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.