Assume rotational velocity is relative to the rotational velocity of the whole universe as according to Mach’s principle. This causes three paradoxes:
1. From the perspective of a non-universal rotating body, the rest of the universe looks as if it’s rotating *around* it. If rotation is relative, this must be true from the perspective of the body. This clearly violates the laws of motion and acceleration (even from the perspective of the body), independently of any axioms about the relativity of rotation per se, because centrifugal force would immediately start pulling the universe apart.
2. If rotation were given its behavior by relation to an absolute rotation (or lack of rotation) of the entire universe, then an object rotating relative to the universe some distance away from the universe’s center would fly away from the universe’s center because, relative to the object’s rotation, the universe as a whole is rotating about its center and is thus generating centrifugal force for the objects within it (or at least for the objects relative to which the universe is rotating).
3. Although the universe is not rotating (that would require under Mach’s principle that it be rotating relative to itself, after all, which would be self-contradictory), measuring an object’s rotation against the rotational velocity (which is 0) of the universe in order to say that the first is relative to the second implies that the universe’s rotational velocity is a variable that could have been something different from 0; for example it could have been whatever velocity it would have to be in order to make the thing that’s rotating relative to it *not* rotating relative to it (and hence, by Mach’s principle, not rotating at all). So it is not simply that rotation doesn’t apply to the universe as a whole, but that the universe’s rotational velocity is a specific variable, and that variable happens to be 0 (zero).
The problem with this is that, if the universe has a specific rational velocity (even if it just happens to be 0), it must have a specific center/axis of rotation, yet, with the universe as a whole necessarily not rotating relative to itself (and hence, according to Mach’s principle, not rotating at all), there is no way even in principle to determine where its axis is. And if there’s no way even in principle to tell where it is, then it doesn’t exist and the notion is meaningless. This clearly contradicts the notion that an axis must logically exist as explained above.
Therefore we can determine, a priori, that rotation cannot be relative.
Rotation is simply a form of constant acceleration, where a centripetal force conserves angular momentum (at each moment the particles “want to” fly off in a direction tangent to the circle of rotation, but they’re accelerated toward the center), so it should be no more mystifying that rotational velocity could exist without being relative to frame of reference than that acceleration in general could exist without being relative to frame of reference.