  We prove global stability of Minkowski space for the Einstein vacuum
equations in harmonic (wave) coordinate gauge for the set of restricted data
coinciding with Schwartzschild solution in the neighborhood of space-like
infinity. The result contradicts previous beliefs that wave coordinates are
"unstable in the large" and provides an alternative approach to the stability
problem originally solved (for unrestricted data, in a different gauge and with
a precise description of the asymptotic behavior at null infinity) by D.
Christodoulou and S. Klainerman.
  Using the wave coordinate gauge we recast the Einstein equations as a system
of quasilinear wave equations and, in absence of the classical null condition,
establish a small data global existence result. In our previous work we
introduced the notion of a eak nul condition and showed that the Einstein
equations in wave coordinates satisfy this condition. The result of this paper
relies on this observation and combines it with the vector field method based
on the symmetries of the standard Minkowski space.
  In a forthcoming paper we will address the question of stability of Minkowski
space for the Einstein vacuum equations in wave coordinates for all "small"
asymptotically flat data and the case of the Einstein equations coupled to a
scalar field.
