Unified treatment of null and spatial infinity III: asymptotically minkowski space-times

Abstract

Abstract The Spi framework provides a 4-dimensional approach to investigate the asymptotic properties of gravitational fields as one recedes from isolated systems in any space-like direction, without reference to a Cauchy surface [1]. It is well suited to unify descriptions at null and spatial infinity because I $$ \mathcal{I} $$ arises as the null cone of i°. The goal of this work is to complete this task by introducing a natural extension of the asymptotic conditions at null and spatial infinity of [2], by ‘gluing’ the two descriptions appropriately. Space-times satisfying these conditions are asymptotically flat in both regimes and thus represent isolated gravitating systems. They will be said to be Asymptotically Minkowskian at i°. We show that in these space-times the Spi group S $$ \mathfrak{S} $$ as well as the BMS group B $$ \mathcal{B} $$ naturally reduce to a single Poincaré group, denoted by p i ° $$ {\mathfrak{p}}_{i^{{}^{\circ}}} $$ to highlight the fact that it arises from the gluing procedure at i°. The asymptotic conditions are sufficiently weak to allow for the possibility that the Newman-Penrose component Ψ 1 ° $$ {\Psi}_1^{{}^{\circ}} $$ diverges in the distant past along I $$ \mathcal{I} $$ +. This can occur in astrophysical sources that are not asymptotically stationary in the past, e.g. in scattering situations. Nonetheless, as we show in the companion paper [5], the energy momentum and angular momentum defined at i° equals the sum of that defined at a cross-section C of I $$ \mathcal{I} $$ + and corresponding flux across I $$ \mathcal{I} $$ + to the past of C, when the quantities refer to the preferred Poincaré subgroup p i ° $$ {\mathfrak{p}}_{i^{{}^{\circ}}} $$ .

Keywords

• Classical Theories of Gravity
• Differential and Algebraic Geometry
• Scattering Amplitudes
• Space-Time Symmetries