Solution for the null-surface formulation in 2+1 dimensions with radiation source

Abstract

The null-surface formulation (NSF) of general relativity has three coupled and highly nonlinear eld equations whose solution determines (a family of) null surfaces indicated by the variable u. This variable is one of a set of intrinsic coordinates that are defined in terms of the surfaces. The first of the three field equations is called the main metricity condition, and it is by far the most complicated. This work considers the NSF in 2+1 dimensions and presents a solution that is founded on the following strategy:
 Simplify the main metricity condition by requiring the uu-component of the metric tensor to be zero and then search for an additively separable solution formed from a sum of separate terms, each term being linked with a term in a certain differential operator ∂ so as to cause a convenient cancellation. The main motivation for this approach is that it may suggest how to find a solution in higher dimensions, where the role of ∂ is played by the eth operator of Newman and Penrose.