BMS-supertranslation charges at the critical sets of null infinity

TL;DR

This paper investigates the relationship between BMS-supertranslation charges at past and future null infinity in asymptotically flat spacetimes, revealing that charges are well-defined and related under certain regularity conditions at spatial infinity.

Contribution

It demonstrates that BMS-supertranslation charges at critical sets are well-defined only for regular initial data and establishes their direct relation via a regularity condition.

Findings

Charges are not well-defined without extra regularity conditions.

Charges at critical sets are determined by initial data.

Relation between charges at past and future null infinity follows from regularity.

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.