Does rotation in Gödel spacetime depend on the frame of reference?

[pervect]

Some general random comments at a broad, non-detailed level that I hope may be helpful.

Tensor quantities, like proper acceleration can (I beleive) be regarded as absolute, because tensors are coordinate independent. Note that this implies that 4-velocity, being a tensor quantity, should be regarded as absolute, even though 3-velocities are famously relative. It's possible this point could be argued, I suppose, I do not have a hard reference that defines the English terms "absolute and relative" mathematically. But I believe that this approach makes sense.

Vorticity is another example (besides proper acceleration) of a tensor quantity. As such, it should be regarded as absolute, because of the coordinate independent property of tensors.

However, there is no guarantee that when someone talks about an object "not rotating", that they are talking about the tensor quantity of vorticity. So some work needs to be done to figure out what a person may be talking about when they talk about non-rotating.

To take a specific example, an object in a Kerr spacetime that is fixed relative to the absolute stars, i.e. a telescope pointed at a distant "guide star", will in general have a non-zero vorticity.

The tensor concept of vorticity applies directly to timelike congruences of worldlines. So some general discussion of what a time-like congruence of worldlines means would seem helpful. If one has a physical object (a bucket full of water, for example), the timelike congruence associated with the physical object can be regarded as the set of 4-velocities of all points in the object at every moment in time. This is just a unit vector field (a timelike unit vector field for a timelike congruence) at every event, since the 4-velocity has unit length.

Worldlines enter into this picture as the integral curves of the above vector field. The integral curves are worldlines of "points on the object", and the tangent vectors of these worldline generate the unit timelike vector field that mathematically defines the timelike congruence.

It'd get too far afield to discuss worldines and tangent vectors, apologies if the terms are not familiar.

 

[PeterDonis]

Tensor quantities, like proper acceleration can (I beleive) be regarded as absolute, because tensors are coordinate independent.

Yes.

this implies that 4-velocity, being a tensor quantity, should be regarded as absolute

Yes. A way of describing it that makes the absolute nature more apparent is that it is the tangent vector to a worldline at a particular point. (Proper acceleration is the path curvature of the worldline at a particular point.)

Vorticity is another example (besides proper acceleration) of a tensor quantity.

Yes.

there is no guarantee that when someone talks about an object "not rotating", that they are talking about the tensor quantity of vorticity.

Yes. If one wants to be more precise, one can describe zero vorticity as "not rotating with respect to local gyroscopes".

an object in a Kerr spacetime that is fixed relative to the absolute stars, i.e. a telescope pointed at a distant "guide star", will in general have a non-zero vorticity.

Yes. Note, however, that this comparison requires the spacetime to be asymptotically flat, and Godel spacetime is not. In other words, in Godel spacetime, there is no analogue to "the distant stars" that can be used to define an alternative sense of "not rotating".