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

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 ${\mathfrak{p}}_{i^{\circ }}^{\mathrm{b}\mathrm{m}\mathrm{s}}$ of the BMS group $\mathfrak{B}$ selected in the companion paper and its restriction to spatial infinity $i^{\circ }$ gives the canonical subgroup ${\mathfrak{p}}_{i^{\circ }}^{\mathrm{s}\mathrm{p}\mathrm{i}}$ of the Spi group $\mathfrak{S}$ there. As a result, one can meaningfully compare angular momentum that has been defined at $i^{\circ }$ using ${\mathfrak{p}}_{i^{\circ }}^{\mathrm{s}\mathrm{p}\mathrm{i}}$ with that defined on ${\mathcal{I}}^{+}$ using ${\mathfrak{p}}_{i^{\circ }}^{\mathrm{b}\mathrm{m}\mathrm{s}}$. 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 $\mathrm{S}\mathrm{O}(3)$ subgroups of the Poincaré group ${\mathfrak{p}}_{i^{\circ }}$.