  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.
