  The purpose of this paper is to find optimal estimates for the Green function
of a half-space of {\it the relativistic $\alpha$-stable process} with
parameter $m$ on $\Rd$ space. This process has an infinitesimal generator of
the form $mI-(m^{2/\alpha}I-\Delta)^{\alpha/2},$ where $0<\alpha<2$, $m>0$, and
reduces to the isotropic $\alpha$-stable process for $m=0$. Its potential
theory for open bounded sets has been well developed throughout the recent
years however almost nothing was known about the behaviour of the process on
unbounded sets. The present paper is intended to fill this gap and we provide
two-sided sharp estimates for the Green function for a half-space. As a
byproduct we obtain some improvements of the estimates known for bounded sets
specially for balls. The advantage of these estimates is a clarification of the
relationship between the diameter of the ball and the parameter $m$ of the
process.
  The main result states that the Green function is comparable with the Green
function for the Brownian motion if the points are away from the boundary of a
half-space and their distance is greater than one. On the other hand for the
remaining points the Green function is somehow related the Green function for
the isotropic $\alpha$-stable process. For example, for $d\ge3$, it is
comparable with the Green function for the isotropic $\alpha$-stable process,
provided that the points are close enough.
