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Current discussions on cosmology are debating the possibility of a rotating universe, see:

I understand that this is far from being accepted in the community and it does not use much cosmological equation. Nevertheless, my question is more general, what are some models of a rotating universe and does that necessarily imply either (a) a center (central axis of rotation) within space-time or (b) closed-universe (imagine the surface of a rotating cylinder)?

Answers to a previous question What if the universe is rotating as a whole? hint at the possibility of having an axis of rotation but delocalized over space (always located at the observer). How is this possible?

Qmechanic
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Mauricio
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2 Answers2

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Does a rotating universe need a center?

No. A canonical example is the Gödel metric. This is a solution of Einstein equations for dust matter with a negative cosmological constant. It is a homogeneous spacetime, so every point of it looks like any other point. So there is neither unique “center of the universe” nor unique axis of rotation but it is not isotropic: at each point there is a distinguished (spatial) direction, local axis of rotation. An observer in comoving frame would see all the matter in the Universe rigidly rotate around him with constant angular velocity as if he was at the center of the Universe. But any other comoving observer at any point in spacetime would have the same experience! Of course, Gödel universe is stationary, so there is no expansion and thus it could not model our Universe. It is also pathological: it contains closed timelike curves through every point of the spacetime.

But the second of Gödel's GR papers (from 1952) provides examples of universes that have both expansion and rotation. This paper (here is a reprinted version from 2000) is less well known than the one from 1949 (describing the Gödel metric), so let's review properties of such solutions. As could be expected from Newtonian reasoning, with expansion the angular velocity of rotation slows. Of course, such universe is no longer homogeneous spacetime but it is spatially homogeneous (as a cosmological model), at a given moment of cosmic time, any spatial point looks like any other and every comoving observer will still see universe rotating as a whole, with him as the center. Expansion can also possibly cure pathologies of the Gödel's metric and if rotation is small enough such spacetime can be arbitrarily close to nonrotating isotropic cosmological solutions. So if observations are compatible with isotropic non-rotating cosmological model they would also be compatible with rotating (anisotropic) cosmology given small enough rotation. Thus such solutions can conceivably describe our Universe.

Answers to a previous question What if the universe is rotating as a whole? hint at the possibility of having an axis of rotation but delocalized over space (always located at the observer). How is this possible?

Equivalence principle. If all the matter in the universe is rotating around one point under the influence of gravity alone, then trajectory of each matter particle would corresponds to a local inertial frame and could serve as new center of rotation. This is straightforward generalization of rigid body kinematics where the angular velocity of rotation is independent on the choice of origin. Also this is analogous to kinematics of the uniform expansion of the universe, where every comoving observes sees himself at a center of expansion.

safesphere
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A.V.S.
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A rotating universe does not need a center. An intuitive perspective on how that works is to think of points on a spinning wheel. From the perspective of any arbitrary point -- not just points on the rotation axis -- everything else is rotating around it. Not only that, the pattern of rotation of surrounding material (e.g., velocity as a function of position relative to your own point) is exactly the same for every point.

For the wheel, you can still tell where the center is because it is not accelerating, whereas all other points are accelerating. But for a rotating universe, the acceleration would be purely gravitational, so it cannot be objectively measured. Every point would be in its own free fall. Consequently there would be no rotational center for the universe as a whole.

Of course, there are lots of gravitationally rotating systems, like galaxies or the Solar System, and they seem to have clearly defined centers. The difference with the universe is that it is uniformly dense. Only in a uniformly dense system does gravity give rise to rigid rotation, for which everything (no matter its distance) orbits with the same period. That's because for uniform density the gravitational acceleration relative to any point is proportional to its distance from that point. It's also for this reason that the universe does not need to have a center of expansion.

Sten
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