On the dynamics of characteristic surfaces

Abstract

We formulate the vacuum Einstein equations as differential equations for two functions, one complex and one real on a six-dimensional manifold, ℳ×S2, with ℳ eventually becoming the space–time and the S2 becoming the sphere of null directions over ℳ. At the start there is no other further structure available: the structure arising from the two functions. The complex function, referred to as Λ[ℳ×S2], encodes information about a sphere’s worth of surfaces through each point of ℳ. From knowledge of Λ one can define a second rank tensor on ℳ which can be interpreted as a conformal metric, so that the ‘‘surfaces’’ are automatically null or characteristics of this conformal metric. The real function, Ω, plays the role of a conformal factor: it converts the conformal metric into a vacuum Einstein metric. Locally, all Einstein metrics can be obtained in this manner. In this work, we fully develop this ‘‘null surface version of general relativity (GR):’’ we display, discuss and analyze the equations, we show that many of the usual geometric quantities of GR (e.g., the Weyl and Ricci tensors, the optical parameters, etc.) can be easily expressed in terms of the Λ and Ω, we study the gauge freedom and develop a perturbation theory. To conclude, we speculate on the significance and possible classical and quantum uses of this formulation.