A definition of asymptotically Minkowskian space–times

A definition of asymptotic flatness is given without the use of Einstein’s equations. For this we examine (a) the possible conditions on ℐ+ and their consequences and relationships, and (b) the coordinate transformations which preserve certain forms of the unphysical metric g̃ on a neighborhood of ℐ+. Roughly, a space–time (ℳ, g) is called asymptotically Minkowskian at future null infinity if it is weakly asymptotically simple, ℐ+ is isometric to ℐ+M (the ℐ+ of the Minkowskian space–time) and (a) Ω;μν=0, Ω−2Ω;μΩ;μ=−1, C̃λμρν=0 on ℐ+ or, equivalently, (b) Ω−2Ω;μΩ;μ=−1, Rμ ν=O (Ω2), Ω;[λRμ] ν +ΩR[λ ν;μ]=O (Ω4) on ℐ+. An equivalent definition is given based on the existence of a coordinate system in which, on a neighborhood of ℐ+, g̃ behaves as the conformal metric of Minkowski’s space–time and can be written in an explicitly given form.