Timelike infinity

Abstract

A new formulation is presented for analyzing the structure of a space–time at timelike infinity. An asymptotically simple space–time is defined as a space–time (ℳ,g) which can be imbedded in a space (ℳ̄,ĝ) with boundary 𝒯,a C∞ metric ĝ and a C∞ scalar field Ω, such that Ω = 0 on 𝒯, Ω≳0 on ℳ−𝒯 andĝμν−ĝμλĝνρΩ‖λ Ω‖ρ = Ω−2gμν−Ω−4gμλ gνρΩ;λΩ;ρ in a neighborhood of 𝒯. Demanding that 𝒯 = 𝒯−U 𝒯+, where each one of 𝒯− and 𝒯+ is isometric to the unit spacelike hyperboloid, and ĝμν Ω‖μ Ω‖ν = Ω−4gμν Ω;μ Ω;ν = 1 on 𝒯, we have an almost asymptotically flat (at timelike infinity) space–time. The group of asymptotic symmetries of (ℳ,g) at timelike infinity is found to be isomorphic to the Lorentz group. Some properties of the space–time near 𝒯 are shown.