Conditions for Schwarzschild Metric to be applicable The Wikipedia article regarding the two-body problem in General Relativity states that the Schwarzschild metric "corresponds to the external gravitational field of a stationary, uncharged, non-rotating, spherically symmetric body of mass $M$".
My question is with regard to what reference frame does the mass $M$ have to be stationary. I mean, this is Relativity, and we can not just assume some preferred 'ether' frame. If we assume just the mass $M$ and some small test mass $m$ in the universe, then the only physical object the motion of $M$ can be referred to is the test mass $m$. But requiring the latter to be at rest with regard to $M$ would be a kind of silly condition. In fact, if we take the equation for the effective potential a bit further down in the article

we would in this case have for the angular momentum $L=0$ and the potential would be the same as the Newtonian gravitational potential.
Am I missing something here?
 A: The Schwarzschild spacetime does not come with any coordinates or frame of reference attached to it. If you want to write down the metric in some coordinates, you can, but the choice is entirely arbitrary. GR allows you to do any smooth change of coordinates.
GR does not have global frames of reference. Therefore there is no a priori reason why we should be able to associate a certain coordinate chart with a frame of reference, as we would have been able to do in newtonian mechanics or SR.
However, the Schwarzschild spacetime is asymptotically flat, so we have the notion of a distant observer. A distant observer in such a spacetime can define the total energy-momentum vector of the spacetime. In the case of a black hole, the distant observer is at rest with respect to the black hole if this energy-momentum vector is tangent to the observer's world-line.
At large distances, the Schwarzschild coordinates are the spherical coordinates that would naturally be chosen by such an observer.
Even without worrying about the technical stuff described above, it should be clear that a distant observer can tell whether the black hole is at rest relative to them. They can do this by exactly the same techniques they would use for any other object.
