Rotating Earth as an inertial frame

[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?

 

[michelcolman]

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?

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.

 

[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.

 

[Naty1]

How can these situations be resolved vis a vis the prohibition against superluminal speeds?

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

 

[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.

 

[ZapperZ]

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.

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

Zz.

 

[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.

 

[ZapperZ]

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.

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.

 

[JustinLevy]

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.

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.

 

[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.

 

[ZapperZ]

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.

 

[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.

 

[ZapperZ]

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.

 

[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?

 

[JesseM]

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?

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".

 

[Al68]

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?

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

 

[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).

 

[DrGreg]

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.

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.)

 

[Al68]

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).

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

 

[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?

 

[Tam Hunt]

AI, thanks for pointing out that the absolute SR prohibition is not present in GR. It seems, however, from my admittedly non-comprehensive survey, that the very limited possibilities for superluminal speeds under GR would not apply to my hypothetical. The Alcubierre drive or postulated wormholes, for example, do not fit my thought experiment because no such phenomena are present with respect to all the fixed stars. And, more generally, superluminal speeds lead to problems with causality due to the time travel that results from superluminal speeds. Or so the theory goes. Thoughts?

 

[Al68]

AI, thanks for pointing out that the absolute SR prohibition is not present in GR. It seems, however, from my admittedly non-comprehensive survey, that the very limited possibilities for superluminal speeds under GR would not apply to my hypothetical. The Alcubierre drive or postulated wormholes, for example, do not fit my thought experiment because no such phenomena are present with respect to all the fixed stars. And, more generally, superluminal speeds lead to problems with causality due to the time travel that results from superluminal speeds. Or so the theory goes. Thoughts?

Superluminal speeds of an object with respect to an inertial frame would lead to causality problems. The fact that the relative velocity of the moon can exceed c with respect to my nose when I turn my head side to side poses no problem at all.

Notice, too that if we look at a distant star's instantaneous velocity at any given time, it's less than c. It's only after we average the velocity over time and with respect to a frame that is itself changing speed and direction between the measurements that the coordinate velocity (relative to a frame that is changing speed and direction between measurements) exceeds c.

Inertial frames by definition aren't changing speed or direction between the measurements, which makes measurements taken relative to them more useful. It doesn't make the measurements more "correct", just more useful.

Al

 

[RandallB]

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

SR requires inertial rectilinear frames.
GR only adds that the frame does not need to be “inertial”
When did it supposedly delete the rectilinear part in place since Galileo & Newton?

 

[Al68]

SR requires inertial rectilinear frames.
GR only adds that the frame does not need to be “inertial”
When did it supposedly delete the rectilinear part in place since Galileo & Newton?

When Einstein first wrote about GR, he refers to a frame K' in any kind of motion whatsoever with respect to system K (reference frame). He obviously includes rotational motion since he specifically mentioned SR's restriction to uniform rectilinear, non-rotary motion just prior.

Al

 

[truhaht]

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?

The answer is simplicity: NONE of those distant objects (eg. stars) are moving with superluminal velocity with respect to any other actual real body. This so-called violation of lightspeed is merely in the mind of the human who constructed that particular (valid) non-inertial frame of reference. Cerebral/concocted violations are perfectly allowed; it's the actual motion of object vis-a-vis object that is at the core of the lightspeed prohibition.

Sillies!

 

[RandallB]

When Einstein first wrote about GR, he refers to a frame K' in any kind of motion whatsoever with respect to system K (reference frame). He obviously includes rotational motion since he specifically mentioned SR's restriction to uniform rectilinear, non-rotary motion just prior.

Al

So your saying the FTL motion of stars a GR frame in rotational motion will obviously see is valid?
How so?
The light speed rule still applies doesn't it?
non-uniform rectilinear motions sure, but how does GR fix the rotational FTL issue?

 

[JustinLevy]

SR requires inertial rectilinear frames.
GR only adds that the frame does not need to be “inertial”
When did it supposedly delete the rectilinear part in place since Galileo & Newton?

GR is written as a coordinate system independent geometric equation. Choose any coordinate system and you can write out the coordinate representation for those geometric quantities and the equations will still hold. Written in coordinate dependent notation, there clearly are some coordinate systems which result in cleaner looking equations for the physics, but the content is the same regardless of the coordinate system.

So your saying the FTL motion of stars a GR frame in rotational motion will obviously see is valid?
How so?
The light speed rule still applies doesn't it?
non-uniform rectilinear motions sure, but how does GR fix the rotational FTL issue?

Even in SR objects can have a coordinate velocity greater than c.
The important restriction (for causality) is that the points on the worldline of any particle are never space-like separated.

Yes, the stars in a rotating frame are traveling faster than c. This does not violate relativity. The points on the world-line of the star are still time-like separated, just like they were according to the inertial frame.


Tam, you are still considerring coordinate velocity to be a physical quantity. Please reread my post here https://www.physicsforums.com/showpost.php?p=2032222&postcount=9
Also, other posters have made similar comments. Please stop asking the same question if you are just going to ignore the answers.

 

[JesseM]

So your saying the FTL motion of stars a GR frame in rotational motion will obviously see is valid?
How so?
The light speed rule still applies doesn't it?
non-uniform rectilinear motions sure, but how does GR fix the rotational FTL issue?

As JustinLevy says, even in a uniformly accelerating, non-rotating coordinate system in SR light beams can have a coordinate velocity different than c (that's true of Rindler coordinates for example). The "light speed rule" is only meant to apply to inertial coordinate systems in SR, or the locally inertial coordinate systems ('locally' meaning a coordinate system that only covers an infinitesimal patch of spacetime) of freefalling observers in GR.

 

[Al68]

So your saying the FTL motion of stars a GR frame in rotational motion will obviously see is valid?
How so?
The light speed rule still applies doesn't it?
non-uniform rectilinear motions sure, but how does GR fix the rotational FTL issue?

Well, others beat me to it, but, yes, distant stars have a coordinate velocity greater than c relative to a rotating reference frame. But the speed of light isn't c relative to this frame. The speed of light is c in inertial frames. So there's no issue to fix.

Even in rectilinearly accelerated frames, the relative velocity of objects can exceed c. No problem there either, since the speed of light is not c in those frames.

Al

 

[RandallB]

SR requires inertial rectilinear frames.
GR only adds that the frame does not need to be “inertial”
When did it supposedly delete the rectilinear part in place since Galileo & Newton?

[GR is written as a coordinate system independent geometric equation. Choose any coordinate system and you can write out the coordinate representation for those geometric quantities and the equations will still hold. Written in coordinate dependent notation, there clearly are some coordinate systems which result in cleaner looking equations for the physics, but the content is the same regardless of the coordinate system.

Even in SR objects can have a coordinate velocity greater than c.
The important restriction (for causality) is that the points on the worldline of any particle are never space-like separated.

Had to think about this one for awhile.
Maybe I’m not understanding how GR is being applied.
I do not see where extending accelerations to SR requires GR to use non- rectilinear frames and also use rotating frames and expect to retain the Postulates.
I understand how SR can have “proper speeds” greater than c and alternate frames with space-like separated locations can be at “wrong” locations in time past or future wrt a given frame. But in any given frame we can chose any point as a reference or starting point and the system works. Can that be true for any point in a rotating frame?

For example using Gravity Probe B the satellite accelerates in a circular orbit around the Earth and rotates as it does.
Using that rotating frame AND picking a local point on that frame at some point out near Pegasus as our reference point.
That local point will immediately see Pegasus locally and physically zoom by FTL by a huge amount.
But using the gyroscopes inside GP-B to define a rectilinear frame undergoing the same acceleration and orbit to define a local point near Pegasus on that frame and no such problems occur. (note that space like separation issues Hubble etc. still apply just as in SR)
Even in the GP-B tests they are comparing gyroscope alignment changes against an accelerating rectilinear frame previously established by the same gyroscope, not a rotating frame are they not?

This is the type of accelerating frame (even moving in circles, ellipses, or randomly) that I thought GR was dealing with. Not one where the frame itself rotates.
If GR really does deal with a truly rotating frame, where and how would it be used or applied?

 

[Dale]

Also, other posters have made similar comments. Please stop asking the same question if you are just going to ignore the answers.

I agree, the question has been answered fully, repeatedly, and completely: a rotating reference frame is by definition not an inertial frame so there is no general limit to the coordinate speed.

 

[Tam Hunt]

The original question has not been answered satisfactorily: the rotating Earth is, in fact, an inertial frame under the standard definition: "an inertial frame is a coordinate system tied to the state of the observer." In this, the inertial reference frame I'm positing is me, on the surface of the earth. As the stars whip by each night, they are, from my reference frame, moving far faster than the speed of light.

No one has satisfactorily answered why this is allowed, other than to say that this is "just" coordinate speed, not "real" speed. This is not the interpretation that Einstein advocated - he believed GR described reality, not just mathematical tricks. The whole point of SR and GR, in my understanding, is that there is no "real" speed, only relative speed. And a key consequence of SR is that nothing can go faster than the speed of light. As we've discussed in this thread, GR doesn't have quite the same prohibition, but there are very limited exceptions that don't apply in this case. From my point of view, the stars are moving, relative to me, far faster than the speed of light, in a radial fashion.

Justin suggests that Poincare symmetry can save us from this problem. I'll admit I don't understand Poincare symmetry very well, but my intuition here (for what it's worth) is that it still doesn't add up. There is a small industry of physicists that work quickly to debunk the various mainstream media stories on alleged phenomena that exceed the speed of light - such as the "motion of effects" phenomena that occasionally crop up in experiments and in the media. Why on Earth (pardon the pun) would these physicists be so actively debunking these stories if the FTL prohibition was not, as Justin suggests, a real prohibition at all? Justin also suggests that there are parts of our universe moving away from us faster than c. This is not the mainstream interpretation, which asserts instead that the furthest objects, quasars, are moving away at almost c. Or are you suggesting that cosmic inflatin is still going on? Justin, perhaps you can suggest some texts for me to read to follow up?

AI suggests that c is constant, and an upper limit on speed, only in inertial frames. This is patently not what SR or GR state, which both have as a postulate the constancy of c to ALL OBSERVERS. This is what results in the malleability of space and time (time dilation, length contraction) as objects approach the speed of light, in both SR and GR.

More generally, perhaps someone could suggest some papers or books to help me clarify the broader issues raised? None of the many books I've read on SR or GR have addressed this particular issue of the rotating Earth as a reference frame (or perhaps I missed it).

 

[D H]

The original question has not been answered satisfactorily: the rotating Earth is, in fact, an inertial frame under the standard definition: "an inertial frame is a coordinate system tied to the state of the observer." In this, the inertial reference frame I'm positing is me, on the surface of the earth. As the stars whip by each night, they are, from my reference frame, moving far faster than the speed of light.

You are (1) using http://en.wikipedia.org/wiki/Inertial_frame" as a definitive source, which it is not; and (2) quoting the wikipedia page out of context, which is a definite no-no. Quoting in context,

In physics, an inertial frame of reference is a reference frame tied to the state of motion of an observer. In inertial frames of reference, each physical law portrays itself in the same form, the most simple form. The contrasting case is the set of non-inertial frames, in which the laws of physics change from frame to frame, and the usual forces governing laws of physics must be supplemented by fictitious forces.

A rotating frame is not an inertial frame.

 

[Tam Hunt]

DH, I'm going a little beyond Wikipedia here. How does one distinguish between an inertial frame and a rotating frame of reference? And please don't appeal to the fixed stars, which suggests instantaneous action at a distance.

 

[D H]

DH, I'm going a little beyond Wikipedia here. How does one distinguish between an inertial frame and a rotating frame of reference? And please don't appeal to the fixed stars, which suggests instantaneous action at a distance.

No, it does not. In an inertial reference system the requisite instantaneous actions at a distance (i.e., fictitious forces) vanish. I suggest you read up on the International Celestial Reference Frame, for example http://aa.usno.navy.mil/faq/docs/ICRS_doc.php" at iers.org.