Light-cone cuts and metricity conditions for a power-law spacetime in 2+1 and 3+1 dimensions

Abstract

The null-surface formulation (NSF) of general relativity differs markedly from the conventional approach. The conventional approach to general relativity is concerned with local fields such as the metric, whereas the NSF focuses on surfaces. The NSF has two distinct but mathematically equivalent interpretations: (a) Future-directed light rays leave a spacetime point and intersect future null-infinity. The resulting surface, known as a light-cone cut, encodes the properties of the spacetime; (b) The angular coordinates (Bondi coordinates) of null-infinity are used to label past light cones, thereby producing a family of null surfaces. These will satisfy the NSF field equations and a solution of these equations provides a description of spacetime. This paper features a new exact solution that, for the first time, directly links the two interpretations, thereby illustrating both approaches and demonstrating their equivalence. The solution and its properties are first explored in 2+1 dimensions, after which, the generalization to 3+1 is outlined.