# Rotating Earth as an inertial frame

1. Jan 12, 2009

### Tam Hunt

According to the principle of relativity - a postulate for Einstein's SR and GR - any frame of reference is as valid as any other for describing phenomena and the laws of physics will be the same in the chosen frame of reference as in any other frame of reference. Taking the rotating Earth as a frame of reference we observe the fixed stars spinning around us at superluminal velocities. Taking this one step further, we can imagine a spinning particle as our frame of reference, in which case the fixed stars are spinning around it at essentially infinite velocity. How can these situations be resolved vis a vis the prohibition against superluminal speeds?

2. Jan 12, 2009

### michelcolman

In SR, inertial frames of reference are not allowed to rotate (the laws of the universe only apply to frames moving at constant speed) so the problem never presents itself.
In GR, frames of reference can do pretty much anything they like, BUT the laws of the universe only hold locally, as observed by someone at that location, over very tiny distances. As soon as you start looking at distant objects, all bets are off.

3. Jan 12, 2009

### JesseM

The principle of relativity in SR is only about the equivalence of inertial frames, a rotating frame is a non-inertial one. And it's a little ambiguous whether the principle of "diffeomorphism invariance" in GR, which puts all coordinate systems on equal footing, is really a physical principle at all or just a feature of the type of mathematics used to describe GR--see my post #8 on this thread.

4. Jan 12, 2009

### Naty1

Superluminal information transmission cannot be made. It just looks that way at first...

5. Jan 12, 2009

### Tam Hunt

Naty1, that is my question: from the point of view of a rotating Earth as the GR frame of reference, the fixed stars ARE moving at superluminal speeds, so how can this apparent contradiction be resolved? You've restated GR's prohibition against superluminal velocity, but that doesn't seem to address my question.

6. Jan 12, 2009

### ZapperZ

Staff Emeritus
Can you list some of these stars that are moving at superluminal velocity?

Zz.

7. Jan 12, 2009

### Tam Hunt

ZZ, almost any fixed star you choose is moving at superluminal velocity with respect to the rotating Earth as a reference frame. This is similar to the lighthouse paradox, but has a key difference. In this gedankenexperiment, there is no lightbeam traveling from the earth. Rather, Earth itself is the reference frame by which the motion of the fixed stars is judged. As Earth rotates each day, the fixed stars complete a full revolution, so depending on how far they are from Earth, they are, from the rotating Earth reference frame's point of view, moving at far higher than c. I'm scratching my head on this one.

8. Jan 12, 2009

### ZapperZ

Staff Emeritus
Really now! If that is true, then the "Z" number they get for even those high red-shifted stars would be utterly wrong, because that explicitly shows that they are NOT at v>c from the Earth's frame. How do you explain that?

Zz.

9. Jan 12, 2009

### JustinLevy

You are taking a coordinate velocity and trying to give physical significance to it.

GR has local poincare symmetry, not global poincare symmetry (like in SR). Therefore the metric can always be locally chosen to be diagonal -1,1,1,1 and look like SR (and measuring in such frames are what we mean by the local speed of light).

Take our universe, and choose a local inertial frame ... if you could look out far enough, there is a point at which the material would be moving away from you faster than the speed of light (due to expansion of the universe). But this does NOT violate relativity because it is not moving faster than the speed of light locally.

Even in SR, I can change my coordinate system by changing my clock synchronization so that objects move faster than the speed of light. This does not contradict SR. Coordinate velocity is not a physical thing. Look at it in coordinate free geometric terms: information cannot be sent from one event to a space-like separated event.

Last edited: Jan 12, 2009
10. Jan 12, 2009

### Tam Hunt

ZZ, the red shift you mention refers to the speed at which the stars are moving away (radially) from Earth. And such calculations don't assume a rotating Earth as the reference frame; rather they assume (I believe, though am not sure on this) the solar system as the reference frame. The key here, which provoked my question, is the rotating Earth as the reference frame.

11. Jan 12, 2009

### ZapperZ

Staff Emeritus
Yes, but the initial post that I asked appears to indicate that almost any fixed star.... I'm looking at Proxima Centauri and, say, Sirius A and B. Are they REALLY moving at v>c? Since when? I'd like to see both papers and calculations that show that they are superluminal.

Zz.

12. Jan 12, 2009

### JustinLevy

Yes, in a rotating frame he is correct.

for example:
w = 2 pi / 24hrs > 1 / yr
R = distance to star > 1 light-yr = c * yr

thus:
coordinate speed = wR > c

This all boils down to Tam expecting coordinate velocity to mean something physical.

13. Jan 12, 2009

### ZapperZ

Staff Emeritus
Oh, I now see what you mean by "coordinate velocity", which is what I was trying to argue that what we "see" isn't necessarily a "straightline shot" at the star at that instant.

Zz.

14. Jan 12, 2009

### Tam Hunt

Justin, I think we may find agreement here ultimately because it seems that the best interpretation of GR is that it does not actually lead to "real" time dilation or "real" length contraction as a result of acceleration or gravitation. Rather, it is perhaps best interpreted as a good mathematical tool for translating between different frames of reference. However, this is not the mainstream interpretation, which holds, to the contrary, that things like time dilation and length contraction are real phenomena. Are you suggesting this, or are you not going this far with your statement that coordinate velocity should not be considered "real" velocity?

15. Jan 12, 2009

### JesseM

In GR certain quantities are coordinate-invariant and others aren't. Velocity depends on your choice of coordinate system, so no coordinate velocity is more "real" than any other (there is no 'real' velocity in GR). On the other hand, the proper time along a given timelike worldline is coordinate-invariant, if each coordinate system uses the correct metric expressed in terms of that system to integrate along the worldline from one point to another they'll all agree on the answer (which means if two observers depart from a common point and reunite at a common point, all coordinate systems agree on who has aged more, and by how much--in this sense time dilation is quite real). There is also an objective notion of distance along spacelike curves, although I can't think of any way to connect that fact with "length contraction".

16. Jan 13, 2009

### Al68

Hi Tam,

The short answer is that there is no general prohibition against superluminal speeds in GR or SR. So there is no contradiction.

There are specific limitations on relative velocity that do not apply to the coordinate velocity of the stars in earth's frame. Obviously, the laws of physics don't prohibit me from turning my head from side to side, even if it results in the velocity of the sun relative to my head being greater than c.

Al

17. Jan 13, 2009

### Tam Hunt

Al, how are you distinguishing coordinate velocity and relative velocity? Einstein's version of the principle of relativity is that any frame is as good as any other frame for describing phenomena AND that the laws of physics are valid in all frames. If this is the case, then it seems that the rotating Earth's frame would also require that all velocities of objects in that frame cannot exceed c, which is, according to everything I have read on this topic, the upper boundary speed limit as a consequence of the basic equations of relativity (mass goes to infinity as velocity approaches c).

18. Jan 13, 2009

### DrGreg

In special relativity any inertial frame is as good as any other inertial frame for describing phenomena and the laws of physics are the same in all inertial frames. But a rotating frame is not an inertial frame. If you are fixed in a rotating frame, you can tell because you will feel "a centrifugal force". (Or to be more precise you'll feel a centripetal force that doesn't cause you to accelerate relative to the frame thus appearing to break Newton's laws relative to the frame.)

Roughly speaking, an inertial frame is one relative to which Newton's laws of motion are valid. And in special relativity, all the inertial frames move at constant velocity relative to each other and do not rotate.

(It gets a bit more complicated in general relativity and the mathematical GR formulation of the laws of physics takes care of any acceleration of frames.)

19. Jan 13, 2009

### Al68

That's simply not true in GR. There is no general upper limit on relative velocity in GR. GR does not say that all frames are equal, just that the laws of physics can be expressed in a way that applies to all frames. Equations that are specific to inertial frames are not such laws. They only apply to inertial frames.

GR doesn't prohibit us from having and using laws that only apply to inertial frames.

GR says we can formulate laws that are generally applicable, not that all laws are generally applicable.

Al

20. Jan 13, 2009

### Tam Hunt

DrGreg, GR applies to frames moving in non-uniform motion relative to each other, which includes rotating frames. It's my understanding, then, that GR applies in my gedankenexperiment (not SR, as you point out). As such the general principle of relativity applies. Einstein defines this principle at p. 69 of Relativity: The Special and General Theory: “All bodies of reference … are equivalent for the description of natural phenomena …, whatever may be their state of motion.” And more technically, in the same book, at page 109: “All Gaussian co-ordinate systems are essentially equivalent for the formulation of the general laws of nature.”

So it seems the problem remains with this thought experiment: how is it that the fixed stars can exceed c, from the frame of the rotating Earth?