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

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 ° .

Carrollian hydrodynamics from symmetries

“…One expects, for instance, that the stretched horizon framework would be useful to describe null and spacelike infinity in the same token. On that front, it would be interesting to understand if our structure connects with the works by Ashtekar and Khera [131,132], which introduced a unified framework to describe null and spatial infinity.…”

“…It also implies that the variational coefficients (132) enter the decomposition of the metric variation as follows…”