Linearized Einstein theory via null surfaces

TL;DR

This paper explores the linearized form of the null surface reformulation of General Relativity, providing insights into its mathematical structure and introducing solution and perturbation schemes.

Contribution

It introduces a linearization of the null surface formulation of GR, clarifies mathematical subtleties, and proposes simple solution and perturbation methods.

Findings

Comparison with standard linear GR enhances understanding.

A simple solution generating scheme is developed.

Initial steps for a perturbation scheme are outlined.

Abstract

Recently there has been developed a reformulation of General Relativity - referred to as {\it the null surface version of GR} - where instead of the metric field as the basic variable of the theory, families of three-surfaces in a four-manifold become basic. From these surfaces themselves, a conformal metric, conformal to an Einstein metric, can be constructed. A choice of conformal factor turns them into Einstein metrics. The surfaces are then automatically characteristic surfaces of this metric. In the present paper we explore the linearization of this {\it null surface theory} and compare it with the standard linear GR. This allows a better understanding of many of the subtle mathematical issues and sheds light on some of the obscure points of the null surface theory. It furthermore permits a very simple solution generating scheme for the linear theory and the beginning of a perturbation scheme for the full theory.