Just to clarify, is the spinning pie a set of particles in the same relative position as with a still pie, but rotating around the origin? Is it a set of masses connected by springs that has reached equilibrium (none of the springs are stretching or compressing) and the whole system is spinning? Is the pie a solid body?
What exactly we're looking at depends on which of the first two you picked. If you picked the third, it is contradictory with special relativity, but there's a lot more evidence for special relativity than there is for the existence of a solid body. Granted, a sufficiently rigid body will still be inconsistent with special relativity, but all that means is that there's a maximum possible rigidity. Large objects are held together by photons, so we wouldn't expect sound to travel through them faster than light.
The spinning set of particles is a toroidal with let say 1 million light years across - the big R. and with the small r of just 1 centimetre. It is painted red and white, differently each metre.
The whole composition starts to slowly rotate on the signal from the centre. And slowly, very slowly accelerate to reach the speed of 0.1 c in a several million years.
Now, do we see any Lorentzian contraction due to the SR, or not due to the GR?
(Small rockets powered by radioactive decay are more than enough to compensate for the acceleration and for the centrifugal force. Both incredibly small. This is the reason why we have choose such a big scale.)
As per a recent comment this thread is meant to voice contrarian opinions, that is anything this community tends not to agree with. Thus I ask you to post your contrarian views and upvote anything you do not agree with based on personal beliefs. Spam and trolling still needs to be downvoted.