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

A bstract 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 $$ \mathcal{I} $$ 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 $$ \mathfrak{S} $$ S as well as the BMS group $$ \mathcal{B} $$ B naturally reduce to a single Poincaré group, denoted by $$ {\mathfrak{p}}_{i^{{}^{\circ}}} $$ p i ° 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 $$ {\Psi}_1^{{}^{\circ}} $$ Ψ 1 ° diverges in the distant past along $$ \mathcal{I} $$ 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 $$ \mathcal{I} $$ I + and corresponding flux across $$ \mathcal{I} $$ I + to the past of C , when the quantities refer to the preferred Poincaré subgroup $$ {\mathfrak{p}}_{i^{{}^{\circ}}} $$ p i ° .

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