Unified treatment of null and spatial infinity IV: angular momentum at null and spatial infinity

Abstract

Abstract In a companion paper [1] 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° in null and space-like directions. These matching conditions are quite weak but suffice to reduce the asymptotic symmetry group to a Poincaré group $$ {\mathfrak{p}}_{i{}^{\circ}} $$ p i ° . Restriction of $$ {\mathfrak{p}}_{i{}^{\circ}} $$ p i ° to future null infinity $$ {\mathcal{I}}^{+} $$ I + yields the canonical Poincaré subgroup $$ {\mathfrak{p}}_{i{}^{\circ}}^{\textrm{bms}} $$ p i ° bms of the BMS group $$ \mathfrak{B} $$ B selected in [2, 3] and that its restriction to spatial infinity i°, the canonical subgroup $$ {\mathfrak{p}}_{i{}^{\circ}}^{\textrm{spi}} $$ p i ° spi of the Spi group $$ \mathfrak{S} $$ S selected in [4, 5]. As a result, one can meaningfully compare angular momentum that has been defined at i° using $$ {\mathfrak{p}}_{i{}^{\circ}}^{\textrm{spi}} $$ p i ° spi with that defined on $$ {\mathcal{I}}^{+} $$ I + using $$ {\mathfrak{p}}_{i{}^{\circ}}^{\textrm{bms}} $$ p i ° bms . We show that the angular momentum charge at i° equals the sum of the angular momentum charge at any 2-sphere cross-section S of $$ {\mathcal{I}}^{+} $$ I + and the total flux of angular momentum radiated across the portion of $$ {\mathcal{I}}^{+} $$ I + to the past of S. In general the balance law holds only when angular momentum refers to SO(3) subgroups of the Poincaré group $$ {\mathfrak{p}}_{i{}^{\circ}} $$ p i ° .