Relativity in rotating frames

[mikeu]

Suppose there is an inertial frame S in which there exists some object A at rest, located at (x,y,z)=(10^8,0,0). Now consider the non-inertial frame S' whose axes are coincident with those of S at t=0, but which is rotating about the common z-axis with constant angular frequency w. If S' has a period of 1s, how do we avoid the conclusion that A appears to be moving (orbiting the origin of S') at v=2pi*10^8 m/s > c? How can we find the gamma factor 1/sqrt(1-v^2/c^2) for an object at rest in S located at r>=w/c?

 

[JesseM]

In a non-inertial frame, my understanding is that it is quite possible for things to be moving faster than c. I'm not sure if the notion of calculating the "gamma factor" even makes sense in a non-inertial frame--it wouldn't be true that the rate a clock is ticking depends only on its velocity in such a frame, for example.

 

[mikeu]

Perhaps the gamma factor is too closely linked to special relativity to make sense in all non-inertial frames, but doesn't GR say that all reference frames, inertial or otherwise, are equally valid? That c is constant in all of them, with nothing able to exceed that speed in any frame?

 

[JesseM]

Perhaps the gamma factor is too closely linked to special relativity to make sense in all non-inertial frames, but doesn't GR say that all reference frames, inertial or otherwise, are equally valid? That c is constant in all of them, with nothing able to exceed that speed in any frame?

I'm not too knowledgeable about GR, but I'm pretty sure that although the laws of physics expressed in tensor form are the same way in any coordinate system, that doesn't mean that light has a constant coordinate velocity in every coordinate system. This post by physicist Steve Carlip seems to say at the end that even the laws of Newtonian physics work the same way in any coordinate system (at least I think that's the rough meaning of the term 'diffeomorphism invariance' which they use there) when expressed in tensor form. I think in GR, it's possible to find a local coordinate system in an arbitrarily small region around a given point in spacetime where the laws of physics work just like they do in an inertial frame in SR, even if you don't express these laws in tensor form, so it makes sense to say that light always travels at the same speed "locally". But since I don't really understand the details of what it means to express a law in tensor form as opposed to the usual non-tensor form I'm used to, I may be misunderstanding what it means to say light travels at the same speed locally.

 

[pervect]

Perhaps the gamma factor is too closely linked to special relativity to make sense in all non-inertial frames, but doesn't GR say that all reference frames, inertial or otherwise, are equally valid? That c is constant in all of them, with nothing able to exceed that speed in any frame?

If you consider "speed" to be the rate of change of a position coordinate with respect to a time coordinate, i.e. v = dx/dt, GR allows one to define totally arbitrary coordinates in which the "speed" of light can be any number you desire.

If you restrict yourselves to coordiante systems that measure distance in meters, and time in seconds and do not incorporate any arbitrary scaling factors, GR only states that the speed of light will be 'c' at (299,792,458 m/s) at or near the origin of the coordinate system. This is what's meant by saying that the geometry of space-time is _locally_ Lorentzian.

Far away from the observer, the "speed" of light might be anything at all. It's only near the origin that the metric coefficients will be Minkowskian, and the speed of light will be equal to 'c'.

[correction- actually, in a rotating coordiante system, your metric won't be Minkowskian even at the origin! But you'll still find that the speed of light is 'c' at the origin, I think.]

To recylce some text from another post of mine: wherever you are, the speed of light is equal to 'c'. While you may think that someplace far far away the speed of light is different than 'c', when you actually go there, and measure it, you'll find that it's still equal to 'c' - assuming you use standard clocks and rods.

 

[EnumaElish]

This is really confusing to me; perhaps because I'm not a physicist. Why is the answer not "One may not assume S' so rotate, because every object in S' would then have infinite (or indeterminate) relativistic mass"?

In other words, how is mikeu's premise different from "Let us assume that two mutually observant galaxies (galaxies within each other's event cone) are escaping from each other at a speed > c" ?

AFAIK, this cannot happen because its happening would imply that the relativistic mass of either galaxy $$\longrightarrow +\infty$$ from the other galaxy's point of view.

OTOH, if the galaxies were outside each other's cone to begin with, then it could happen.

Suppose there is an inertial frame S in which there exists some object A at rest, located at (x,y,z)=(10^8,0,0). Now consider the non-inertial frame S' whose axes are coincident with those of S at t=0, but which is rotating about the common z-axis with constant angular frequency w. If S' has a period of 1s, how do we avoid the conclusion that A appears to be moving (orbiting the origin of S') at v=2pi*10^8 m/s > c? How can we find the gamma factor 1/sqrt(1-v^2/c^2) for an object at rest in S located at r>=w/c?

 

[pervect]

Rotating coordinate systems are not allowed in SR, but they are perfectly valid in GR. But as I review the math, you (EnumaElish) do have a good point - see below

One reason that one might want to use a rotating coordiante system is if one happens to be unfortunate enough to live on a rotating planet :-).

If you don't mind some math

In a non-rotating coordinate system in flat space-time (special relavity) with c=1 (for simplicity) we have the following expression for the invariant "Lorentz interval" , which is the same for all observers.

ds^2 = dx^2 + dy^2 + dz^2 - dt^2

In a rotating coordinate system, we just make the substitution

xx = x*cos(w*t)-y*sin(w*t)
yy = y*cos(w*t)+x*sin(w*t)

We wind up with

ds^2 = dxx^2+dyy^2+dz^2 + 2*w*yy*dxx*dt - 2*w*xx*dyy*dt + (w^2*(xx^2+yy^2)-1) dt^2

[add]
For anyone wishing to veryify this equation, remember that

dxx = dxx/dx * dx + dxx/dy * dy + dxx/dt * dt
dyy = dyy/dy * dy + dyy/dx * dx + dyy/dt * dt

a good symbolic algebra program helps a lot in carrying out a messy calculation like this
[end add]

Hmm, it does look like we need to require that w^2*(xx^2+yy^2) < 1, so it looks like the coordinates are only good near the origin, and break down when the velocities would equal the speed of light. So "don't do that" is in fact good advice with regards to attempting to extend the rotating system of coordinates too far, they do in fact break down sufficiently far away from the origin.

 

[JesseM]

This is really confusing to me; perhaps because I'm not a physicist. Why is the answer not "One may not assume S' so rotate, because every object in S' would then have infinite (or indeterminate) relativistic mass"?

Because the usual rules of SR, such as the increase of relativistic mass with increased velocity, only apply to inertial reference frames--coordinate systems where the origin and spatial axes are not accelerating.

 

[Chronos]

Nice try, mikeu. But once you impose a coordinate system upon any arbitrarily chosen reference frame, you automatically assume the burden of proof. Specifically, you must propose an observation or experiment that defies the GR prediction. Logic is not an acceptable substitute for quantifiable results.

 

[mikeu]

In a rotating coordinate system, we just make the substitution

xx = x*cos(w*t)-y*sin(w*t)
yy = y*cos(w*t)+x*sin(w*t)

$$\vdots$$

Hmm, it does look like we need to require that w^2*(xx^2+yy^2) < 1, so it looks like the coordinates are only good near the origin, and break down when the velocities would equal the speed of light. So "don't do that" is in fact good advice with regards to attempting to extend the rotating system of coordinates too far, they do in fact break down sufficiently far away from the origin.

I've looked at making that substitution and arrived at the same conclusion... The substitution seems to assume rigid rotation though, and since there isn't really any absolute concept of rigidity in GR I thought there might be some other method, perhaps a retarded rotation along the lines of letting t -> t-r/c in the rotation equation?

Also, in an early post you said that the metric won't be Minkowskian even at or near the origin (which isn't too surprising to me) but what will it look like?

 

[mikeu]

Nice try, mikeu. But once you impose a coordinate system upon any arbitrarily chosen reference frame, you automatically assume the burden of proof. Specifically, you must propose an observation or experiment that defies the GR prediction. Logic is not an acceptable substitute for quantifiable results.

I think you've misunderstood the point of my question... I'm trying to examine a system in a rotating frame from the assumption that GR is correct, not attempting to disprove the theory.

I'm hoping to find a coordinate transformation from a Minkowski spacetime to a spacetime rotating about the common origin, without having to cut off the rotating spacetime at r=c/w. Either that or find a metric for the rotating frame...

 

[pervect]

I've looked at making that substitution and arrived at the same conclusion... The substitution seems to assume rigid rotation though, and since there isn't really any absolute concept of rigidity in GR I thought there might be some other method, perhaps a retarded rotation along the lines of letting t -> t-r/c in the rotation equation?

Also, in an early post you said that the metric won't be Minkowskian even at or near the origin (which isn't too surprising to me) but what will it look like?

Unfortunately I blew that one - when xx=0 and yy=0, the metric is diagonal, making it Minkowskian, so at the origin the metric is Minkowskian.

The problem appears to be in the 't' coordinate, so one might try the approach used by Eddington-Finklestein coordiantes , or Krustkal coordiantes - replace one or more of the problem coordinates with a null coordinate. Replacing time with a null coordinate based on the geodesics of an outgoing light beam would be the first thing that springs to mind.

I'll think about this some more when I have some time.

 

[εllipse]

This thread is confusing me.

Far away from the observer, the "speed" of light might be anything at all. It's only near the origin that the metric coefficients will be Minkowskian, and the speed of light will be equal to 'c'.

Does this mean, also, that far away from the observer, the speed of bodies with mass can be greater than c?

 

[pervect]

This thread is confusing me.



Does this mean, also, that far away from the observer, the speed of bodies with mass can be greater than c?

Yes, this can happen. But note that light itself moves faster than 'c' far away from the observer, so that the bodies with mass will still be moving slower than a light-beam at the same location. The speed of light becomes different from 'c' far enough away from the origin in some coordinate systems (most notably the Rindler coordinate system of an accelerated observer).

if you actually go to that location, so that it's at the origin of your coordinate system, and if you use standard clocks and rulers, you'll still measure the speed of light to be 'c', and the speed of any material body will be slower than 'c'.

 

[pervect]

I just stumbled across a lead that might answer the question of how to expand the region of validity of rotating coordiantes to all of space-time. While it's only a pointer to a piece of literature that I don't have, I thought I would share it, just in case the original poster is still around and still interested.

Source:

http://arxiv.org/PS_cache/gr-qc/pdf/0311/0311058.pdf

Finally there is the enormous amount of bibliography, reviewed in Ref.[45], about the
problems of the rotating disk and of the rotating coordinate systems. Independently from
the fact whether the disk is a material extended object or a geometrical congruence of
time-like world-lines (integral lines of some time-like unit vector field), the idea followed by
many researchers [11, 12, 46] is to start from the Cartesian 4-coordinates of a given inertial
system, to pass to cylindrical 3-coordinates and then to make a either Galilean (assuming a
non-relativistic behaviour of rotations at the relativistic level) or Lorentz transformation to
comoving rotating 4-coordinates, with a subsequent evaluation of the 4-metric in the new
coordinates. In other cases [47] a suitable global 4-coordinate transformation is postulated,
which avoids the so-called horizon problem (the points where all the previous 4-metrics
have either vanishing or diverging components, when the rotational frequency reaches the
velocity of light).

Thus getting a hold of reference [47] might answer the question.

[47] G.Trocheris, Electrodynamics in a Rotating Frame of Reference, Philos.Mag. 40, 1143 (1949).
H.Takeno, On Relativistic Theory of Rotating Disk, Prog.Thor.Phys. 7, 367 (1952).

 

[mikeu]

Thus getting a hold of reference [47] might answer the question.

[47] G.Trocheris, Electrodynamics in a Rotating Frame of Reference, Philos.Mag. 40, 1143 (1949).
H.Takeno, On Relativistic Theory of Rotating Disk, Prog.Thor.Phys. 7, 367 (1952).

In case anyone's still interested in this... I tracked down the two papers above in the campus library and will attempt a quick summary here. If there's interest I could post more complete derivations or possibly scan the papers...

Takeno uses a group-theoretic approach based on a few assumptions (which all seem reasonable). He first assumes that whatever 'uniform rotation with constant angular velocity w' means, the set of such rotations over all w forms a group. He then considers three discs, one with w=0 and the other two with arbitrary w. Then by looking at how an arbitrary event is expressed in the three different coordinate systems (of the three discs) and assuming that the instantaneous velocities of those points obey the special relativistic velocity addition law (since the two non-zero velocities are assumed parallel as they are perpendicular to the radial direction at the same point) he determines, using infintessimal w, the generator of a Lie group from which he obtains the following transformation equations between two cylindrical coordinate systems in uniform rotation by w relative to each other:
$$t' = t\cosh\left(\frac{r\omega}{c}\right)-\frac{r\phi}{c}\sinh\left(\frac{r\omega}{c}\right)$$
$$r'=r$$
$$\phi' = \phi\cosh\left(\frac{r\omega}{c}\right)-\frac{ct}{r}\sinh\left(\frac{r\omega}{c}\right)$$
$$z'=z$$

Trocheris uses a different approach, which I have not looked at as closely. He seems to say that since the traditional, rigid transformation (i.e. $\phi\to\phi-\omega t$) reduces to Galileian transformations we should instead just look for ones that reducs to Lorentz transformations. He says that "the simplest transformation to a first order approximation in w is"
$$t=t'-\frac{\omega r'^2}{c^2}\phi'$$
$$r=r'$$
$$\phi=\phi'-\omega t'$$
$$z=z'$$

He then imposes the group property that "if two frames of reference are both in uniform rotation about the same axis then they are in uniform rotation with respect to each other" and says that the only transformation which satisfies this requirement and reduces to his previous one is the one arrived at by Takeno.