Charge and Antipodal Matching across Spatial Infinity

We derive the antipodal matching relations used to demonstrate the equivalence between soft graviton theorems and BMS charge conservation across spatial infinity. To this end we provide a precise map between Bondi data at null infinity $\mathscr{I}$ and Beig-Schmidt data at spatial infinity $i^0$ in a context appropriate to the gravitational scattering problem and celestial holography. In addition, we explicitly match the various proposals of BMS charges at $\mathscr{I}$ found in the literature with the conserved charges at $i^0$.
Comment: 26 pages + appendices. v2: added references. v3: added references and comments. v4: corrected analysis for the antipodal matching of the angular momentum aspect, in agreement with recent observations made by Compere et al. in 2303.17124. v5: added funding acknowledgment