Spatial and null infinity via advanced and retarded conformal factors

TL;DR

This paper introduces a refined conformal approach to space-time asymptotics, unifying null and spatial infinity through advanced and retarded conformal factors, leading to improved definitions of asymptotic flatness and finite energy fluxes.

Contribution

It proposes a new conformal factor as a product of advanced and retarded factors, refining the description of infinity in space-time and enabling better analysis of gravitational radiation.

Findings

Conformal boundary is locally a light cone with spatial infinity as the vertex.

Asymptotic regularity conditions ensure finite total radiated energy.

Bondi-Sachs and ADM energy-momenta are shown to coincide at spatial infinity.

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. Generalizing examples such as flat and Schwarzschild space-times, it is proposed that the Penrose conformal factor be a product of advanced and retarded conformal factors, which asymptotically relate physical and conformal null (light-like) coordinates and vanish at future and past null infinity respectively, with both vanishing at spatial infinity. A correspondingly 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 simple joint transformation of the spin-metric and spin-basis (or metric and tetrad). The advanced and retarded conformal factors may be used as expansion parameters near the respective null infinity, together with a dependent expansion parameter for both spatial and null infinity, essentially inverse radius. Asymptotic regularity conditions on the spin-coefficients 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.