  Asymptotically flat spacetimes have been studied in five separate regions:
future/past timelike infinity $i^\pm$, future/past null infinity
$\mathcal{I}^\pm$, and spatial infinity $i^0$. We formulate assumptions and
definitions such that the five infinities share a single Bondi-Metzner-Sachs
(BMS) group of asymptotic symmetries and associated charges. We show how
individual ingoing/outgoing massive bodies may be ascribed initial/final BMS
charges and derive global conservation laws stating that the change in total
charge is balanced by the corresponding radiative flux. This framework provides
a foundation for the study of asymptotically flat spacetimes containing ingoing
and outgoing massive bodies, i.e., for generalized gravitational scattering.
Among the new implications are rigorous definitions for quantities like
initial/final spin, scattering angle, and impact parameter in multi-body
spacetimes, without the use of any preferred background structure.
