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?
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.