Rotating Spacetime - a question

[neopolitan]

It seems that there are at least some who want to talk about rotating spacetime, so here is an opportunity.

In SR we can consider any inertial frame to be at rest and all other frames to be in motion relative to that frame (ie all other observers who are not at rest relative to the reference rest frame).

In GR do we actually make the same sort of claim, that any frame, which could be rotating or undergoing acceleration, could be considered to be a rest frame?

I think there is a problem with considering a rotating frame to be a rest frame.

If I decided that I am at rest, do I not have problems with explaining the motion of distant stars? Relative to me they are orbiting around me in such a way that they travel one orbit around me in approximately 24 hours or 365.24 complete orbits in a year.

This would mean they have a velocity (relative to me) of 365.24*2*(pi)*r, where r is in light years and the result is in terms of light-years per year (ie c=1). For any star more distant than the sun, this is a superluminal velocity.

It seems to me that I cannot choose any frame whatever as my "rest frame" and use it indiscriminately.

Is there something wrong with this thinking or am I just reinventing the wheel?

cheers,

neopolitan

 

[JesseM]

In SR we can consider any inertial frame to be at rest and all other frames to be in motion relative to that frame (ie all other observers who are not at rest relative to the reference rest frame).

I've mentioned this to you before, but this is not the standard way of approaching problems in SR. In a given problem, it's not as if you normally label one frame "the rest frame" and the other "the moving frame"; the phrase "rest frame" is just used in the context of specific objects that arise in the problem, like "the rest frame of the spaceship" vs. "the rest frame of the space station". This whole notion of picking out one frame and labeling it "nominally at rest" or "nominally in motion" is something you rarely see in textbook problems nowadays, although Einstein did do something like this in his original 1905 paper.

In GR do we actually make the same sort of claim, that any frame, which could be rotating or undergoing acceleration, could be considered to be a rest frame?

Again, the phrase "a rest frame", as opposed to a more typical frame like "the rest frame of object X", doesn't really have a well-defined meaning in SR or GR (and in the context of GR, even talking about 'the' rest frame of a given object would probably be ambiguous, because you have the freedom to pick pretty much any type of coordinate system you like, unlike with inertial coordinate systems in SR where once you specify which object is at rest in your coordinate system, the velocity of every other object, as well as the definition of simultaneity, is uniquely determined). But to try to answer the question I think you're getting at, just as in SR all inertial coordinate systems are "created equal" in the sense that the equations representing the laws of physics should be the same in all inertial coordinate systems, something similar is true in GR where the tensor equations of GR actually hold in any coordinate system you draw on curved spacetime, a property called "diffeomorphism invariance". You can see a short discussion of this here:

http://www.einstein-online.info/en/spotlights/background_independence/index.html

I think there is a problem with considering a rotating frame to be a rest frame.

If I decided that I am at rest, do I not have problems with explaining the motion of distant stars? Relative to me they are orbiting around me in such a way that they travel one orbit around me in approximately 24 hours or 365.24 complete orbits in a year.

This would mean they have a velocity (relative to me) of 365.24*2*(pi)*r, where r is in light years and the result is in terms of light-years per year (ie c=1). For any star more distant than the sun, this is a superluminal velocity.

Yes, the star will have a coordinate speed faster than c, but of course photons can also have a coordinate speed faster than c in this coordinate system too, so this isn't really a "superluminal" velocity. The claim that light always moves at c is only meant to apply to the inertial coordinate systems of SR, or to a "locally inertial" coordinate system in an arbitrarily small region of curved spacetime in GR (small enough that the effects of curvature go to zero). In non-inertial coordinate systems light can move at different coordinate speeds, and the non-tensor equations of SR like the time dilation equation should no longer be expected to hold. But as long as you state the laws of physics using tensor equations (which I don't claim to understand very well), they should be the same in all coordinate systems.

 

[nanobug]

In GR do we actually make the same sort of claim, that any frame, which could be rotating or undergoing acceleration, could be considered to be a rest frame?

Any frame can be considered a 'rest frame', that is, a the frame in which one is at rest. The question is other inertial forces are felt or not. Obviously, someone at rest in a rotating frame is going to feel these forces.