What I understand about Mach's principle VS modern physics:
According to classical physics, there are ways to distinguish weather a body is rotating or not. For example if it is rotating, the Coriolis force experienced by an observer at the surface of the body is a function of the location, and the body will exhibit an equatorial bulge.
Some people find this is odd when we consider a rotating body in an otherwise empty universe because this imply the existence of preferred reference frames for rotation, namely the inertial frames, independent of all the universe matter.
Mach proposed (Mach's principle) that a body inertia is actually caused by its interaction with the entire universe mass distribution. A rotating sphere experience Coriolis forces because of its acceleration with respect to the rest of the universe mass. However, in a universe containing only a rotating sphere, since the sphere constitutes the total universe mass, it does not make sense to claim that the sphere is rotating. As a consequence, Coriolis forces and the equatorial bulge will vanish and no experiment will be able to demonstrate a rotating motion.
Einstein found Mach's principle attractive but couldn't include it in his General Relativity (GR): Mach's principle is incompatible with GR.
I wish to understand why, and here is my problem: what is the prediction of GR for a rotating sphere in an empty universe? According to frame-dragging, shouldn't the spacetime be "dragged" by the rotating body, and eventually "catch-up", so that at steady-state the whole body+spacetime is rotating as a solid block? In such case it becomes meaning less to say that the sphere is rotating. Where do I get it wrong? I have a very superficial GR knowledge.