A recent question Rotation of our Galaxy's inertial frame is about an observational evidence of the space rotation. My question is if such a rotation is conceptually possible in GR. Can we assume that for whatever reason spacetime has evolved in such a way that an empty region of space now rotates relative to the universe? If so, what would be the evolution of this rotation in time? Would the rotation continue forever, stop instantly, or slow down gradually and how fast? Would this region expand or contract in the process?
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$\begingroup$ Comments to the post (v1): If the region is empty, how do you know it is rotating? Do you allow singularities in the empty rotating region? $\endgroup$– Qmechanic ♦Commented Oct 10, 2017 at 15:46
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$\begingroup$ I do not know if that is possible, but if so, it can possibly explain the extra mass needed for uniform rotation curve of spiral galaxies, currently designated as dark matter. A Spiral can hint a space rotation, just like a hurricane spiral. $\endgroup$– kpvCommented Oct 10, 2017 at 16:16
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$\begingroup$ @Qmechanic We would know the region is rotating by putting a test object there small enough to not affect the spacetime geometry. The object could be something like two rocks connected by a rope. They would rotate without the rope stretched by a centrifugal force. Does this make sense? No singularities in this question. Perhaps a separate question can be asked if singularities can create a stable rotating region. $\endgroup$– safesphereCommented Oct 10, 2017 at 19:46
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$\begingroup$ @kpv This is a natural thought, but it seems problematic. If the space rotation is in the direction of the galaxy rotation, then the galaxy is not rotating in this frame and would collapse under its own gravity. If the space rotation is in the opposite direction and is faster closer to the center, then the galaxy rotation is more uniform, as observed, but slower than observed (unless I am missing anything). $\endgroup$– safesphereCommented Oct 10, 2017 at 19:50
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$\begingroup$ @safesphere: It would be in the direction of the galaxy rotation, just enough to count for the uniform curve. Galaxy itself would also be rotating per general gravity/relativity, so that it does not collapse (and does not fly away). The space rotation would only count for the missing mass. $\endgroup$– kpvCommented Oct 10, 2017 at 20:19
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The answer might surprise you, but this exactly what the Kerr metric is.
The Kerr metric is a vacuum solution i.e. the stress energy tensor is zero everywhere, except at the singularity where it is undefined (we normally remove the singularity from the manifold anyway). So the Kerr metric is exactly a bit of empty space rotating.
But of course the Kerr metric has a parameter $M$ with the dimensions of a mass, so what then is this? Well, it's a geometric property called the ADM mass. For suitable geometries we find there is a mass (or equivalently an energy) associated with the geometry even for a vacuum solution where we have put in no mass. The same is true of the Schwarzschild metric. It too is a vacuum solution but has an ADM mass.
So the answer is that yes we can have areas of rotating vacuum in GR, but unfortunately you'll find they always have a mass associated with them even when no matter is present.
answered Oct 10, 2017 at 14:58
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1$\begingroup$ and the ADM mass, unlike stress-energy $T_{00}$, can be negative arxiv.org/abs/1407.1457 $\endgroup$– lurscherCommented Oct 10, 2017 at 15:30
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$\begingroup$ +1 Thanks John, a good insight to ponder. This however is frame dragging and so doesn't exactly answer the question :) $\endgroup$ Commented Oct 13, 2017 at 5:50