Solutions in the \(2+1\) Null Surface Formulation
The null surface formulation of general relativity (NSF) differs from the standard approach by featuring a function \(Z\), describing families of null surfaces, as the prominent variable, rather than the metric tensor. It is possible to reproduce the metric, to within a conformal factor, by using \(Z\) (entering through its third derivative, which is denoted by \(\varLambda \)) and an auxiliary function \(\varOmega \). The functions \(\varLambda \) and \(\varOmega \) depend upon the spacetime coordinates, which are usually introduced in a manner that is convenient for the null surfaces, and also upon an additional angular variable. A brief summary of the (\(2+1\))-dimensional null surface formulation is presented, together with the NSF field equations for \(\varLambda \) and \(\varOmega \). A few special solutions are found and the properties of one of them are explored in detail.
This work was supported by the Mount Saint Vincent University Dean of Arts and Science Travel Fund. Discussions with Dr. Ted Newman and Dr. Simonetta Frittelli during the authors’ visits to Pittsburgh are gratefully acknowledged.
- 6.Tanimoto, M. : On the null surface formalism—formulation in three dimensions and gauge freedom, ArXiv:e-prints arXiv:gr-qc/9703003 (1997)
- 8.Harriott, T., Williams, J.: Light cone cut solution in the \(2+1\) null surface formulation. In: Damour, T., Jantzen, R., Ruffini, R. (eds.) Proceedings of the Twelfth Marcel Grossmann Meeting on General Relativity, vol. 12, pp. 1896–1898. World Scientific, Singapore; Hackensack, NJ (2012). doi: 10.1142/9789814374552_0355