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 \( \mathcal{I} \) 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 \( \mathfrak{S} \) as well as the BMS group \( \mathcal{B} \) naturally reduce to a single Poincaré group, denoted by \( {\mathfrak{p}}_{i^{{}^{\circ}}} \) 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 \( {\Psi}_1^{{}^{\circ}} \) diverges in the distant past along \( \mathcal{I} \)+. 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 \( \mathcal{I} \)+ and corresponding flux across \( \mathcal{I} \)+ to the past of C, when the quantities refer to the preferred Poincaré subgroup \( {\mathfrak{p}}_{i^{{}^{\circ}}} \).
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Acknowledgments
We thank Kartik Prabhu for discussions, Amitabha Veermani for his comments on the manuscript, and Gabriele Veneziano for motivating us to write up these results. AA also thanks Stephen Hawking for a discussion in which he suggested that the asymptotic symmetry group for isolated systems in general relativity should be the Poincaré group. This work was supported in part by the Eberly and Atherton research funds of Penn State and the Distinguished Visiting Research Chair program of the Perimeter Institute. N.K. is supported by the Natural Sciences and Engineering Council of Canada.
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Ashtekar, A., Khera, N. Unified treatment of null and spatial infinity III: asymptotically minkowski space-times. J. High Energ. Phys. 2024, 210 (2024). https://doi.org/10.1007/JHEP02(2024)210
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DOI: https://doi.org/10.1007/JHEP02(2024)210