Null surfaces formulation in 3D

TL;DR

This paper extends the Null Surface Formulation of General Relativity to 2+1 dimensions, simplifying the equations and analyzing null surfaces in asymptotically flat spacetimes.

Contribution

It develops a 3D version of the Null Surface Formulation, providing new approaches to derive field equations and analyzing null surfaces in lower-dimensional gravity.

Findings

Derived 3D field equations with geometric meaning

Constructed null surface families in asymptotically flat spacetimes

Analyzed a representative example illustrating generic features

Abstract

The Null Surface Formulation of General Relativity is developed for 2+1 dimensional gravity. The geometrical meaning of the metricity condition is analyzed and two approaches to the derivation of the field equations are presented. One method makes explicit use of the conformal factor while the other only uses conformal information. The resulting set of equations contain the same geometrical meaning as the 4-D formulation without the technical complexities of the higher dimensional analog. A canonical family of null surfaces in this formulation, the light cone cuts of null infinity, are constructed on asymptotically flat space times and some of their kinematical aspects discussed. A particular example, which nevertheless contains most of the generic features is explicitly constructed and analyzed, revealing the behavior predicted in the full theory.