Localizing solutions of the Einstein constraint equations

We perform an optimal localization of asymptotically flat initial data sets and construct data that have positive ADM mass but are exactly trivial outside a cone of arbitrarily small aperture. The gluing scheme that we develop allows to produce a new class of $N$-body solutions for the Einstein equation, which patently exhibit the phenomenon of gravitational shielding: for any large $T$ we can engineer solutions where any two massive bodies do not interact at all for any time $t\in (0,T)$, in striking contrast with the Newtonian gravity scenario.