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

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