  Mach's principle applied to rotation can be correct if one takes into account
the rotation of the quantum vacuum together with the Universe. Whether one can
detect the rotation of the vacuum or not depends on its properties. If the
vacuum is fully relativistic at all scales, Mach's principle should work and
one cannot distinguish the rotation: in the rotating Universe+vacuum, the
co-rotating bucket will have a flat surface (not concave). However, if there
are "quantum gravity" effects which violate Lorentz invariance at high energy,
then the rotation will become observable. This is demonstrated by analogy in
condensed-matter systems, which consist of two subsystems: superfluid
background (analog of vacuum) and "relativistic" excitations (analog of
matter). For the low-energy (long-wavelength) observer the rotation of the
vacuum is not observable. In the rotating frame, the "relativistic"
quasiparticles feel the background as a Minkowski vacuum, i.e. they do not feel
the rotation. Mach's idea of the relativity of rotational motion does indeed
work for them. But rotation becomes observable by high-energy observers, who
can see the quantum gravity effects.
