  In this article and its sequel we discuss the asymptotic structure of
space-times representing isolated bodies in General Relativity. Such
space-times are usually required to be asymptotically flat (AF), and thus to
have a prescribed type of asymptotic. Despite all the "reasonable" that the
requirement is, it seems to be against the spirit of General Relativity where
the global structure of the space-time should be also considered as a variable.
It is shown here that, even eliminating from the definition any a priori
reference or assumption about the asymptotic, the space-times of isolated
bodies are unavoidably and a posteriori AF. In precise terms, between the two
articles it is proved that any vacuum strictly stationary space-time end whose
(quotient) manifold is diffeomorphic to R^3 minus a ball and whose Killing
field has its norm bounded away from zero is necessarily AF with
Schwarzschidian fall off. The "excised" ball would contain (if any) the actual
material body, but this information or any other is not necessary to reach the
conclusion. Physical and mathematical implications are also discussed.
