For asymptotically flat spacetimes, a conjecture by Strominger states that asymptotic BMS-supertranslations and their associated charges at past null infinity can be related to those at future null infinity via an antipodal map at spatial infinity. 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 to a 2-sphere. The cylinder touches past and future null infinities 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. 

The asymptotic structure of null and spatial infinities of asymptotically flat spacetimes plays an essential role in discussing gravitational radiation, gravitational memory effect, and conserved quantities in General Relativity. Bondi, Metzner and Sachs established that the asymptotic symmetry group for asymptotically simple spacetimes is the infinite-dimensional BMS group. Given that null infinity is divided into two sets: past null infinity and future null infinity, one can identify two independent symmetry groups: BMS- at past null infinity and BMS+ at future null infinity. Associated with these symmetries are the so-called BMS charges. A recent conjecture by Strominger suggests that the generators of BMS- and BMS+ and their associated charges are related via an antipodal reflection map near spatial infinity. To verify this matching, an analysis of the gravitational field near spatial infinity is required. This task is complicated due to the singular nature of spatial infinity for spacetimes with non-vanishing ADM mass. Different frameworks have been introduced in the literature to address this singularity, e.g., Friedrich's cylinder, Ashtekar-Hansen's hyperboloid and Ashtekar-Romano's asymptote at spatial infinity. This paper reviews the role of Friedrich's formulation of spatial infinity in the investigation of the matching of the spin-2 charges on Minkowski spacetime and in the full GR setting. 

The subject of asymptotic symmetries and charges is of interest due to its relation to the gravitational memory effect which gives a direct link to observations. In this work, we make use of Friedrich's cylinder at spatial infinity to evaluate the asymptotic charges for spin-1 and spin-2 fields at the critical sets where spatial infinity meets null infinity on Minkowski spacetimes. The spin-1 and spin-2 fields are expanded near spatial infinity and their initial data are expanded in terms of spin-weighted spherical harmonics. Given this set of initial data, we show that the asymptotic charges are, in general, not well-defined at the critical sets unless one imposes extra regularity conditions. If the initial data satisfy the regularity conditions, the asymptotic charges for the spin-1 and spin-2 fields are shown to be fully expressed in terms of the initial data on a Cauchy hypersurface and there exists a natural correspondence between the charges at past and future null infinity.  

The study of the asymptotic behaviour of fields near spatial infinity is challenging partly due to the singular nature of spatial infinity in the regular point compactification for spacetimes with non-vanishing ADM mass. In this article, we study the relation between two different formulations that address this challenge, namely; Friedrich's cylinder at spatial infinity and Ashtekar's definition of asymptotically Minkowskian spacetimes at spatial infinity that give rise to the 3-dimensional asymptote at spatial infinity. We initially consider the relation between Friedrich's cylinder at spatial infinity and Ashtekar's asymptote on Minkowski spacetime. It can be shown that the two constructions are conformally related. For general spacetimes satisfying Ashtekar's definition, the conformal factor relating the two formulations cannot be determined explicitly. However, we show that a solution to the conformal geodesic equations on the asymptote can be extended to a small neighbourhood of spatial infinity by making use of the stability theorem for ordinary differential equations. This solution can be used to construct a conformal Gaussian system in a neighbourhood of the asymptote. Associated with this conformal Gaussian system is a conformal factor that provide the relation between Friedrich's and Ashtekar's representations of spatial infinity.