Solution for the 3+1 null-surface formulation with a power-law spacetime

Abstract

The null-surface formulation (NSF) of general relativity differs from the usual approach by treating the spacetime metric as a derivable quantity instead of regarding it as fundamental. The NSF has two mathematically equivalent interpretations: (a) Light rays leave a spacetime point and intersect null-infinity to form a ‘light-cone cut,’ which encodes the properties of the spacetime; (b) At null-infinity, angular coordinates (Bondi coordinates) label past light cones. Being null surfaces, these past light cones will satisfy the NSF field equations, the solution of which will provide a description of spacetime. In an earlier work, the present authors gave an exact solution for the NSF field equations in 2+1 dimensions, showing how the solution directly linked the two NSF interpretations. The present paper expands on that work by constructing the corresponding (3+1)-dimensional solution and then, as in 2+1 dimensions, linking the two interpretations so as to illustrate their equivalence. The functions relevant to the 3+1 NSF are calculated, and the field equations are shown to be satisfied. This is the first time that a nontrivial (3+1)-dimensional NSF solution has been found and its properties examined.