  We study the initial value problem for the Einstein-Klein-Gordon system and
establish the global nonlinear stability of massive matter in the
near-Minkowski regime when the initial geometry is a perturbation of an
asymptotically flat, spacelike hypersurface in Minkowski spacetime and the
metric enjoys the harmonic decay 1/r (in term of a suitable distance function r
at spatial infinity). Our analysis encompasses matter fields that have small
energy norm and solely enjoys a slow decay at spacelike infinity. Our proof is
based on the Euclidean-hyperboloidal foliation method recently introduced by
the authors, and distinguishes between the decay along asymptotically
hyperbolic slices and the decay along asymptotically Euclidean slices. We
carefully analyze the decay of metric component at the harmonic level 1/r,
especially the metric component in the direction of the light cone. In presence
of such a slow-decaying matter field, we establish a global existence theory
for the Einstein equations expressed as a coupled system of nonlinear wave and
Klein-Gordon equations.
