GR via Characteristic Surfaces

TL;DR

This paper introduces a reformulation of Einstein's equations using characteristic surfaces on a four-manifold, constructing Einstein metrics from families of null surfaces parameterized over space-time.

Contribution

It presents a novel approach to Einstein's equations by expressing them as equations for surface families on a four-manifold, linking conformal factors and null surfaces.

Findings

Reformulation of Einstein equations as surface equations

Construction of Einstein metrics from surface families

Use of characteristic null surfaces in metric formulation

Abstract

We reformulate the Einstein equations as equations for families of surfaces on a four-manifold. These surfaces eventually become characteristic surfaces for an Einstein metric (with or without sources). In particular they are formulated in terms of two functions on R4xS2, i.e. the sphere bundle over space-time, - one of the functions playing the role of a conformal factor for a family of associated conformal metrics, the other function describing an S2's worth of surfaces at each space-time point. It is from these families of surfaces themselves that the conformal metric - conformal to an Einstein metric - is constructed; the conformal factor turns them into Einstein metrics. The surfaces are null surfaces with respect to this metric.