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 p i ° $$ {\mathfrak{p}}_{i{}^{\circ}} $$ . Restriction of p i ° $$ {\mathfrak{p}}_{i{}^{\circ}} $$ to future null infinity I + $$ {\mathcal{I}}^{+} $$ yields the canonical Poincaré subgroup p i ° bms $$ {\mathfrak{p}}_{i{}^{\circ}}^{\textrm{bms}} $$ of the BMS group B $$ \mathfrak{B} $$ selected in [2, 3] and that its restriction to spatial infinity i°, the canonical subgroup p i ° spi $$ {\mathfrak{p}}_{i{}^{\circ}}^{\textrm{spi}} $$ of the Spi group S $$ \mathfrak{S} $$ selected in [4, 5]. As a result, one can meaningfully compare angular momentum that has been defined at i° using p i ° spi $$ {\mathfrak{p}}_{i{}^{\circ}}^{\textrm{spi}} $$ with that defined on I + $$ {\mathcal{I}}^{+} $$ using p i ° bms $$ {\mathfrak{p}}_{i{}^{\circ}}^{\textrm{bms}} $$ . 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 I + $$ {\mathcal{I}}^{+} $$ and the total flux of angular momentum radiated across the portion of I + $$ {\mathcal{I}}^{+} $$ to the past of S. In general the balance law holds only when angular momentum refers to SO(3) subgroups of the Poincaré group p i ° $$ {\mathfrak{p}}_{i{}^{\circ}} $$ .

Keywords

• Classical Theories of Gravity
• Differential and Algebraic Geometry
• Space-Time Symmetries
• Scattering Amplitudes