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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.

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    $\begingroup$ Possible duplicate of Is Mach's Principle Wrong? See also other questions in the Related column. $\endgroup$ – sammy gerbil Jul 10 '18 at 18:53
  • $\begingroup$ That question is in a similar vein but my question is primarily focused on if we can escape the idea of absolute rotation in even the simplest possible case, using Mach's principle as the most common formulation of a relative rotation. The primary question in my post is: is there a way we can think about the spaceship is not rotating? $\endgroup$ – Keefer Rowan Jul 10 '18 at 18:58
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    $\begingroup$ Have you looked at the other questions in the Related column, eg Why does rotation simulate gravity if motion is relative? $\endgroup$ – sammy gerbil Jul 10 '18 at 19:08
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    $\begingroup$ As an observer, I don't care if this question is close to a duplicate. It's perfectly asked and frames the question in a different way than I've read it before. $\endgroup$ – BuckFilledPlatypus Jul 10 '18 at 19:21
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    $\begingroup$ Yeah, I think I've look at all other pertinent questions. In the question you linked for example, everyone gives one of two answers either a. rotation is absolute or b. rotation is relative, but relative to a background universe of bodies. The first answer still applies if that's what you want to claim, but if you want to claim rotation is relative (which is to me the more appealing option) my though experiment introduces new problems by removing the background of stars/galaxy to give as reference. It seems the notion of relative rotation can't survive the simplest example. $\endgroup$ – Keefer Rowan Jul 10 '18 at 19:23
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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.

No, this is just a philosophical bias, which is not borne out at all by the actual math.

In the very early days, it was thought that velocity was absolute. Then Galilean relativity came along and said the opposite. If one didn't pay attention, one might think that Galilean relativity means nothing is absolute: that is, "absolute acceleration cannot be right because it seems to put us right back to the days before Galileo". But that simply isn't true. You can't just say that because one thing isn't absolute, a completely different thing isn't absolute either -- that is lazy philosophizing.

The same thing holds for angular velocity. You might argue that angular velocity is also called a velocity, so it has to be relative like linear velocity. But that's a rather superficial resemblance. In my book angular velocity is not a velocity at all, but rather a particular kind of periodic acceleration. And we know acceleration is absolute.

To put it another way: we go out and observe certain symmetries of the universe. Translation invariance tells us position isn't absolute, boost invariance tells us velocity isn't absolute, and rotational invariance tells us angular orientation isn't absolute. There is no such observed symmetry for angular velocity.

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"? [...] 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.

In the formalism of general relativity, the structure of rotating and non-rotating frames is already put in from the outset, in the form of the Levi-Civita connection. This is prior to the notion of any observer or any particular matter content. This makes general relativity not obey Mach's principle, though Einstein himself didn't like this.

Specifically, suppose we are in Minkowski spacetime, where the connection is flat. An inertial frame is one where the connection coefficients are all zero. This is preserved by Lorentz transformations, but not by going to a rotating frame. Since the connection coefficients may be measured locally, an observer can find which frames are inertial even if they have no angular reference whatsoever. (The stress-energy tensor is found in just the usual way, but its conservation law $D_\mu T^{\mu\nu} = 0$ depends directly on the connection. The same goes for the geodesic equation.)

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