BMS-supertranslation charges at the critical sets of null infinity
Mariem Magdy, Kartik Prabhu, Juan A. Valiente Kroon
TL;DR
The paper uses Friedrich's spatial infinity framework to examine the Strominger antipodal matching of BMS supertranslation charges, showing that generic initial data yield ill-defined charges at the critical sets unless a regularity condition is satisfied. By enforcing a parity condition on the freely specifiable initial data function $oldsymbol{\xi}$ on $ olinebreak olinebreak $, the BMS charges at the critical sets become well-defined and entirely determined by initial data, with an explicit antipodal relation between $\ I^+$ and $\nI^-$. The regularity condition effectively removes odd-parity multipoles for $l\ge2$, leaving only even-parity contributions, and ties the matching to a geometric regularity at spatial infinity rather than a purely dynamical feature. The analysis relies on a spinor formulation of the extended conformal field equations, a careful NP-to-F gauge transformation, and transport along conformal geodesics to relate initial data to charges at $\mathscr{I}^{\pm}$, establishing a concrete link between initial-data regularity and asymptotic charge conservation with global significance for gravitational memory and soft theorems.
Abstract
For asymptotically flat spacetimes, a conjecture by Strominger states that asymptotic BMS-supertranslations and their associated charges at past null infinity $\mathscr{I}^{-}$ can be related to those at future null infinity $\mathscr{I}^{+}$ via an antipodal map at spatial infinity $i^{0}$. We analyse the validity of this conjecture using Friedrich's formulation of spatial infinity, which gives rise to a regular initial value problem for the conformal field equations at spatial infinity. A central structure in this analysis is the cylinder at spatial infinity representing a blow-up of the standard spatial infinity point $i^{0}$ to a 2-sphere. The cylinder touches past and future null infinities $\mathscr{I}^{\pm}$ at the critical sets. We show that for a generic class of asymptotically Euclidean and regular initial data, BMS-supertranslation charges are not well-defined at the critical sets unless the initial data satisfies an extra regularity condition. We also show that given initial data that satisfy the regularity condition, BMS-supertranslation charges at the critical sets are fully determined by the initial data and that the relation between the charges at past null infinity and those at future null infinity directly follows from our regularity condition.