FTL in General Relativity

[melech]

I don't understand how it is possible for an object to be measured as moving through space at FTL speeds. The classic example I've seen is an observer on a rapidly spinning top. The stars will appear to be moving at speeds exceeding light.
This raises all kinds of questions to me:
Doesn't this violate the rules of SR? Does the constant-speed-of-light principle not apply to accelerating objects, but only to uniform motion? Wouldn't other consequences of SR prevent this measurement, e.g., foreshortening in the direction of motion and increase of the mass of the moving object to infinity as it reached or exceeded the speed of light?
If GR does permit these speeds in certain cases, are there any potential applications for communicating information? (I can't think of any.)

[macaw]

I don't understand how it is possible for an object to be measured as moving through space at FTL speeds. The classic example I've seen is an observer on a rapidly spinning top. The stars will appear to be moving at speeds exceeding light.

...but they don't. Rotation is absolute, so it is the observer (not the stars) that is moving. And, there is no object moving, it is just his line of sight sweeping around at angular speed . The tip of the line of sight vector seeps at tangential speed where is the radial distance from the center. In the example, is not the speed of an object, so, for very large you can have without contradicting relativity.

Doesn't this violate the rules of SR?

No, see above.

[worzel]

But GR allows you to use the rapidly spinning top's frame of reference as well as any other. And in that frame of reference distant stars will move at speeds far exceeding the speed of light.

I don't know how to word it correctly, but the reason it doesn't violate SR is because it is only very distant things that can exceed c, and only as an artefact of the choice of coordinates. Still nothing can move faster than c relative to anything nearby, where whatever GR coordinates are in use increasingly look like SR coordinates.

[Ivan Viehoff]

The earth can be your spinning top. So Alpha Centauri, 4 light years away, apparently travels about 25 light years a day in a great circle around an observer standing on the earth. But if you actually consider the relative motion of the observer on the surface of the earth at any moment in time, and Alpha Centauri, we discover it is not faster than light. It is the vector sum of the instantaneous velocity of the surface of the earth, the instantaneous velocity of the earth's rotation around the sun, and the relative motion of the sun and Alpha centauri. All of these are small vectors with magnitude much less than the speed of light, and there is no way of summing them to make them faster than the speed of light.

Patterns can move faster than light. That is what the earthbound astronomer seeing in the apparent daily motion of the stars across the sky is seeing, the motion of a pattern. But it is not true motion, you can't transmit information faster than light by such a method.

[worzel]

It's not just spinning frames of reference, though. Superluminal galaxies are moving away from you at faster than the speed of light, again, precisely because of the choice of coordinate system.

[melech]

Thanks all for thought-provoking answers. Per usual on Baut, the replies raise more questions in my mind:

lMacow: you said the rotating motion of the top (or the earth) is absolute. This issue is really fundamental to my understanding. I thought Einstein's basic concept was that ALL motion is relative, and the inertial forces felt as proof of rotation could just as easily be explained as a gravitational tug by the universe as it rotates around the stationary top. In that case the universe could be legitimately seen as rotating at FTL, no?

Worzel: You said that nothing can move faster than c only when compared to a nearby object. Does this mean that SR is only valid locally? If I could observe a far distant object moving at great relative speeds to me, would the normal effects of SR not be measured, such as foreshortening, time dilation, etc? I have the same questions about superluminal galaxies, if we could somehow observe than.

[macaw]

Thanks all for thought-provoking answers. Per usual on Baut, the replies raise more questions in my mind:

lMacow: you said the rotating motion of the top (or the earth) is absolute. This issue is really fundamental to my understanding. I thought Einstein's basic concept was that ALL motion is relative,

Nope, this is not what relativity says, accelerated motion is absolute.

In that case the universe could be legitimately seen as rotating at FTL, no?

No.

[caveman1917]

Nope, this is not what relativity says, accelerated motion is absolute.

In SR yes, but the question is framed in GR.
The resolution lies not in the absoluteness of proper acceleration, but in the non-existence of global Lorentz invariance.

melech's answer to the question seems correct to me. Distant stars can be legitimately seen as moving at FTL, but this is not a problem since Lorentz invariance only applies locally.

In any case, if you would try to resolve the question by pointing to proper acceleration, you'd still have the problem of the FTL recessional velocity of distant galaxies to explain, where no proper acceleration is felt for either us or the galaxy.

[macaw]

In SR yes, but the question is framed in GR.

Nope, the OP opens with "..... an observer on a rapidly spinning top. The stars will appear to be moving at speeds exceeding light.". There is no GR, the problem is not involving any calculations dependent on gravitation, it is a purely well-known kinematic "paradox", very much like the "rotating searchlight" or the "superluminal scissor blades". This is simply SR applied to rotating frames.

In any case, if you would try to resolve the question by pointing to proper acceleration, you'd still have the problem of the FTL recessional velocity of distant galaxies to explain, where no proper acceleration is felt for either us or the galaxy.

I didn't point to any acceleration. I pointed out that rotation is absolute in SR (not so in GR but this is irrelevant since the problem does not involve any gravitation, so it is an SR problem).

[caveman1917]

Nope, the OP opens with "..... an observer on a rapidly spinning top. The stars will appear to be moving at speeds exceeding light.". There is no GR. This is simply SR applied to rotating frames.

Which, as we both know, is a problem that rests within GR to treat fully. Take the metric for an "observer on a rapidly spinning top" and calculate the Christoffel symbols, you'll find that not all of them vanish. We're in curved space here.

But i sense an earlier discussion coming back up
Might we just conclude that there are two ways to solve the problem?
1. In SR acceleration is absolute, and Lorentz invariance only applies to inertial frames. No problem with the question.
2. In GR all frames (inertial and accelerating) are equivalent, but Lorentz invariance applies only locally. Again no problem with the question.

[macaw]

Which, as we both know, is a problem that rests within GR to treat fully.

No, it isn't. SR handles rotating motion just fine. The GR machinery comes in only when you have gravitating bodies.
We have been over this in the past. I do not mind being corrected when I am wrong but this is not the case. This is a simple SR "paradox", you can find it in Taylor and Wheeler "Spacetime Physics".

Might we just conclude that there are two ways to solve the problem?
1. In SR acceleration is absolute, and Lorentz invariance only applies to inertial frames. No problem with the question.

The problem is a "classical" physics problem, has nothing to do with Lorentz invariance. It is explained simply that for "large enough" and / or "large enough" you can have .

2. In GR all frames (inertial and accelerating) are equivalent, but Lorentz invariance applies only locally. Again no problem with the question.

This is not a GR question, it isn't even an SR question if you think about it, it is a "classical" physics question, just like the explanation of the Sagnac effect. You don't even need to bring in any metric, rotating frames, etc.

[chornedsnorkack]

We know how gravitational fields affect energy of zero rest mass particles emitted outside a Schwarzschild black hole (redshifted to infinite redshift at event horizon) and inside the black hole (do not escape, all light rays terminate at singularity).

If a (nonzero rest mass) body emitting tachyons of fixed energy and fixed imaginary rest mass (small compared to the energy when emitted), how is the energy of the tachyons for a recipient at infinity affected?

[caveman1917]

No, it isn't. SR handles rotating motion just fine. The GR machinery comes in only when you have gravitating bodies.

Note that i specifically said "treat fully".
You will hear no argument from me that SR cannot explain this simple paradox, it can. However saying that it "handles rotating motion just fine" is not completely correct either, Ehrenfest's paradox remains unsolved (and perhaps unsolvable) within SR, rotating motion requires GR to treat fully.

The problem is a "classical" physics problem, has nothing to do with Lorentz invariance. It is explained simply that for "large enough" and / or "large enough" you can have .

Which, if you note that Lorentz invariance is what is giving the expectation that "nothing can exceed c", is the same thing as what i said in different words: "Lorentz invariance only applies to inertial frames, no problem with the question".

This is not a GR question, it isn't even an SR question if you think about it, it is a "classical" physics question, just like the explanation of the Sagnac effect. You don't even need to bring in any metric, rotating frames, etc.

Perhaps, but note that i wasn't saying that this specific paradox requires the full machinery. What i was responding to was that you said melech's interpretation was incorrect, i was only saying that assertion wasn't true. His answer was acceptable, that's what i was mainly responding to.

[caveman1917]

In any case, since melech opened this thread with the title specifying an answer in GR, i think it is safe to say that what he said in post 6 is correct in his given context.

[macaw]

Note that i specifically said "treat fully".

The OP doesn't need any GR in order for it to be explained.

You will hear no argument from me that SR cannot explain this simple paradox, it can.

Then we are done. As I explained, this is not even an SR problem, it is a "classical" kinematics problem.

However saying that it "handles rotating motion just fine" is not completely correct either, Ehrenfest's paradox remains unsolved (and perhaps unsolvable) within SR, rotating motion requires GR to treat fully.

To my best knowledge, Ehrenfest paradox has nothing to do with GR. The reason is simple, there are no gravitating bodies in the paradox, hence GR machinery does nothing for it. To properly treat Ehrenfest paradox you need to give up on ideally rigid bodies. All modern treatments are in FLAT spacetime.

What i was responding to was that you said melech's interpretation was incorrect, i was only saying that assertion wasn't true. His answer was acceptable, that's what i was mainly responding to.

In other words, you think that the paradox "violates SR", as melech claimed? This is the precise statement I took issue with. Do you think that his statement is correct and my taking issue with it is incorrect?

[macaw]

I don't understand how it is possible for an object to be measured as moving through space at FTL speeds. The classic example I've seen is an observer on a rapidly spinning top. The stars will appear to be moving at speeds exceeding light.
This raises all kinds of questions to me:
Doesn't this violate the rules of SR?

1. Your scenario has nothing to do with GR. The name of your thread is incorrect.
2. The scenario does not "violate SR".
3. As a matter of fact, your scenario has nothing to do with SR either, perhaps with the exception that it involves the light speed limit in the inequality
4. The one way light speed isotropy is maintained locally in rotating frames, see chapter 6 of this paper.

[caveman1917]

The OP doesn't need any GR in order for it to be explained.

Maybe he doesn't, but he specifies he wants to understand FTL in general relativity, so why not provide him with that?

Then we are done. As I explained, this is not even an SR problem, it is a "classical" kinematics problem.

But at that stage you have not addressed the problem he stated. He already knows that for large enough and large enough you'd get . That's why he said

The stars will appear to be moving at speeds exceeding light.

His question that you left unanswered by reducing it to a simple kinematics problem is

Doesn't this violate the rules of SR?

It is obvious that the "rule of SR" he is thinking of is global Lorentz invariance.
His question is thus: is inconsistent with global lorentz invariance? And the answer to that is yes, that's why he's having the "paradox".
The actual answer to resolve his paradox is: it's not a problem because the observer is not in an inertial frame and lorentz invariance only applies in inertial frames.
All of this is staying within SR. In fact, if you want to stay in SR, he answered his own question in the very next sentence:

Does the constant-speed-of-light principle not apply to accelerating objects, but only to uniform motion?

To my best knowledge, Ehrenfest paradox has nothing to do with GR.

It has to do with GR in that it can only be resolved completely within GR, not SR. Please see for example Nikolic 1999.

The reason is simple, there are no gravitating bodies in the paradox, hence GR machinery does nothing for it.

This does not follow. While gravitating bodies must be considered in GR, this does not mean that GR is only applicable when you have gravitating bodies, think for example of de Sitter space.

To properly treat Ehrenfest paradox you need to give up on ideally rigid bodies.

I'm sorry, but in the context of this discussion, this makes no sense to me. We are talking about stars that seem to rotate around the observer as if they are fixed on an ideally rigid rotating body. The fact that there is no real rigid body, just a "virtual" one, makes no difference whatsoever to the problem at hand. We must still treat it as if they are fixed on one.

In other words, you think that the paradox "violates SR", as melech claimed?

Come on, we both know that is not what i think, in fact i am saying the complete opposite and showing why it doesn't "violate SR".

Do you think that his statement is incorrect and my taking issue with it is incorrect?

I think his statement was correct, and you taking issue with it was incorrect.
For reference here is the post i took issue with:

lMacow: you said the rotating motion of the top (or the earth) is absolute. This issue is really fundamental to my understanding. I thought Einstein's basic concept was that ALL motion is relative

Nope, this is not what relativity says, accelerated motion is absolute.

In that case the universe could be legitimately seen as rotating at FTL, no?

No.

I see that perhaps a misunderstanding could have arisen in that he is talking about GR, and you about SR (neither of you specify in this post).
If you read both his assertions again, considering him to be talking in a GR framework, do you still disagree with them?

[macaw]

Maybe he doesn't, but he specifies he wants to understand FTL in general relativity, so why not provide him with that?

Because, like you, he mixes SR and GR.

But at that stage you have not addressed the problem he stated. He already knows that for large enough and large enough you'd get .

I cannot read his mind, I do not know what he knows and what he doesn't, this is precisely why I put the problem in its mathematical formalism.

His question that you left unanswered by reducing it to a simple kinematics problem

It is a "classical" problem, so it has a simple classical answer. If you want to complicate it unnecessarily, you can bring in any other theory you want.

It is obvious that the "rule of SR" he is thinking of is global Lorentz invariance.
His question is thus: is inconsistent with global lorentz invariance? And the answer to that is yes, that's why he's having the "paradox".

The correct answer, is no, it isn't. There is no "paradox", can take any value. The spotlight of a rotating lighthouse can sweep arcs at speeds far exceeding c. See Taylor and Wheeler, "Spacetime Physics" top of page 72 on this subject. This is precisely what the OP is talking about. The fact that he conflates it with GR is throwing a lot of people off.

It has to do with GR in that it can only be resolved completely within GR, not SR. Please see for example Nikolic 1999.

I know Nikolic personally and I have used his papers (and this particular one extensively). Nikolic makes it quite clear, several times that the treatment is FLAT spacetime, meaning NO GR. Whoever wrote the wiki reference claiming that Nikolic resorts to GR, doesn't know what he's talking about.

I think his statement was correct, and you taking issue with it was incorrect.

Well, I explained several times why his statement is not correct.

If you read both his assertions again, considering him to be talking in a GR framework, do you still disagree with them?

His whole thread has nothing to do with GR. Heck, it has nothing to do with SR either.

[caveman1917]

Because, like you, he mixes SR and GR.

Where have i mixed SR and GR?

The correct answer, is no, it isn't.

The statement is correct as it stands. I know it can go a lot faster than c.
Note, again, that his question wasn't "can it go faster than c?" but "does this violate SR?". Showing that it can go faster than c does not answer the question, the question is answered by showing why it doesn't violate SR.

I know Nikolic personally and I have used his papers (and this particular one extensively). Nikolic makes it quite clear, several times that the treatment is FLAT spacetime, meaning NO GR. Whoever wrote the wiki reference claiming that Nikolic resorts to GR, doesn't know what he's talking about.

His result is that different observers with no relative motion can not be considered in the same lorentz frame, each is in its own frame that only locally approximates minkowski. This certainly looks a lot like GR to me.

Well, I explained several times why his statement is not correct.

So you are claiming that not all frames are equivalent in GR?

His whole thread has nothing to do with GR. Heck, it has nothing to do with SR either.

How can the question "does this violate SR" have nothing to do with SR?

[macaw]

Where have i mixed SR and GR?



The statement is correct as it stands. I know it can go a lot faster than c.
Note, again, that his question wasn't "can it go faster than c?" but "does this violate SR?". Showing that it can go faster than c does not answer the question, the question is answered by showing why it doesn't violate SR.

What in , "it does not violate SR since it has nothing to do with SR" did you not understand?

His result is that different observers with no relative motion can not be considered in the same lorentz frame,

...exactly as in the treatment of hyperbolic motion (SR)

each is in its own frame that only locally approximates minkowski. This certainly looks a lot like GR to me.

It might look like GR to you but it isn't. Flat spacetime ain't GR. Never was, never will.

How can the question "does this violate SR" have nothing to do with SR?

Because this is a "classical" physics problem, it has nothing to do with SR. Did you look it up in Taylor and Wheeler? They give the same exact explanation I gave, with a few more words sprinkled around.

[Ken G]

In my opinion, the semantic issue here is that there are really three flavors of relativity, but only two labels to hang on them (SR and GR), and this causes endless confusion. The three flavors are motion as seen by inertial observers (standard SR), motion as seen by accelerated observers or in non-inertial coordinates (to which only a careful and nontypical version of SR axioms apply), and motion under the influence of gravity (the standard meaning of GR). There's really no name for the second flavor of relativity, which is GR with no curvature sources.

Now, I already know one individual who is thinking "but you don't need GR if there is no curvature." I know this, so does caveman1917. Needing GR isn't the point at all. The axioms of SR apply only for an inertial observer, and Einstein didn't like that, he viewed it as a flaw in the theory that he knew he would need to correct to get gravity right, but even if there was no such thing as gravity, Einstein still wouldn't have liked axioms that only work for inertial observers. So he would have pressed on to the curvature-free version of GR, the Einstein equation with no curvature sources, all the same. But since when he did this, he also put in gravity, we will forever have people claim that GR only applies to gravity sources, and they will solve problems of that second flavor by referring to a chain of inertial observers who are instantaneously comoving with any noninertial observers we wish to interrogate. That certainly works, but it isn't an axiomatic theory that works for all observers. GR is.

Another point to bear in mind is that in the limit of an arbitrarily small observer, say a point observer (as a gedankenexperiment), then such an observer can rotate without being in an accelerated frame. It is then purely an issue of coordinates, which is where it belongs. But SR axioms don't work in just any old coordinates, the way they are usually expressed (such as, the speed of light always equals c). That is not a coordinate-free theory because it requires a certain construction of synchronized clocks and so forth. This is fixed in more formal treatments of SR, and it is fixed in GR too, but it doesn't matter the label we hang on it-- what matters is that the coordinatization of motion is not the physical version of motion. Coordinate speeds can exceed c, the physical meaning of a speed that cannot exceed c is a relative speed of two objects passing each other at the same event-- i.e., something local.

[macaw]

In my opinion, the semantic issue here is that there are really three flavors of relativity, but only two labels to hang on them (SR and GR), and this causes endless confusion. The three flavors are motion as seen by inertial observers (standard SR), motion as seen by accelerated observers or in non-inertial coordinates (to which only a careful and nontypical version of SR axioms apply), and motion under the influence of gravity (the standard meaning of GR). There's really no name for the second flavor of relativity, which is GR with no curvature sources.

Now, I already know one individual who is thinking "but you don't need GR if there is no curvature." I know this, so does caveman1917. Needing GR isn't the point at all. The axioms of SR apply only for an inertial observer, and Einstein didn't like that, he viewed it as a flaw in the theory that he knew he would need to correct to get gravity right, but even if there was no such thing as gravity, Einstein still wouldn't have liked axioms that only work for inertial observers. So he would have pressed on to the curvature-free version of GR, the Einstein equation with no curvature sources, all the same. But since when he did this, he also put in gravity, we will forever have people claim that GR only applies to gravity sources, and they will solve problems of that second flavor by referring to a chain of inertial observers who are instantaneously comoving with any noninertial observers we wish to interrogate. That certainly works, but it isn't an axiomatic theory that works for all observers. GR is.

Yeah, one of those guys is called Rindler. And he knows his stuff. One of the funny things is that, whenever possible, I reduce GR problems to Rindler coordinate problems (using EP) since they are much easier to solve this way. So, the idea is to do things exactly opposite, not bring in the machinery of GR (unless absolutely necessary) and use SR (in accelerated frames) instead.
Besides, this thread is not about GR (and all the mistaken claims that Ehrenfest paradox is solved using GR). It is not even SR, it is a simple problem of classical kinematics. I can direct you as well to the exercise on top of page 72 in Taylor-Wheeler "Spacetime Physics". On the other hand, I pretty much explained the solution to the exercise already, so you don't have to check Taylor-Wheeler, they are essentially saying the same exact thing I have already posted.

But SR axioms don't work in just any old coordinates, the way they are usually expressed (such as, the speed of light always equals c).

Hmm, they work just fine in uniformly rotating frames. Nikolic shows that in a very nice way, he extended the Rindler coordinates to rotating frames.

Coordinate speeds can exceed c, the physical meaning of a speed that cannot exceed c is a relative speed of two objects passing each other at the same event-- i.e., something local.

Sure.

[worzel]

1. Your scenario has nothing to do with GR. The name of your thread is incorrect.

Surely every physically realizable scenario has everything to do with GR if GR is our best theory. How could we possibly know that GR is consistent with other theories if we limit ourselves to only thinking in terms of GR when no other theory will do?

[macaw]

Surely every physically realizable scenario has everything to do with GR if GR is our best theory.

Not if the scenario is a kinematics one, see the Taylor-Wheeler reference.

How could we possibly know that GR is consistent with other theories if we limit ourselves to only thinking in terms of GR when no other theory will do?

This thread is not a means of testing GR. It provides absolutely nothing that connects the scenario with GR.

[worzel]

Not if the scenario is a kinematics one, see the Taylor-Wheeler reference.

Are you saying that GR doesn't work for kinematic scenarios?

This thread is not a means of testing GR.

Indeed. But for the sake of understanding a theory, as well as to test that it is consistent with previous theories where the difference should be negligible, it is perfectly valid to ask how particular scenarios are modelled in that theory, even when the scenario does not require it.

It provides absolutely nothing that connects the scenario with GR.

If GR is our best model to date of the universe at large, then I would say that every physically realizable scenario does connect with GR. But moreover, in GR there are many examples (of which a spinning frame of reference is one) where we have distant objects who's coordinate speed is FTL seemingly in contradiction to SR. So the OP's question is a very good one in my opinion.

[macaw]

Are you saying that GR doesn't work for kinematic scenarios?

If you read my posts you'd have seen that this is not what I am saying. What I am saying is that there is no point in bringing in the heavy machinery of GR when you are dealing with a simple problem absent any gravitating bodies. Just the same as not worth bringing in the machinery of SR at low speeds.

Indeed. But for the sake of understanding a theory, as well as to test that it is consistent with previous theories where the difference should be negligible, it is perfectly valid to ask how particular scenarios are modelled in that theory, even when the scenario does not require it.

OK, I have explained the outcome using the mathematical formalism of classical kinematics, why don't you explain it using the mathematical formalism of GR? This should be more interesting than bickering about methodology.

If GR is our best model to date of the universe at large, then I would say that every physically realizable scenario does connect with GR. But moreover, in GR there are many examples (of which a spinning frame of reference is one) where we have distant objects who's coordinate speed is FTL seemingly in contradiction to SR.

...but there is no contradiction with SR. I have already explained that. If you don't like my explanation, look up page 72 in Taylor-Wheeler, it tells you essentially the same, in more words.

So the OP's question is a very good one in my opinion.

OK, can you use the mathematical formalism of GR to answer the question? Bring in whatever you like, Kerr metric, covariant derivatives, anything that you see fit. But use math, please.

[Ken G]

Yeah, one of those guys is called Rindler. And he knows his stuff. One of the funny things is that, whenever possible, I reduce GR problems to Rindler coordinate problems (using EP) since they are much easier to solve this way.

Fine, but the point is, gravity acts locally just like acceleration (the EP is a GR axiom), and so instead of treating acceleration with a chain of inertial observers, you can roll it all up in the package of Rindler coordinates, and this is just the same as applying GR machinery locally. The EP assures us that we can use either approach and arrive at the same thing as Rindler coordinates, and we can do it all without curvature sources because we are not dealing in any tidal effects.

So, the idea is to do things exactly opposite, not bring in the machinery of GR (unless absolutely necessary) and use SR (in accelerated frames) instead.

I agree that the machinery of GR is nice to avoid, but that's just the point of the EP-- it is the crossover point between GR and SR when you don't have to worry about tidal effects, i.e., exactly when there is no real gravity around-- the "flavor 2" I talked about above.

Besides, this thread is not about GR (and all the mistaken claims that Ehrenfest paradox is solved using GR).

That depends on what one means by "about" GR-- one does not need gravity, that's true, and even your mention of accelerated observers was a red herring too. It is not about gravity, it is not even about acceleration-- it is just about coordinates. But GR is the dynamical theory that makes no assumptions at all about coordinates, which is not the case for the standard way SR gets explained. I've mentioned before that this is why I think people should never be taught SR in the first place-- just teach GR, but start with no curvature sources, and that would lead right to Rindler if one wanted coordinates, or really do it right and leave it all completely coordinate-free at the level of the theory, and bring in Rindler when you are ready to compare to some quantitative experiments involving multiple clocks.

[macaw]

That depends on what one means by "about" GR-- one does not need gravity, that's true, and even your mention of accelerated observers was a red herring too.

I did not mention any accelerated observers, what I said textually was:

"...but they don't. Rotation is absolute, so it is the observer (not the stars) that is moving. And, there is no object moving, it is just his line of sight sweeping around at angular speed . The tip of the line of sight vector seeps at tangential speed where is the radial distance from the center. In the example, is not the speed of an object, so, for very large you can have without contradicting relativity."

I've mentioned before that this is why I think people should never be taught SR in the first place-- just teach GR, but start with no curvature sources, and that would lead right to Rindler if one wanted coordinates, or really do it right and leave it all completely coordinate-free at the level of the theory, and bring in Rindler when you are ready to compare to some quantitative experiments involving multiple clocks.

Well, at this point I would ask you the same thing I asked worzel, use the mathematical formalism of GR in order to explain away the OP "paradox" . Please do not substitute prose for math and don't tell me that your prose is "actually math".

[worzel]

...but there is no contradiction with SR. I have already explained that.

I never said there was.

The OP's question was clear, follows naturally from learning SR and then discovering FTL coordinate speeds in GR, and is easily answerable in the terms it was asked.

In your determination to find the question ill-conceived you are completely missing the fundamental point: in GR a spinning frame of reference is as good as any other, and in that frame of reference those distant stars do actually have FTL speeds. While that doesn't contradict SR, it may initially seem like it does, and therefore require an explanation as to why it does not.

Simply saying, "well, here's how angular velocity works in classical kinematics", completely fails to acknowledge the actual question asked, let alone answer it.

[macaw]

I never said there was.

The OP's question was clear, follows naturally from learning SR and then discovering FTL coordinate speeds in GR, and is easily answerable in the terms it was asked.

In your determination to find the question ill-conceived you are completely missing the fundamental point: in GR a spinning frame of reference is as good as any other,

If the spinning frame is GR material (according to you), why don't you use the mathematical formalism of GR and resolve the "paradox"?

and in that frame of reference those distant stars do actually have FTL speeds. While that doesn't contradict SR, it may initially seem like it does, and therefore require an explanation as to why it does not.

To which I gave a perfectly valid explanation many posts ago. Why don't you give an explanation using GR? This should be a lot more instructive.

Simply saying, "well, here's how angular velocity works in classical kinematics", completely fails to acknowledge the actual question asked, let alone answer it.

So, if you think that neither I , nor Taylor-Wheeler answered the question to your satisfaction, why don't you answer it yourself? Please use math.