Spatial and null infinity via advanced and retarded conformal factors

Abstract

A new approach to space-time asymptotics is presented, refining Penrose's idea of conformal transformations with infinity represented by the conformal boundary of space-time. It is proposed that the Penrose conformal factor be a product of advanced and retarded conformal factors, which asymptotically relate physical and conformal null coordinates and vanish at future and past null infinity respectively. A refined definition of asymptotic flatness at both spatial and null infinity is given, including that the conformal boundary is locally a light cone, with spatial infinity as the vertex. It is shown how to choose the conformal factors so that this asymptotic light cone is locally a metric light cone. The theory is implemented in the spin-coefficient (or null-tetrad) formalism by a joint transformation of the spin-metric and spin-basis (or metric and tetrad). Asymptotic regularity conditions are proposed, based on the conformal boundary locally being a smoothly embedded metric light cone. These conditions ensure that the Bondi-Sachs energy-flux integrals of ingoing and outgoing gravitational radiation decay at spatial infinity such that the total radiated energy is finite, and that the Bondi-Sachs energy-momentum has a unique limit at spatial infinity, coinciding with the uniquely rendered ADM energy-momentum.