Einstein's resolution of the clock paradox

[Al68]

Einstein's resolution of the "clock paradox"

I have read that most physicists believe that Einstein's resolution of the "clock paradox" or "twins paradox" is flawed and just plain wrong. I'm curious about what everyone here thinks about it.

And I believe Einstein generally was interested in the more profound aspect of the "paradox", ie why inertial frames must be treated "special", from his writings, not the trivial exercise found here and elsewhere just showing how to do the math in SR from the inertial frame's point of view.

Another example of a related idea might be the spinning globes. For those unfamiliar, Einstein referred to two adjacent liquid globes in deep space each one spinning relative to the other about a common axis, and only one of them had a bulging equator, so we conclude that it's the one "really" spinning while the other is stationary. But, he asks, it's spinning relative to what? He finally concluded that since each globe was spinning relative to the other, the cause of the one's bulging equator must lie outside the system containing the globes, since nothing locally could explain why one equator bulged but not the other.

Any thoughts?

Thanks,
Al

 

[jtbell]

Which method of resolving the clock paradox is "Einstein's"? I don't remember.

 

[George Jones]

I have read that most physicists believe that Einstein's resolution of the "clock paradox" or "twins paradox" is flawed and just plain wrong. I'm curious about what everyone here thinks about it.

And I believe Einstein generally was interested in the more profound aspect of the "paradox", ie why inertial frames must be treated "special", from his writings, not the trivial exercise found here and elsewhere just showing how to do the math in SR from the inertial frame's point of view.

Another example of a related idea might be the spinning globes. For those unfamiliar, Einstein referred to two adjacent liquid globes in deep space each one spinning relative to the other about a common axis, and only one of them had a bulging equator, so we conclude that it's the one "really" spinning while the other is stationary. But, he asks, it's spinning relative to what? He finally concluded that since each globe was spinning relative to the other, the cause of the one's bulging equator must lie outside the system containing the globes, since nothing locally could explain why one equator bulged but not the other.

From the Physics Forum Posting Guidelines:

Posting the same topic (or homework question) across multiple forums or multiple threads is considered spamming and is not allowed.

You have already mentioned Einstein's position on the twin paradox and Mach's principle in another thread

https://www.physicsforums.com/showthread.php?p=1767579#post1767579

You might feel that that these questions were not discussed adequately in that thread, so, for now, I will allow this thread to continue, but

It is against our Posting Guidelines to discuss, in most of the PF forums, new or non-mainstream theories or ideas that have not been published in professional peer-reviewed journals or are not part of current professional mainstream scientific discussion."

you need to give specific references. You need to back up the claim

I have read that most physicists believe that Einstein's resolution of the "clock paradox" or "twins paradox" is flawed and just plain wrong.

with more than anecdotal evidence. If you cannot do so, the thread will be locked.

Discussing endlessly the twin parardox and Mach's principle is not the purpose of the relativity forum.

 

[Al68]

Which method of resolving the clock paradox is "Einstein's"? I don't remember.

The one using GR to analyze the scenario from the point of view of the ship's observer.

 

[Mentz114]

Al68,

Another example of a related idea might be the spinning globes. For those unfamiliar, Einstein referred to two adjacent liquid globes in deep space each one spinning relative to the other about a common axis, and only one of them had a bulging equator, so we conclude that it's the one "really" spinning while the other is stationary. But, he asks, it's spinning relative to what? He finally concluded that since each globe was spinning relative to the other, the cause of the one's bulging equator must lie outside the system containing the globes, since nothing locally could explain why one equator bulged but not the other.

I don't believe Einstein thought anything of the kind. References please.

Acceleration and rotation are absolute in the sense that they can always be detected and measured in the local frame. There is no mysyery/paradox or puzzle involved.
You do not need to define rotation 'relative' to something.

Also, note that rotation cannot be defined except for an extended object, in which case there's no problem in defining mass-centred frames.

 

[Al68]

From the Physics Forum Posting Guidelines:

Posting the same topic (or homework question) across multiple forums or multiple threads is considered spamming and is not allowed.

You have already mentioned Einstein's position on the twin paradox and Mach's principle in another thread

https://www.physicsforums.com/showthread.php?p=1767579#post1767579

You might feel that that these questions were not discussed adequately in that thread, so, for now, I will allow this thread to continue, but


you need to give specific references.

Einstein, A. (1918) "Dialog über Einwände gegen die Relativitätstheorie", Die Naturwissenschaften 48, pp697-702, 29 November 1918 (English translation: dialog about objections against the theory of relativity). I thought that a new thread was appropriate, and would maybe help prevent "endlessly discussing" the same aspects of the twins paradox that are discussed endlessly all over the net.

You need to back up the claim

I have read that most physicists believe that Einstein's resolution of the "clock paradox" or "twins paradox" is flawed and just plain wrong.

with more than anecdotal evidence. If you cannot do so, the thread will be locked.

Well, I'm only claiming that I've read this, on an older version of the Wikipedia entry http://en.wikipedia.org/w/index.php...5960#Einstein.27s_general_relativity_solution, among other places. I'm not claiming anything else. I was asking what others thought of this resolution.

Discussing endlessly the twin paradox and Mach's principle is not the purpose of the relativity forum.

It's not endless. We all die eventually.

Thanks,
Al

 

[Al68]

Al68,I don't believe Einstein thought anything of the kind. References please.

Einstein, A. (1916) "The Foundation of the General Theory of Relativity".

Referring to the globes S1 and S2: The only satisfactory answer must be that the physical system consisting of S1 and S2 reveals within itself no imaginable cause to which the differing behaviour of S1 and S2 can be referred. The cause must therefore lie outside this system. We have to take it that the general laws of motion, which in particular determine the shapes of S1 and S2 , must be such that the mechanical behaviour of S1 and S2 is partly conditioned in quite essential respects, by distant masses which we have not included in the system under consideration. These distant masses and their motions relative to S1 and S2 must then be regarded as the seat of the causes (which must be susceptible to observation) of the different behaviour of our two bodies S1 and S2.

Acceleration and rotation are absolute in the sense that they can always be detected and measured in the local frame. There is no mysyery/paradox or puzzle involved.
You do not need to define rotation 'relative' to something.

Also, note that rotation cannot be defined except for an extended object, in which case there's no problem in defining mass-centred frames.

Well, I won't argue with that. Please see above.

Thanks,
Al

 

[matheinste]

Hello Al68

The following is an extract from your, partial, paraphrased extract of the quoted source:-

""But, he asks, it's spinning relative to what? He finally concluded that since each globe was spinning relative to the other, the cause of the one's bulging equator must lie outside the system containing the globes, since nothing locally could explain why one equator bulged but not the other."""

Just to clear up some ambiguity do you think he is saying that one will bulge and not the other. Or is he is presenting a scenario in which one bulges and not the other and then asking with regards to this scenario, to what can we attribute this bulge.

Matheinste.

 

[Al68]

It is against our Posting Guidelines to discuss, in most of the PF forums, new or non-mainstream theories or ideas that have not been published in professional peer-reviewed journals or are not part of current professional mainstream scientific discussion."

Is it safe to assume that Einstein's published writings are fair game, even if some of them are not part of "current professional mainstream scientific discussion"?

Some of Einstein's writings are not currently discussed much, some even discounted or ignored, which is why I ask about them. I thought this was the appropriate forum for that purpose.

I now realize that I need to include references for my sources instead of just assuming familiarity with them. I apologize for omitting them earlier.

Thanks
Al

 

[Al68]

Hello Al68

The following is an extract from your, partial, paraphrased extract of the quoted source:-

""But, he asks, it's spinning relative to what? He finally concluded that since each globe was spinning relative to the other, the cause of the one's bulging equator must lie outside the system containing the globes, since nothing locally could explain why one equator bulged but not the other."""

Just to clear up some ambiguity do you think he is saying that one will bulge and not the other. Or is he is presenting a scenario in which one bulges and not the other and then asking with regards to this scenario, to what can we attribute this bulge.

Matheinste.

The latter. Sorry if that wasn't clear.

Thanks, Al

 

[matheinste]

Hello Al68.

Thanks, it is the wording in the original ( translation ) that was not clear to me although it should have been obvious what was meant.

Matheinste.

 

[Mentz114]

Al,
thanks for the reference etc. I think the great man was just plain wrong, or I don't understand the scenario.

For those unfamiliar, Einstein referred to two adjacent liquid globes in deep space each one spinning relative to the other about a common axis, and only one of them had a bulging equator,...

The reason one body has a bulging equator is that it has sufficient intrinsic spin ( rotation about an axis through the body) to deform itself. Intrinsic spin can be defined without relation to outside influences, so I don't follow why E says "... the cause of the one's bulging equator must lie outside the system containing the globes, since nothing locally could explain why one equator bulged but not the other."

M

 

[Fredrik]

Acceleration and rotation are absolute in the sense that they can always be detected and measured in the local frame. There is no mysyery/paradox or puzzle involved.
You do not need to define rotation 'relative' to something.

These things seem obvious to us because we've already made up our minds about what sort of mathematical structure to use as a model of space and time, and what identifications we have to make between things in the mathematical model and things in the real world in order to turn the mathematical model into a testable theory of physics. (An example of the latter is that the quantity that's measured by a clock in the real world is the integral of $\sqrt{-g_{\mu\nu}dx^\mu dx^\nu}$ along the curve in the spacetime manifold that represents the clock's path through real-world space and real-world time). What you're saying is true in the mathematical model you have in mind, but Einstein was probably still trying to figure out what sort of theory he should be looking for. So maybe his statements make a lot more sense than they seem to, but it's hard to tell without knowing more about the context.

 

[Mentz114]

Hi Frederik,
well, acceleration in rotating frames is a fact of life and not model dependent.
I think Einstein was talking about the origin of inertia at some point. The rotating body would not bulge if its parts did not have inertia. Anyhow, as you say, without reading the work in question one cannot be sure.

The OP says he wants to find out why Einstein thought inertial frames are a special class - I can tell him that the reason is that they are connected by the Lorentz transformations, and proper intervals are preserved by the LT. This is what the word 'special' in special relativity means.

M

 

[Al68]

Al,
thanks for the reference etc. I think the great man was just plain wrong, or I don't understand the scenario.



The reason one body has a bulging equator is that it has sufficient intrinsic spin ( rotation about an axis through the body) to deform itself. Intrinsic spin can be defined without relation to outside influences, so I don't follow why E says "... the cause of the one's bulging equator must lie outside the system containing the globes, since nothing locally could explain why one equator bulged but not the other."

M

I seems obvious to me that E was pointing out that no local measurement could explain the cause of the bulging equator(along with other results of rotation) of one globe but not the other. Of course we can use the equatorial bulge(along with other results of rotation) to say which globe is rotating, but that would just be circular logic.

Thanks,
Al

 

[Al68]

These things seem obvious to us because we've already made up our minds about what sort of mathematical structure to use as a model of space and time, and what identifications we have to make between things in the mathematical model and things in the real world in order to turn the mathematical model into a testable theory of physics. (An example of the latter is that the quantity that's measured by a clock in the real world is the integral of $\sqrt{-g_{\mu\nu}dx^\mu dx^\nu}$ along the curve in the spacetime manifold that represents the clock's path through real-world space and real-world time). What you're saying is true in the mathematical model you have in mind, but Einstein was probably still trying to figure out what sort of theory he should be looking for. So maybe his statements make a lot more sense than they seem to, but it's hard to tell without knowing more about the context.

You can read more here: http://hem.bredband.net/b153434/Works/Einstein.htm .

Warning, Einstein obviously had no problem defining and using accelerated reference frames. Of course he had to consider them differently in SR, which he considered a shortcoming of SR.

Thanks,
Al

 

[Al68]

The OP says he wants to find out why Einstein thought inertial frames are a special class

You must have misunderstood me, I don't see this in any of my posts. I did mention that Einstein questioned why they should be considered privileged in his writings, and concluded that they are not (in GR). [References provided in earlier post].

Thanks,
Al

 

[Al68]

Acceleration and rotation are absolute in the sense that they can always be detected and measured in the local frame.

Well, we frequently refer to a ball thrown up in the air as "accelerating" toward Earth and turning around to return to earth, but that acceleration of the ball is only coordinate acceleration relative to the "stationary" observer at Earth's surface. A tiny observer attached to the ball would justifiably consider the ball at rest wrt an inertial frame, this tiny observer would feel no acceleration, could say that the Earth's surface "really" accelerated to turn around wrt his frame, and also point out that the large observer at Earth's surface "felt" the (proper) acceleration. And the tiny observer would be right, and the cause of the "turnaround" of the ball relative to Earth is the same as the cause of the proper acceleration felt by the large observer on earth.

My point here is that an observer can be considered stationary although he "feels" the acceleration, and an object can be said to accelerate relative to this accelerated obserever, although that object was in inertial freefall and felt no acceleration.

Einstein's clock paradox resolution [http://en.wikisource.org/wiki/Dialog_about_objections_against_the_theory_of_relativity] basically considers the ship's twin "stationary", and the Earth to "turnaround" (in the same sense that a ball thrown up in the air will turnaround relative to an "accelerated" observer at Earth's surface), and uses the equivalence principle to claim that the ship's acceleration causes its clock to run slow compared to the Earth clock, in the same way that (proper) acceleration due to gravity effects clocks.

Right or wrong, it actually attempts to resolve what Einstein considered to be an issue with the clock paradox. (Unlike the SR resolutions which only "resolve" a problem that never existed to begin with).

But my initial question was, what do others think of Einstein's resolution?

Thanks,
Al

 

[Fredrik]

I'm too lazy to read all that, but it's definitely possible to define an accelerating point of view in this case, and there's nothing wrong with using it to prove that there's no paradox. It's kind of pointless though, since you can do it without considering accelerating frames.

 

[Al68]

I'm too lazy to read all that, but it's definitely possible to define an accelerating point of view in this case, and there's nothing wrong with using it to prove that there's no paradox. It's kind of pointless though, since you can do it without considering accelerating frames.

Sure, if you only want to prove that there's no internal paradox in SR. Einstein apparently wanted a solution that resolved the problem he saw with it, which was that if we analyze the situation from the ship's twin's point of view (frame in which he is considered stationary), we should be able to get a consistent answer. And I believe that was the whole point of the "clock paradox" to begin with.

Obviously, the SR "resolutions" resolve that issue by saying it's not an issue in SR. I would agree it's not a problem or paradox in SR, but it's the whole point of the clock paradox.

Does nobody have an opinion on E's resolution? It almost seems like I'm the only person around who's even read it, which I know can't be true.

 

[Mentz114]

Al68,
Why do you keep on using the word 'paradox' in this context ? There is no paradox, no resolution is required - so what is your point ?

Obviously, the SR "resolutions" resolve that issue by saying it's not an issue in SR. I would agree it's not a problem or paradox in SR, but it's the whole point of the clock paradox.

You have put quotes on "resolutions". Does this mean you think they are not resolutions ? What do you mean by "it's the whole point of the ..." ?

 

[Al68]

Al68,
Why do you keep on using the word 'paradox' in this context ? There is no paradox, no resolution is required - so what is your point ?



You have put quotes on "resolutions". Does this mean you think they are not resolutions ? What do you mean by "it's the whole point of the ..." ?

I use the word paradox just because it's commonly used that way.

I put quotes on "resolutions" because they resolve a supposed paradox that doesn't exist.

And they don't even address the apparent paradox pointed out by Einstein, that if you consider the ship stationary (in SR), you will get a different result. Of course this isn't a real paradox, because the ship cannot be considered stationary in SR. It's just an "inherent epistemological defect" in SR, in E's words, due to it's preferential treatment of inertial frames. Einstein's resolution also doesn't resolve any real paradox, but it does resolve this "defect", by showing (with GR) that the ship's twin will age less even if we consider the ship stationary and the Earth twin "accelerating" (during the turnaround) relative to the frame in which the ship is at rest.

 

[Mentz114]

Al68,
OK. you should put 'paradox' in quotes, in that case.

And they don't even address the apparent paradox pointed out by Einstein, that if you consider the ship stationary (in SR), you will get a different result. Of course this isn't a real paradox, because the ship cannot be considered stationary in SR. It's just an "inherent epistemological defect" in SR, in E's words, due to it's preferential treatment of inertial frames. Einstein's resolution also doesn't resolve any real paradox, but it does resolve this "defect", by showing (with GR) that the ship's twin will age less even if we consider the ship stationary and the Earth twin "accelerating" (during the turnaround) relative to the frame in which the ship is at rest.

The thing is, the ageing is given by the proper length of the worldlines, and this is agreed on by all inertial observers ( IOW, invariant under LT) and I don't see how E thinks this is a defect. It works even if both travellers have accelerated world-lines.

Anyhow, I'm pretty bored with this subject so if you don't mind I'll bow out for now.

M

 

[Fredrik]

And they don't even address the apparent paradox pointed out by Einstein, that if you consider the ship stationary (in SR), you will get a different result. Of course this isn't a real paradox, because the ship cannot be considered stationary in SR. It's just an "inherent epistemological defect" in SR, in E's words, due to it's preferential treatment of inertial frames.

Your claim that the ship can't be considered stationary in SR is only true if we define SR so that that the mathematical model of space and time is ($\mathbb R^4$, the set of all inertial frames, the Minkowski metric). If we instead take the mathematical model to be Minkowski space (which includes other coordinate systems), there's no problem.

I really don't see why we would want to exclude non-inertial coordinate systems from a definition of SR. The only reason to do that that makes any kind of sense to me is that it might (I'm not even sure about this) let us get away with making a smaller number of postulates about identifications between the model and the real world. (Such postulates are necessary to make the theory testable. See my thread with the title "identifications between the model and the real world" for an explanation of what I'm talking about). In other words, an exclusion of inertial frames might make those postulates of SR that no one ever mentions simpler.

Let's call that version of the theory "simple SR" for now, and the version based on Minkowski space and a full set of identifications between the model and the real world "actual SR". I think it probably took a while for Einstein to understand "actual SR" after he had fully understood "simple SR". The step from "simple SR" to "actual SR" must have been a very important step towards GR, but that doesn't mean that "actual SR" has more in common with GR than with "simple SR. It's just a very minor modification of "simple SR", so to me it makes a lot more sense to think of "simple SR" as an unfinished product, as a work in progress, and "actual SR" as the theory of space and time that we now call special relativity.

 

[dubsed]

Hello all,

I am fairly new to this but I find if extreamly interesting. So please pardon me if I sound ignorant. But before you would perform such an experiment how would you know which twin would age more or less than the other? For simplicity let's remove the Earth from the picture. Let's say you have two ships in deep space (no reference points except the other ship). The two ships are not movin relative to each other. However they are infact moving (absolute velocity of zero is at very least highly improbable). One ship accelerates away from the other then accelerates back. How do you know which twin is older eg the one who changed his velocity or the one who didn't. I would assume that the answer to this question is dependant on the direction that the one moved relative to his inital (and unknowable) velocity. So if you have forgotten my original question by now let me restate it. Until you perfore the experiment how do you know which twin will be older.

 

[Mentz114]

Hi dubsed,

But before you would perform such an experiment how would you know which twin would age more or less than the other?

You don't need to know. The point of the experiment is to find out. You could use clocks instead of twins, or rats or even twin rats.

M

 

[dubsed]

So you are saying that the entire point of this excercise is a possible method for determining an absolute velocity?

 

[Fredrik]

how do you know which twin will be older.

It actually follows immediately from the fact that what a clock measures is the integral of $\sqrt{dt^2-dx^2-dy^2-dz^2}$ along the path in spacetime that represents the clock's motion. This form of this line element implies that any movement in space will make the total time measured by the clock shorter. So the real question is, "how do we know that this is what a clock measures?" and the answer is that we can't know that without doing some sort of experiment. This knowledge about what a clock measures should be considered a postulate of the special theory of relativity.

By the way, there are lots of threads about the twin paradox already. See e.g. the post I made here. That spacetime diagram explains a lot.

 

[Mentz114]

So you are saying that the entire point of this excercise is a possible method for determining an absolute velocity?

What ? That does not make sense. I didn't mention velocity, absolute or otherwise.

 

[Fredrik]

Einstein apparently wanted a solution that resolved the problem he saw with it, which was that if we analyze the situation from the ship's twin's point of view (frame in which he is considered stationary), we should be able to get a consistent answer. And I believe that was the whole point of the "clock paradox" to begin with.

Obviously, the SR "resolutions" resolve that issue by saying it's not an issue in SR. I would agree it's not a problem or paradox in SR, but it's the whole point of the clock paradox.

And they don't even address the apparent paradox pointed out by Einstein, that if you consider the ship stationary (in SR), you will get a different result. Of course this isn't a real paradox, because the ship cannot be considered stationary in SR.

I just read what you said again. It seems to me that your definition of SR has led you to some strange ideas about what the problem is and what its resolution is. (Those ideas are strange to me, but they are perfectly consistent with your definition of SR, so it seems to me that it's your definition of SR that has made you miss the point of the twin paradox). This seems to be what you're thinking:

[Incorrect] Problem: From the ship's point of view the other twin is younger.
[Incorrect] Resolution: The "ship's point of view" is undefined in SR.

As I pointed out in my previous post, the most natural definition of SR includes all coordinate systems, and if you use one (of many possible) coordinate systems that we can think of as representing the ship's point of view, you will definitely find that the twin on the ship is younger when they meet again. This is a perfectly valid resolution of the "paradox".

If you instead choose to define SR such that non-inertial frames aren't included in the model, then I agree that the problem, as you have interpreted it, doesn't need to be resolved within SR. It is by definition not a problem in SR. It's a problem in some other theory (that I call SR and you call GR). Of course that means that we still have to find the resolution of the problem in that theory. (I described one way to do it in the preceding paragraph, and I will describe one more way to do it below).

This is how I would describe the problem and its resolution:

Problem: The time dilation formula says that both twins are right when they claim that the other twin is aging slower, both when the ship's moving away from Earth and when it's on its way back. (Note that this illusion of a paradox is present even in the kind of "SR" that you have in mind).

Resolution: The twins are both right, but the fact that the twin on the ship can conclude correctly that the other twin is aging slower during both parts of the trip doesn't imply that the other twin will be younger when the ship gets back to Earth. You will only think that it does if you forget to take into account that the simultaneity lines get tilted in the opposite direction when the rocket turns around. (See my spacetime diagram. You will find if you follow the link in my previous post).

 

[Al68]

I just read what you said again. It seems to me that your definition of SR has led you to some strange ideas about what the problem is and what its resolution is. (Those ideas are strange to me, but they are perfectly consistent with your definition of SR, so it seems to me that it's your definition of SR that has made you miss the point of the twin paradox). This seems to be what you're thinking:

[Incorrect] Problem: From the ship's point of view the other twin is younger.
[Incorrect] Resolution: The "ship's point of view" is undefined in SR.

As I pointed out in my previous post, the most natural definition of SR includes all coordinate systems, and if you use one (of many possible) coordinate systems that we can think of as representing the ship's point of view, you will definitely find that the twin on the ship is younger when they meet again. This is a perfectly valid resolution of the "paradox".

If you instead choose to define SR such that non-inertial frames aren't included in the model, then I agree that the problem, as you have interpreted it, doesn't need to be resolved within SR. It is by definition not a problem in SR. It's a problem in some other theory (that I call SR and you call GR). Of course that means that we still have to find the resolution of the problem in that theory. (I described one way to do it in the preceding paragraph, and I will describe one more way to do it below).

This is how I would describe the problem and its resolution:

Problem: The time dilation formula says that both twins are right when they claim that the other twin is aging slower, both when the ship's moving away from Earth and when it's on its way back. (Note that this illusion of a paradox is present even in the kind of "SR" that you have in mind).

Resolution: The twins are both right, but the fact that the twin on the ship can conclude correctly that the other twin is aging slower during both parts of the trip doesn't imply that the other twin will be younger when the ship gets back to Earth. You will only think that it does if you forget to take into account that the simultaneity lines get tilted in the opposite direction when the rocket turns around. (See my spacetime diagram. You will find if you follow the link in my previous post).

My statements here have been about Einstein's papers on the twins paradox and SR, not my own views. I was asking what others thought of his view, not mine.

And it seems strange to me that the way Einstein viewed the issue would be indicative of "strange ideas about what the problem is and what its resolution is", "missing the point of the twin paradox", since he is the one who had the point to make initially (though a different point than most here consider important).

You do make a very good point about defining SR, as Einstein's "GR resolution" of the twins paradox is based on the equivalence principle and gravitational time dilation, which predate GR. It has been called the "GR resolution" by others (and me) perhaps incorrectly.

 

[Al68]

Al68,
OK. you should put 'paradox' in quotes, in that case.



The thing is, the ageing is given by the proper length of the worldlines, and this is agreed on by all inertial observers ( IOW, invariant under LT) and I don't see how E thinks this is a defect. It works even if both travellers have accelerated world-lines.

Anyhow, I'm pretty bored with this subject so if you don't mind I'll bow out for now.

M

Thanks for all your responses,

Al

 

[Dale]

I always prefer the spacetime geometric resolution. It works in both SR and GR contexts, with gravity or without, for inertial or non-inertial observers, for arbitrary paths, and for more complicated "paradox" scenarios. Plus, it is, IMO, useful for pedagogy as well, and gives quantitative answers.