Definition of a rotating frame in GR?

[bcrowell]

What is a definition in GR that correctly captures the concept that a frame is rotating? Is it enough to say that it's stationary but not static?

 

[atyy]

There is a definition of rotating for a family of observers. Roughly, if they are not rotating, then there is space at a time for them (or something like that). Try Eq (2.3.5) and (2.3.6) of Eric Poisson's http://www.physics.uoguelph.ca/poisson/research/agr.pdf.

 

[edpell]

One observes no fictitious centrifugal force. ?

 

[bcrowell]

There is a definition of rotating for a family of observers. Roughly, if they are not rotating, then there is space at a time for them (or something like that). Try Eq (2.3.5) and (2.3.6) of Eric Poisson's http://www.physics.uoguelph.ca/poisson/research/agr.pdf.

Thanks for the link, atyy! Hmm...I think what he's saying with the timelike congruences is essentially equivalent to the idea that a particular observer can check whether the Sagnac effect exists. For instance, say you have a rotating disk. You can make a timelike congruence consisting of world-lines at rest relative to the axis, or a congruence consisting of world-lines at rest relative to the disk. In the latter case, you get a Sagnac effect at every point in space.

I guess my question was awfully vague, but this may help to point me in the right direction to make it more well defined. It seems straightforward to define the right notion for a local observer: do you get a Sagnac effect? I had in mind more the question of whether there was any way to say anything globally.

One observes no fictitious centrifugal force. ?

I don't think this works, because by the equivalence principle a centrifugal force is equivalent to a gravitational force.

 

[atyy]

Thanks for the link, atyy! Hmm...I think what he's saying with the timelike congruences is essentially equivalent to the idea that a particular observer can check whether the Sagnac effect exists. For instance, say you have a rotating disk. You can make a timelike congruence consisting of world-lines at rest relative to the axis, or a congruence consisting of world-lines at rest relative to the disk. In the latter case, you get a Sagnac effect at every point in space.

I guess my question was awfully vague, but this may help to point me in the right direction to make it more well defined. It seems straightforward to define the right notion for a local observer: do you get a Sagnac effect? I had in mind more the question of whether there was any way to say anything globally.

So I googled a bit and came across Ashtekar and Magnon, 1975 about the Sagnac effect in GR. They discuss two definitions of rotation which are absolute. One is the rotation of a timelike vector field, the other is the rotation of a Fermi transported tetrad. And somehow the Sagnac effect links both of them, and they also say rotation is only a "local" concept in GR. I haven't read the paper beyond that.

 

[bcrowell]

And somehow the Sagnac effect links both of them, and they also say rotation is only a "local" concept in GR. I haven't read the paper beyond that.

Cool, thanks! That makes sense to me. The Sagnac effect is something you can measure locally, and the absence of a Sagnac effect (locally) is equivalent to staticity (locally). So I think the answer to my original question is probably that there is no way to say in general whether a frame is globally rotating, but you can do it locally, and my proposed definition works.