Null surfaces formulation in three dimensions

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 contains the same geometrical meaning as the four-dimensional 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 are 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.