomy and light cone cut functions, leading to a new set of coupled equations that describe the spacetime's conformal structure.

Contribution

It introduces a novel non-local formulation of Einstein's equations in terms of holonomy and light cone cuts, reducing the problem to two coupled equations for the conformal structure.

Findings

Derived differential equations for holonomy and light cone cut functions.

Reduced equations to a coupled system involving conformal structure and factor.

Proposed a perturbative scheme around Minkowski space.

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.