I have been trying to understand how we can talk about absolute rotation in general relativity. I get that it is an area of active debate with some adherents of Mach's Principle and others believing that there simply exists absolute rotation. I think the best way of confronting the issue is trying to work with the simplest situation I can think of, and it seems to me that Mach's Principle cannot survive this situation. So here is the thought experiment:
You are a on a cylindrically symmetric spaceship without any other objects in the universe. You start with everything at rest: you feel no forces, motion is described by the Minkowski metric. Then you start a large flywheel in the center of the ship rotating quite quickly. To preserve angular momentum, the ship will rotate in the opposite direction. You are now rotating with the ship, so you feel "artificial gravity", a force which forces you to the outer rim of the ship (you would call it a centrifugal force classically).
We can perform an easy experiment that seems to show that we are rotating and in which direction: simply throw one ball in each tangential direction, one will fall slower and one will fall faster than a dropped ball. But given a relativistic framework it seems in bad taste to appeal to an absolute spacetime which we are rotating relative to, so why can't we claim that we on the spaceship are at rest and the flywheel in the center is rotating very quickly? Is there a way we could write a stress-energy tensor which would accurately describe motion in the spaceship without claiming a distinct "non-rotating frame"? Machians seem to be able to avoid absolute rotation by claiming all rotation is relative to distant bodies, but without any other bodies in the universe, what is our reference? This leads some to conclude that Newton's Bucket would not have the surface of the water become concave by the "rotation" in a universe without other bodies, but in our universe we started with a stationary ship, in a frame where we could use the Minkowski metric. Transforming the metric into the new (relatively rotating) frame would predict geodesic motion that give the effects of "artificial gravity", so there must clearly be rotational effects at play in this example. But if there were to be an observer which only came into existence after the ship had already started rotating, she could not know that in the past both the ship and the wheel had been at relative rest and the Minkowski metric applied, so how could she have a reference for the rotation.
The only way all this seems possible to explain to me is to claim an absolute rotation which is not in reference to any other bodies. How can Mach's Principle survive this? Is there a valid way to write a stress-energy tensor in a cooridinate system which "thinks of" the spaceship as at rest and the rotating flywheel and/or the mass energy of the ship as giving all the odd effects we would like to attribute to rotation? More simply: is there any way to think of the spaceship as not rotating?
It is my inclination that absolute rotation cannot be right as it seems to put us right back to the days before Einstein, but the conclusions seem difficult to escape.