Null surfaces formulation in 3D

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.

arXiv comment15 pp., 4 figures, to appear in J. Math. Phys

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• Two-dimensional Riemannian and Lorentzian geometries from second order ODEs
• Cartan's equivalence method and null coframes in General Relativity
• On the Riemannian and Einstein-Weyl Geometry in Theory of the Second Order Ordinary Differential Equations
• The Riemann Extensions in Theory of Ordinary Differential Equations and their Applications