Unified Treatment of null and Spatial Infinity IV: Angular Momentum at Null and Spatial Infinity

Abstract

In a companion paper we introduced the notion of asymptotically Minkowski spacetimes. These space-times are asymptotically flat at both null and spatial infinity, and furthermore there is a harmonious matching of limits of certain fields as one approaches $i^{\circ }$ in null and space-like directions. These matching conditions are quite weak but suffice to reduce the asymptotic symmetry group to a Poincaré group ${\mathfrak{p}}_{i^{\circ }}$. Restriction of ${\mathfrak{p}}_{i^{\circ }}$ to future null infinity ${\mathcal{I}}^{+}$ yields the canonical Poincaré subgroup ${\mathfrak{p}}_{i^{\circ }}^{\mathrm{bms}}$ of the BMS group $\mathfrak{B}$ selected in the companion paper and its restriction to spatial infinity $i^{\circ }$ gives the canonical subgroup ${\mathfrak{p}}_{i^{\circ }}^{\mathrm{spi}}$ of the Spi group $\mathfrak{S}$ there. As a result, one can meaningfully compare angular momentum that has been defined at $i^{\circ }$ using ${\mathfrak{p}}_{i^{\circ }}^{\mathrm{spi}}$ with that defined on ${\mathcal{I}}^{+}$ using ${\mathfrak{p}}_{i^{\circ }}^{\mathrm{bms}}$. We show that the angular momentum charge at $i^{\circ }$ equals the sum of the angular momentum charge at any 2-sphere cross-section $S$ of ${\mathcal{I}}^{+}$ and the total flux of angular momentum radiated across the portion of ${\mathcal{I}}^{+}$ to the past of $S$. In general the balance law holds only when angular momentum refers to $\mathrm{SO}(3)$ subgroups of the Poincaré group ${\mathfrak{p}}_{i^{\circ }}$.

arXiv commentVersion to appear in JHEP. Slightly edited to improve the presentation. Four references added. 30 pages, 1 figure

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Cited by in corpus (8)

• ADM, BMS, and some puzzling interconnections
• Projective and Carrollian geometry at time/space-like infinity on projectively compact Ricci flat Einstein manifolds
• Supertranslation ambiguity in post-Minkowskian expansion
• On supertranslation invariant Lorentz charges
• Ti and Spi, Carrollian extended boundaries at timelike and spatial infinity
• Quasi-local spin-angular momentum and the construction of axial vector fields
• Supertranslations at Spatial and Timelike Infinities in the First-Order Formalism
• Note on post-Minkowskian expansion and Bondi coordinates