Structure of the gravitational field at spatial infinity. II. Asymptotically Minkowskian space–times

A new formulation is established for the study of the asymptotic structure at spatial infinity of asymptotically Minkowskian space–times. First, the concept of an asymptotically simple space–time at spatial infinity is defined. This is a (physical) space–time (ℳ,g) which can be imbedded in an unphysical space–time (ℳ̂,ĝ) with a boundary 𝒮, a C∞ metric ĝ and a C∞ scalar field Ω such that Ω=0 on 𝒮, Ω≳0 on ℳ̂−𝒮, and ĝμν + ĝμλ ĝνρ Ω‖λ Ω‖ρ=Ω−2 gμν +Ω−4 gμλ gνρ Ω;λ Ω;ρ on ℳ. Then an almost asymptotically flat space–time (AAFS) is defined as an asymptotically simple space–time for which 𝒮 is isometric to the unit timelike hyperboloid and ĝμν Ω‖μ Ω‖ν =Ω−4 gμν Ω;μΩ;ν=−1 on 𝒮. Equivalent definitions are given in terms of the existence of coordinate systems in which gμν or ĝμν have simple explicitly given forms. The group of asymptotic symmetries of (ℳ,g) is studied and is found to be isomorphic to the Lorentz group. The asymptotic behavior of an AAFS is studied. It is proven that the conformal metric g̃μν=Ω2gμν gives C̃λμρν=0, Ω−1 C̃λμρν Ω;μ =0, Ω−2 Cλμρν Ω;μ Ω;ν=0 on 𝒮.