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.

 

[Dale]

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."

Ahh, that explains the confusion. This is most certainly not the definition of an inertial frame.

You can use anyone of several equivalent definitions of an inertial reference frame:
1) Newtonian definition: a reference frame where dp/dt=F.
2) Fictitious forces: a reference frame with no fictitious forces.
3) SR - standard form: a reference frame where all laws take take their "textbook" form.
4) GR - accelerometers: an observer* where an attached ideal accelerometer reads 0.

For a rotating reference frame you have:
1) dp/dt = F - 2mω x v - mω x (ω x r) - m dω/dt x r ≠ F
2) the fictitious Coriolis, centrifugal, and Euler forces exist
3) laws don't take their "textbook" form, e.g. see 1) for the form of Newton's 2nd law
4) an ideal accelerometer on the surface of the Earth reads g - 2ω x v - ω x (ω x r)

*Note, in GR there is no distinction between inertial and non-inertial frames (global frames), just between inertial and non-inertial observers (local frames).

 

[matheinste]

Hello Tam Hunt

Rindler describes an inertial frame as non-rotating uniformly moving.

Hartle says inertial frames are non-ritating

Schutz says that in an inertial frame all points are at rest with relation to the origin. This is not true of a rotating frame. (My favourite)

Weinberg (Gravitation) says the surface of the Earth is not an inertial frame.

These are just the better known authors. I could go on but what is the point

When i have some time, if nobody else does it first i will look up a reference for the apparent superluminal movement of stars
Matheinste

 

[RandallB]

... , in GR there is no distinction between inertial and non-inertial frames (global frames), just between inertial and non-inertial observers (local frames).

But that does not clear up my question/point in post #30.
Clearly you can have an accelerating (and therefore non-inertial) observer that does not rotate positioned at any coordinate in a non-rotating frame with rectilinear accelerations and “jerks” (accelerations of acceleration) frame.

The question is; if you have a rotating frame does GR require that center of rotation, which may also move with non-inertial rectilinear accelerations, be the only place a valid GR non-inertial observer can be positioned?

If that is true – then there is no issue for Tam.
I do not see how GR can use any position on a rotating frame as an obsevation point – but I’m not saying to Tam that this is true and a resolution to his issue
– because I do not know GR that well that is why I ask:

Does anyone here know GR well enough to say if it allows placing a non-inertial observer at any distance on a radial of a rotating frame or does it require it be at the center of rotation.?

 

[Dale]

The question is; if you have a rotating frame does GR require that center of rotation, which may also move with non-inertial rectilinear accelerations, be the only place a valid GR non-inertial observer can be positioned?

An observer can follow any arbitrary (timelike) worldline in any arbitrary coordinate system in GR. They are not restricted to be stationary in the coordinate system nor are they restricted to a "special" location. As long as you know the metric you can do the physics.

 

[atyy]

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.

Thanks for the links. I remember looking them up once before after reading a previous post of yours, and read something about a specification of a metric tensor. If there's a metric tensor, how can it be an inertial frame? Or is it a Minkowski metric written in some unfamiliar coordinates?

 

[D H]

Thanks for the links. I remember looking them up once before after reading a previous post of yours, and read something about a specification of a metric tensor. If there's a metric tensor, how can it be an inertial frame? Or is it a Minkowski metric written in some unfamiliar coordinates?

The ICRF is a pseudo inertial frame. First off, there is no such thing as a true inertial frame (name one!). Secondly, the frame origin is at the solar system barycenter, so it is an accelerating frame. Thirdly, it is a Cartesian frame. Better stated, the ICRF is a non-rotating frame. The gory details are in IERS Technical Note 29. The five "Comparision of 'Old' and 'New' Concepts" position papers starting on PDF page 25 are particularly relevant. IERS Technical Note 29 can be accessed (no fee) from http://www.iers.org/MainDisp.csl?pid=46-25773. You can download the entire document or just download the relevant sections. The five comparison position papers can be downloaded individually (in which case, ignore the comment about PDF page 25).

 

[JesseM]

The question is; if you have a rotating frame does GR require that center of rotation, which may also move with non-inertial rectilinear accelerations, be the only place a valid GR non-inertial observer can be positioned?

This question doesn't really make sense to me--what do you mean by "observer", exactly? Coordinate systems aren't linked to "observers" at unique positions, you can have a human observer at absolutely any position in any coordinate system, just like you can have an asteroid at absolutely any position in any coordinate system, human observers aren't treated any differently from any other physical object in relativity. Of course if an observer measures the velocity of some nearby object (like a light beam) in his own local neighborhood using a locally inertial coordinate system, the velocity in these locally inertial coordinates may be totally different from the velocity of the same object in a non-inertial coordinate system like a rotating frame (in particular, an observer will always measure light to move past him at c in locally inertial coordinates, even though in a non-inertial frame the light may move past him at some velocity very different from c).

 

[truhaht]

The original question has not been answered satisfactorily
... 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
... From my point of view, the stars are moving, relative to me, far faster than the speed of light, in a radial fashion.

Sure, I answered it justly in post #25. And you don't mean "radial"; you mean "tangentially", which is quite the opposite. If a star were moving faster than the speed of light radially from you, then that would really be news! That would be a violation. No actual object/body/particle can move toward or away from another at light speed. In the rotating Earth frame, no celestial object moves superluminally with regard to you or with regard to any other actual material thing. That's essential. The superluminal speeds to which you refer are motions of real bodies with respect to the coordinate lines that you've defined in your mind's eye, and those aren't actual physical things. This is the essence -- TRUST me!

 

[RandallB]

An observer can follow any arbitrary (timelike) worldline in any arbitrary coordinate system in GR. They are not restricted to be stationary in the coordinate system nor are they restricted to a "special" location. As long as you know the metric you can do the physics.

but they canot move in the frame they are in.
(Edit; better to state “in” as used here; as the reference frame defined as the “rest Frame” for the observer’s “location”; location understood as a “point particle” location.)
Do they have to be at the center of rotation in their frame?

 

[JesseM]

but they canot move in the frame they are in. Do they have to be at the ceneter of rotation in their frame?

Observers aren't "in" one frame or another, all frames make the same predictions about all physical observations they make. For inertial observers you can talk about their unique rest frame, for non-inertial ones there are an infinite variety of coordinate systems in which they remain at rest.

 

[Dale]

but they canot move in the frame they are in. Do they have to be at the ceneter of rotation in their frame?

No, for example, an observer located on the end of a turbine blade would be at rest in a rotating frame centered on the axle. Such an observer would obviously be non-inertial.

However, I would like you to pay attention to JesseM's point. Observers aren't "in" one frame, they "have" a frame (or rather many frames) where they are at rest. For an observer O this is called "O's rest frame" or simply "O's frame". But they are "in" all frames.

 

[Tam Hunt]

I'm making some progress here, so thanks to everyone for the great information.

From the various posts and from some additional research (Stanford Enc. of Philosophy has a great entry on inertial frames), I see now that there is no such thing as an inertial frame. Einstein actually pointed this out in his 1938 book The Evolution of Physics, but I didn't fully internalize the reasoning.

So, let's dispense with inertial frames and any discussion of SR and focus instead on frames of reference more generally, and GR.

I'm still having trouble understanding how GR explains the apparent superluminal motion of the stars with respect to the rotating Earth as a reference frame (not an inertial frame). JustinLevy hasn't chimed in further re Poincare symmetry, but perhaps DaleSpam or someone else could chime in on this?

Truhaht, thanks for pointing out my error in terminology - you are right that I meant tangential motion, not radial. However, it seems that your response still doesn't explain the apparent superluminal motion. Just as the Earth rotates about the center of gravity of the solar system, we validly describe the fixed stars as rotating around the Earth (regardless of the fact that the Earth is most definitely not the center of gravity of the fixed stars). As such, this apparent motion is, according to the general principle of relativity, as 'real' as the motion of the Earth around the Sun. All motion is relative and, in my hypothetical, I am describing the motion of the fixed stars relative to the rotating earth. And in this description, it seems to me still that superluminal velocities are being observed.

The Stanford Enc. entry suggests at the end that GR describes space as not a fixed coordinate system, as in SR or classical mechanics, but a variable spacetime system deformed by matter and energy throughout. As such, is this what JustinLevy has described as the distinction between local poincare symmetry and global poincare symmetry? If so, I'd like to know how the distinction is made. At what extent does local give way to global? It seems that such a distinction must necessarily be rather arbitrary and thus rather dubious epistemologically and ontologically.

 

[truhaht]

Truhaht, thanks for pointing out my error in terminology - you are right that I meant tangential motion, not radial. However, it seems that your response still doesn't explain the apparent superluminal motion. Just as the Earth rotates about the center of gravity of the solar system, we validly describe the fixed stars as rotating around the Earth (regardless of the fact that the Earth is most definitely not the center of gravity of the fixed stars). As such, this apparent motion is, according to the general principle of relativity, as 'real' as the motion of the Earth around the Sun.

Real, sure, I suppose, just not superluminal. But if you're so convinced it is real then why did you just then above refer to them as merely "apparent motion", hmmm?

All motion is relative and, in my hypothetical, I am describing the motion of the fixed stars relative to the rotating earth. And in this description, it seems to me still that superluminal velocities are being observed.

Well then you're wrong because superluminal velocities are NOT being observed and in general, superluminal velocities (obviously) CANNOT be observed. Superluminal velocities will not be observed by anyone at any time. You can define your observing frame as the rotating Earth surface alright, but those stars aren't moving superluminally "relative to the rotating earth" (your words). Inertial frame SR math is simple and so it can readily be applied to the entirety of the observer's X-Y-Z space. From an inertial frame observatory, if something moves transverse to your line of sight, then that viewed object's speed is still computed as being relative to your defined coordinate axes and the speed will not exceed light speed. But in actuality, if you can just twist your neck and thereby follow the movement of that transverse-moving body, then its speed approximates zero, relative to your twisting (rotating) head. In order to address the hard core crux of the superluminal prohibition without such ambiguities, one must talk about the most direct case, and that is the case of a body moving straight toward or straight away from the observer. That speed CANNOT be superluminal. And since any tangible body/object/particle can potentially be an observatory, you will never ever encounter superluminal speed of one object directly receding from or directly closing in on another material object.

All the above is no different from what I asserted in my other two posts: a mere cerebral concoction of coordinate lines in superluminal motion with respect to the paths of tangible bodies does not a violation make.

 

[JesseM]

I'm making some progress here, so thanks to everyone for the great information.

From the various posts and from some additional research (Stanford Enc. of Philosophy has a great entry on inertial frames), I see now that there is no such thing as an inertial frame. Einstein actually pointed this out in his 1938 book The Evolution of Physics, but I didn't fully internalize the reasoning.

There certainly is such a thing as an inertial frame in the flat spacetime SR. And in GR there are local inertial frames, meaning that if you zoom in on an arbitrarily small neighborhood of a point in curved spacetime, the spacetime in that neighborhood gets arbitrarily close to flat. This is the basis for the equivalence principle.

I'm still having trouble understanding how GR explains the apparent superluminal motion of the stars with respect to the rotating Earth as a reference frame (not an inertial frame). JustinLevy hasn't chimed in further re Poincare symmetry, but perhaps DaleSpam or someone else could chime in on this?

Again, light only is guaranteed to travel at c in inertial frames, and a rotating frame is not one. In the local inertial frame of an observer who passes right next to a light beam, that light beam is moving at c.

The Stanford Enc. entry suggests at the end that GR describes space as not a fixed coordinate system, as in SR or classical mechanics, but a variable spacetime system deformed by matter and energy throughout. As such, is this what JustinLevy has described as the distinction between local poincare symmetry and global poincare symmetry? If so, I'd like to know how the distinction is made. At what extent does local give way to global? It seems that such a distinction must necessarily be rather arbitrary and thus rather dubious epistemologically and ontologically.

My understanding is that global Poincare symmetry only applies in the case of flat SR spacetime, which is what GR would predict in the case of a universe with no mass or energy anywhere (or infinitesimal amounts). And local Poincare symmetry is the idea I described above that GR reduces to SR in the limit as you pick smaller and smaller neighborhoods of a point in spacetime (the equivalence principle article I linked to may help you to understand this idea, I recommend reading it all the way through).

 

[D H]

From the various posts and from some additional research (Stanford Enc. of Philosophy has a great entry on inertial frames), I see now that there is no such thing as an inertial frame. Einstein actually pointed this out in his 1938 book The Evolution of Physics, but I didn't fully internalize the reasoning.

Whoa now. You are once again using faulty references and taking those faulty references out of context. The Stanford Encyclopedia of Philosophy is a great resource -- if you want to study philosophy, that is. Using it as the basis for reasoning about physics is faulty. The basis for your claim "there is no such thing as an inertial frame" is apparently this from http://plato.stanford.edu/entries/spacetime-iframes/: "the variable curvature of spacetime makes the imposition of a global inertial frame impossible". You omitted the key word global. Moreover, the referenced page does not address the issue of rotation whatsoever, so it is quite out-of-context.

Cherry-picking, quoting out of context, use of faulty sources, and use of logical fallacies might be acceptable in environmental law, it is not acceptable in physics. The preferred language for physics is mathematics. The mathematics of general relativity is embodied in the metric tensor. The metric tensor for a non-rotating frame differs from that for versus rotating frames versus a rotating frame are distinguishable. All local Lorentz frames are non-rotating and have an origin that follows a geodesic: They are a local inertial frame. In such a frame the distant stars will not have superluminal velocity.