# Proper accelerations and rigid bodies.

• Boustrophedon
In summary, the proper acceleration of the rear end of an accelerated rigid rod must be greater than that of the front end. This is illustrated by a Minkowski diagram showing two differently curved hyperbolae for the trajectories of the ends. However, according to sources cited in this thread, this is incorrect. The proper acceleration is the Lorentz-Invariant acceleration that is measured by an un-accelerated, inertial observer.

#### Boustrophedon

"Proper" accelerations and rigid bodies.

In Special Relativity, the term "proper" in front of a property means, as I understand it, the value of that property as measured by an observer with test equipment at rest relative to the object being measured. Thus "proper length" is the length determined by an observer in the same frame as the object and the "proper time" is the time according to a clock that is carried with the observer throughout the relevant period.

However, in textbooks, articles and research papers in SR, I frequently come across the statement that "the "proper acceleration" of the rear end of an accelerated rigid rod must be greater than that of the front end." This is often illustrated by a Minkowski diagram showing two differently curved hyperbolae for the trajectories of the ends.

This is very strange since it is clearly the accelerations of the rod ends estimated by an un-accelerated, inertial observer that are being represented - quite contrary to the established meaning of the term "proper". From consistency of meaning "proper acceleration" should refer to accelerations measured by co-moving observers.

Since co-moving observers verify a constant "proper" length for the rod, and an observer at each end will record the same "proper" time, then by definition the "proper" acceleration must surely in fact be identical at each end of the rod, whether they are measured by inertial accelerometers or by reference to the background inertial frame.

Boustrophedon said:
However, in textbooks, articles and research papers in SR, I frequently come across the statement that "the "proper acceleration" of the rear end of an accelerated rigid rod must be greater than that of the front end." This is often illustrated by a Minkowski diagram showing two differently curved hyperbolae for the trajectories of the ends.

To make sure that this thread doesn't end up in a never-ending loop, please make exact citation to such sources so that others can check up and see if what you understood here is identical to what is claimed in these sources.

Since this thread is in the same vein as a previously-closed thread, I will right out front tell everyone that wants to be involved in this that the Moderators will give it a very short leash. If we intend to repeat the same mess in that previous thread, this thread will be shut down and this topic will be banned for good.

Zz.

The meaning of "proper" in physics is the same as the german prefix "eigen-", meaning "belonging to". It is always an invariant quantity, belonging to the object in question independently of coordinate transforms. Thus proper acceleration of an observer means the Lorentz-Invariant acceleration that this observer actually measures locally, e.g. using a customary accelerometer. It never refers to coordinate accelerations in some specific system, "comoving" or not.

It never refers to coordinate accelerations in some specific system
Exactly my point : that that is how it's being mis-used !

When there is acceleration, the rest systems of the two ends will not be the same. "Proper" does mean in the rest frame, but that will be at different times for the two ends if the time is the same in the system in which both are moving.

I'm afraid it won't - the proper times will be the same unless by "if the time is the same" you mean clocks have been re-synchronised and therefore no longer carry proper time.
Synchronisation and simultaneity are not the same thing. Simultaneity shifts "by itself" as a result of the motion, but synchronisation requires manual intervention. Proper time is defined by clock readings.
So the proper time at one end is not what observer at the other end perceives, but what is actually elapsed on an in situ un-adjusted clock. It is that the clocks are unaffected by acceleration and maintain the same rate that causes each observer to claim a differing "simultaneous" time as the simultaneity shifts with acceleration, and thus arises the necessity for re-synchronising.

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If the four peer-reviewed papers by five authors cited in the last thread haven't convinced B. that he's wrong, what will? If showing B. the evidence, in the literature, that he's wrong doesn't convince him, I don't see what else we can reasonably do.

Is there any particular person (preferably on this board) that B would listen to or is interested in hearing from? He's already blown off one professional in the field, one with many published papers.

For another citation, see for instance, Misner, Thorne, Wheeler, "Gravitation", pg 165 I've posted this before on another forum

"At distances l away from his world line, strange things of dimensionless magnitude gl happen to his (ed: the accelerated observer's) lattice - e.g, the acceleration measured by accelerometers differs from g by a fractioanal amount gl (exercise. 6.7); also, clocks intially synchronized with the clock on his world line get out of step (tick at different rates) by a fractional amount ~gl (exercise 6.6).

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To add to what pervect has mentioned, *I* am still waiting for the citations that I've asked for earlier. Please do not ignore this request.

Zz.

Boustrophedon said:
Exactly my point : that that is how it's being mis-used !
Well, I think that misuse is exactly what you do. I'll try to give a line of reasoning why the proper accelerations have to be different:
In the t-x-diagram of an inertial observer A, the worldline of a particle with constant proper acceleration follows a defined hyberbola.
It would follow the same (but translated) hyperbola if it started from a different point.
So in A's frame, front and rear end are always at a constant distance.
Inevitably, in every other frame, including a comoving one, the distance must be greater.
Thus, in a comoving frame, the rocket appears longer.
When the acceleration ceases, this distance can be identified with the proper length of the rocket.
So the rocket actually grew longer. It cannot be 'rigid'.

Boustrophedon said:
In Special Relativity, the term "proper" in front of a property means, as I understand it, the value of that property as measured by an observer with test equipment at rest relative to the object being measured. Thus "proper length" is the length determined by an observer in the same frame as the object and the "proper time" is the time according to a clock that is carried with the observer throughout the relevant period.

However, in textbooks, articles and research papers in SR, I frequently come across the statement that "the "proper acceleration" of the rear end of an accelerated rigid rod must be greater than that of the front end." This is often illustrated by a Minkowski diagram showing two differently curved hyperbolae for the trajectories of the ends.

Correct, force does not propagate instantaneously in a solid, it propagates at the speed of sound. This reality has an effect on how different parts of a body are accelerated differently. For example, a rocket experiences different speeds during launch, with the rear (where the engine is) having a higher speed than the tip.

This is very strange since it is clearly the accelerations of the rod ends estimated by an un-accelerated, inertial observer that are being represented - quite contrary to the established meaning of the term "proper". From consistency of meaning "proper acceleration" should refer to accelerations measured by co-moving observers.
"Unsubstantiated": ZZ asked you to produce proof that this is the interpretation as published in the references. Yo can scan the pages you are referring to and attach them to your next post in a JPEG form.

Proof that what I am saying follows standard SR detail

Just a few typical examples of the point I made in the #1st post about regarding the accelerations measured from a different inertial frame as "proper" accelerations of the moving system - in reversal of the accepted meaning of the term "proper"
www.csupomona.edu/~ajm/professional/talks/relacc.ppt[/URL]
[url]http://www.mathpages.com/home/kmath422/kmath422.htm[/url]
[PLAIN]http://arxiv.org/PS_cache/gr-qc/pdf/0301/0301050.pdf [Broken]
I can't however find the "four peer-reviewed papers" Pervect refers to - I can only find the Nicolic one which I wholeheartedly disagree with simply because it's faulty.

The narrow point at issue here, following Meir Achuz's comment, concerns "proper time" and is easily clarified by paying attention to the following absolutely stock-standard features of Special Relativity.

For inertial systems in constant relative v motion, each with lines of synchronised clocks laid out along the direction of motion, they each find that comparison shows that the clocks in the other relatively moving system are set out-of-synch but run at the same rate as each other.
To be more specific using the train+platform for illustration, the platform observer regards the clocks toward the front of the train as increasingly set behind (in time) those at the back. Or in other words, the rear train clocks appear set ahead (in time) of the forward clocks.
Reciprocally, the train rider sees the platform clocks out of synch in the same way - that is, the platform clocks at the far end as he approaches appear to be set ahead (in time) of the clocks at the "near" end that he would pass first.
Neither observer claims that the other's clocks are running at different rates - merely that they have been set "out-of-synch".

This is understood in SR as due to each observer's "lines of simultaneity" having become tilted with respect to the other at a specific angle for any given velocity.
So although the platform clocks are perfectly synchronised to the guy on the platform, the train rider's judgement of "simultaneity" has tilted "upward" or forward in time ahead of him while dipping backward in time behind, so that "now" ahead of him is later in time than for the platform in that direction, and he therefore regards the clocks as set forward in time.

Now what happens as the train accelerates from rest ? The "lines of simultaneity" at every point of the train start parallel and smoothly rotate so as to increasingly tilt forward in time towards the front and backwards in time towards the rear.
Note especially that nothing physical happens at any given point - all that happens is that the simultaneity of other places has changed with respect to the given point.
So nothing changes at the rear of the train except that events "later" in time at the front of the train have become simultaneous, whereas nothing has happened at the front of the train except that rear-train events "earlier" in time have become simultaneous.
Since this is progressive during acceleration, it's as if the rear train rider sees time passing faster at the front and the front rider sees time passing slower at the rear.

The train clocks themselves of course, have not changed and continue to show the same (proper) time as each other, as the "stationary" platform observer will verify.
The rear train rider will, however, see as simultaneous with his clock "now", the reading of the front clock a bit later in time. The front rider perceives as simultaneous the rear clock as it was a bit earlier.

It is thus quite incorrect to claim that the front clock "speeds up" or "goes faster" than the rear clock during acceleration.

For example, if the de-synchronisation was caused by, say, the front clock going faster, then when it is turned back during re-synchronisation at the end of acceleration period, the effect would be to restore agreement between the train clocks and the platform clocks - exactly contrary to accepted constant velocity SR.

However, if as I am pointing out, the clocks continue to read the same (proper) time during acceleration, then when say, the front clock is turned back during re-synchronisation at the end of the acceleration period, it will become de-synchronised with the rear from the point of view of the platform observers, who will find it set behind (in time) the rear train clocks ---- exactly as constant velocity SR says it should be.

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Boustrophedon said:
The narrow point at issue here, following Meir Achuz's comment, concerns "proper time" and is easily clarified by paying attention to the following absolutely stock-standard features of Special Relativity.

For inertial systems in constant relative v motion, each with lines of synchronised clocks laid out along the direction of motion, they each find that comparison shows that the clocks in the other relatively moving system are set out-of-synch but run at the same rate as each other.
To be more specific using the train+platform for illustration, the platform observer regards the clocks toward the front of the train as increasingly set behind (in time) those at the back. Or in other words, the rear train clocks appear set ahead (in time) of the forward clocks.
Reciprocally, the train rider sees the platform clocks out of synch in the same way - that is, the platform clocks at the far end as he approaches appear to be set ahead (in time) of the clocks at the "near" end that he would pass first.
Neither observer claims that the other's clocks are running at different rates - merely that they have been set "out-of-synch".
That last statement is hardly consistent with "absolutely stock-standard" Special Relativity. To be clear: Track-frame observers make measurements that allow them to conclude that (1) the train clocks are out of synch, and that (2) the train clocks run slowly compared to their track-frame clocks. Of course, these effects are perfectly symmetric so the Train-frame observers draw the same conclusions about the track-frame clocks.

This is standard relativity.

Boustrophedon said:
Just a few typical examples of the point I made in the #1st post about regarding the accelerations measured from a different inertial frame as "proper" accelerations of the moving system - in reversal of the accepted meaning of the term "proper"
www.csupomona.edu/~ajm/professional/talks/relacc.ppt[/URL]
...
[/QUOTE]

Just to clarify what you are not agreeing to: is it this? (from slide 12 of the referenced presentation by Mallinckrodt, dealing with 'rigid rod' acceleration):
[quote]
From Mallinckrodt presentation slide 12:
- Since the vertex distances are different, so are the accelerations
- The “front” object, B, has a smaller proper acceleration than the “rear” object, A
- A and B agree at all times on matters of simultaneity
- A and B agree at all times on their common velocity
- A and B agree that their proper separation is constant
- A and B agree that B’s clock runs faster in direct proportion to their respective vertex distances.
[/quote]
I read it that the first item refers to coordinate accelerations, which differs from each other and also from the proper accelerations (the second item).

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To pick up on Doc Al's comment - It is actually stock-standard SR. Although I was trying hard to be as unambiguous as possible without long-windedness, I could have been more precise by including "...running at different rates from each other - merely that they...", which is what I meant. That is, all the train clocks run at the same rate but appear set progressively out-of-synch.

Jorrie - I think it's clear that both refer to what he means by proper accelerations as depicted by differently shaped hyperbolae in the diagram (eg. see 9). I agree with you they are actually coordinate accelerations perceived from an inertial frame and the way he elides from just "acceleration" to "proper acceleration" in referring to the same thing is just what one has to watch out for.

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Boustrophedon said:
Just a few typical examples of the point I made in the #1st post about regarding the accelerations measured from a different inertial frame as "proper" accelerations of the moving system - in reversal of the accepted meaning of the term "proper"
www.csupomona.edu/~ajm/professional/talks/relacc.ppt[/URL]
[url]http://www.mathpages.com/home/kmath422/kmath422.htm[/url]
[PLAIN]http://arxiv.org/PS_cache/gr-qc/pdf/0301/0301050.pdf [Broken]

Remember what you said:

Boustrophedon said:
However, in textbooks, articles and research papers in SR, ...

I would loosely accept those as "articles" (I would prefer something more valid than those that are used commonly in physics). However, they are certainly not textbooks and research papers, unless you think anything appearing in ArXiv are automatically "research papers". Please cite actual physics textbooks and actual peer-reviewed physics papers to back up your claim before you proceed any further.

Remember the short leash that I'm giving this thread.

Zz.

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ZapperZ said:
Remember what you said:

I would loosely accept those as "articles" (I would prefer something more valid than those that are used commonly in physics). However, they are certainly not textbooks and research papers, unless you think anything appearing in ArXiv are automatically "research papers". Please cite actual physics textbooks and actual peer-reviewed physics papers to back up your claim before you proceed any further.

Remember the short leash that I'm giving this thread.

Zz.

http://www.iop.org/EJ/article/0143-0807/24/2/361/ej3261.pdf

with the same title and by the same authors. (And I've seen the name Tartaglia before, he's done some papers that I like on the rotating disk).

so it would seem that this preprint was actually published, in the European Journal of physics.

As far as I know, this is a reputable peer-reviewed journal. I'm afraid I can't confirm that the arxiv paper is the same as the published paper, though I see no reason to expect any differences.

Oddly enough, (or perhaps not so oddly), this paper comes to the same conclusion that Nikolic's paper,

http://arxiv.org/abs/physics/9810017

also published in a peer reviewed journal (Am. J of physics)
http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS000067000011001007000001&idtype=cvips&gifs=yes [Broken]

and to the same conclusion that MTW does in "Gravitation", namely that the proper acceleration is different at two ends of a "long" rod, and that two observers, who acclerate with the same proper acceleration, will increase the distance between them, so that a string connecting them will break.

There were a couple of more papers cited in Nikolic's paper as well, which also came to similar conclusions.

Having seen the number of papers which say that the string does break in the Bell spaceship paradox, perhaps we should now ask for papers similarly published in peer reviewed journals that show that the string does not break?

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pervect said:
.. two observers, who accelerate with the same proper acceleration, will increase the distance between them, so that a string connecting them will break.
Well that is obvious, since that is a characteristic of a Minkowski space-time. Except, of course, for the trivial case where the observers start from the same space-time location. Note that it is not just the distance that gets affected also the times go out of phase.

In a way one you could argue, but obviously not prove, that the origin of inertia is exactly the change in distance and the shift in times inside any non-zero length object when it undergoes acceleration.

Also consider the situation where all objects in a Minkowski space-time are at relative rest with each other. If all objects accelerate at the same time (Einstein synchronized) then distances do increase and clocks will become desynchronized.
A kind of expansion of the universe.

Isn't relativity a wonderful thing?

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pervect said:
http://www.iop.org/EJ/article/0143-0807/24/2/361/ej3261.pdf

with the same title and by the same authors. (And I've seen the name Tartaglia before, he's done some papers that I like on the rotating disk).

so it would seem that this preprint was actually published, in the European Journal of physics.

As far as I know, this is a reputable peer-reviewed journal. I'm afraid I can't confirm that the arxiv paper is the same as the published paper, though I see no reason to expect any differences.

Oddly enough, (or perhaps not so oddly), this paper comes to the same conclusion that Nikolic's paper,

http://arxiv.org/abs/physics/9810017

also published in a peer reviewed journal (Am. J of physics)
http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS000067000011001007000001&idtype=cvips&gifs=yes [Broken]

and to the same conclusion that MTW does in "Gravitation", namely that the proper acceleration is different at two ends of a "long" rod, and that two observers, who acclerate with the same proper acceleration, will increase the distance between them, so that a string connecting them will break.

There were a couple of more papers cited in Nikolic's paper as well, which also came to similar conclusions.

Having seen the number of papers which say that the string does break in the Bell spaceship paradox, perhaps we should now ask for papers similarly published in peer reviewed journals that show that the string does not break?

If you read between the lines, it appears that Boustrophedon believes that force propagates instantaneously (with infinite speed), which is the only way to explain why both ends of the rod would have the same acceleration profile as outlined in the OP. Of course, the exact opposite is the truth.

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nakurusil said:
If you read between the lines, it appears that Boustrophedon believes that force propagates instantaneously (with infinite speed), which is the only way to explain why both ends of the rod would have the same acceleration profile as outlined in the OP. Of course, the exact opposite is the truth.

Does this mean that you still hold the opinion that the overriding reason why the two ends of a length-wise accelerated rod have different proper accelerations is caused by the finite speed of sound in the rod (or the finite speed of propagation of the force applied at one end)?

My understanding is that the mainstream literature (as adequately quoted by Pervect and others) disagrees with your view (if it is in fact how you see it) and that the difference in proper acceleration at the two ends is a purely relativistic effect, valid even if the rod would be "perfectly rigid" (or accelerated as Born-rigid for that matter).

I am asking this because I think it is quite important to clear this one, perhaps by an answer from a science advisor. Robust debate on it is fine, but so far it has been difficult to find the real answer.

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In addition to Pervect's helpful tracing of the Euro.J.of Phys. reference, I could also refer to Wolfgang Rindler's "Essential Relativity" section 3.8 or A.P.French & E.F.Taylor "Limitation on proper length in SR" Am.J.Phys. 51 p.890 (1983) or J.E.Romain "A Geometrical Approach to Relativistic Paradoxes" Am.J.Phys. 31 p.580 (1963).

However, Pervect's assertion :
...to the same conclusion that MTW does in "Gravitation", namely that the proper acceleration is different at two ends of a "long" rod, and that two observers, who acclerate with the same proper acceleration, will increase the distance between them, so that a string connecting them will break.
- is factually incorrect. The quote he used earlier supports no such particular construction, since it does not even involve a second observer, let alone any connecting string. The quote is in fact highly ambiguous and almost certainly wrong if taken literally. The comments refer to the observer's "lattice" when of course no such lattice ( of rods & synchronised clocks etc.) is feasible during acceleration.

Since Pervect has introduced the so-called "Bell spaceship problem" I should point out that the falsity of Bell's claim follows inevitably from the demonstration that both time dilation and length "contraction" effects follow in a elementary way from the divergence in simultaneity without any "extra" assumptions that a moving rod or clock is any different from a "stationary" one.
It is easy to see that the observers disagree about the length of a rod simply and solely because they disagree about whether each other recorded the ends simultaneously. This is a non-physical, purely kinematical effect and cannot possibly give rise to any "stresses", "forces", "tensions" or "string-breaking". The same causes ( differing simultaneity ) would give the same effects ( differing length estimates ) if the ends of the rod were not even connected.

As I showed earlier, the shift in simultaneity that accompanies acceleration cannot affect the clocks themselves ( as otherwise the same clock would be read as two different times by the two momentarily adjacent observers ). It is after acceleration ceases that, for instance, the forward obs. turns his clock back to synchronise with his "tilted" perspective of the rear clock ( and rear obs.'s similar but contrary shift in perspective of the front clock ) thus de-synchronising them from POV of obs. who had remained unaccelerated. The differently synchronised clocks now conform to the different simultaneity in each frame and observers in either frame will measure lengths in the other frame to be shorter.
In SR the diminishing measurements of the length of an accelerated rod are not due to any change in rod length but actually due to a growing disparity between the simultaneity of each frame.

Once it is realized that "constant to&fro speed of light" >>> "shifted or non-parallel simultaneity between relatively moving inertial frames" >>> "each frame sees the other's 'synchronised' clocks as desynchronised >>> "moving clocks appear to run at a retarded rate" + "moving lengths appear shorter"...SR is seen as elegant and obvious as Einstein is justly famous for making it.

Nakrusil seems to be bogged down in consideration of inertial forces, which are not relevant. It is invariably implicit in such thought experiments that accelerations are sufficiently gentle and/or evenly distributed that they can be rendered negligable and disregarded.
It might help to read the lines themselves instead of looking for messages between them !

Jorrie - Why do you ask if I "still" hold an opinion that I have never held, nor would ever give
the slightest consideration to.

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This is WHY I asked for the exact citation. Note that once you have the SOURCE that you are using to illustrate your conclusion, we can now all go to it and see if you've read what others have understood.

And the fact that you still say this:

Boustrophedon said:
is factually incorrect. The quote he used earlier supports no such particular construction, since it does not even involve a second observer, let alone any connecting string. The quote is in fact highly ambiguous and almost certainly wrong if taken literally. The comments refer to the observer's "lattice" when of course no such lattice ( of rods & synchronised clocks etc.) is feasible during acceleration.

.. while pervect and others have a different conclusion implies that there IS a different take on the SAME sources. If you had not shown your sources, we could all be arguing about some undefined heresay in which no one has any clue on.

So now, go back to the SOURCES and argue if they has been interpreted correctly. Because if they haven't, then all of these arguments are MOOT, because your original premise would have been wrong.

Note that if this is going to be something that isn't part of what has already been published and accepted, it will have to go into the IR forum. So just be aware that it is being monitored, unless you want to cut it right to the chase and declare it as such. That would make it a lot simpler to handle.

Zz.

Jorrie said:
Does this mean that you still hold the opinion that the overriding reason why the two ends of a length-wise accelerated rod have different proper accelerations is caused by the finite speed of sound in the rod (or the finite speed of propagation of the force applied at one end)?

The above is purely your misinterpretation about what I said in an earlier thread. There are TWO effects at work that should be considered :

1. The relativistic effect: during the accelerated motion, the distance between the rockets, computed in a frame comoving with any of the two rockets increases. The increase can be computed rigurously by using the equations of hyperbolic motion (no Lorentz transforms, remember? )

2. In ADDITION to the distance increase calculated at point 1 there is the effect due to finite speed of forces propagating at the speed of sound. The finite speed of force propagation FURTHER increses the distance between the tip of the trailing rocket and the rear of the leading rocket thus FURTHER stretching the rope.

BOTH effects need to be calculated in order to obtain the complete and correct solution to the problem.

I can send you a link to the classnotes of a very good class at U of Texas where they solve the problem rigurously.

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Boustrophedon said:
Nakrusil seems to be bogged down in consideration of inertial forces, which are not relevant. It is invariably implicit in such thought experiments that accelerations are sufficiently gentle and/or evenly distributed that they can be rendered negligable and disregarded.
It might help to read the lines themselves instead of looking for messages between them !
.

Umm, no. Read my answer to Jorrie. There are TWO effects at work, you need to consider both in order to solve the problem correctly. As an aside, I would not call the accelration at rocket launch "gentle". BTW, would you try to quantify the acceleration values for which "they can be rendered negligable and disregarded."?

nakurusil said:
1. The relativistic effect: during the accelerated motion, the distance between the rockets, computed in a frame comoving with any of the two rockets increases. The increase can be computed rigurously by using the equations of hyperbolic motion (no Lorentz transforms, remember? )

2. In ADDITION to the distance increase calculated at point 1 there is the effect due to finite speed of propagation of forces, at the speed of sound, in solids. The finite speed of force propagation FURTHER increses the distance between the tip of the trailing rocket and the rear of the leading rocket thus FURTHER stretching the rope.

My question concerned the acceleration of a rod, not two rockets at constant proper accelerations. I think that was the topic under discussion, or at least, that is what this thread is about. I think I stated relatively () clearly that a "rigid rod" was under consideration.
Would you mind answering the question as politely as possible?

Jorrie said:
My question concerned the acceleration of a rod, not two rockets at constant proper accelerations. I think that was the topic under discussion, or at least, that is what this thread is about. I think I stated relatively () clearly that a "rigid rod" was under consideration.
Would you mind answering the question as politely as possible?

The rod is connected between two rockets in Bell's paradox, have you forgotten? Let's not start this nonsense again, I gave you a very detailed answer, please leave it at that.

Nakrusil - you are wrong. Why you insist on dragging in inertial effects when they are completely irrelevant I do not know. Acceleration at "rocket launch" can be as gentle as one likes - that's why Bell refers to "spaceships" instead because many people can't get the idea of a Saturn 5 taking off out of their heads in order to focus on the relevant issue - the relativistic effects. The two sorts of effect are quite independant which means one can always arrange the thought conditions to make the undesirable one negligable.

The unconnected rod ends are equivalent to the two rockets or the two spaceships for the purpose in hand. The point that the ends alone give the same effects as the whole rod is saying that the two spaceships give the same results as the string ends, without any tension.
It's called analogy.

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Boustrophedon said:
Jorrie - Why do you ask if I "still" hold an opinion that I have never held, nor would ever give the slightest consideration to.

No Boustrophedon, I did not ask you, I asked Narkurusil, who was referring to you. I know it is not your opinion - as far as I'm aware, Narkurusil is the only one with that view.

Sorry if my intention was unclear!

Boustrophedon said:
Nakrusil - you are wrong. Why you insist on dragging in inertial effects when they are completely irrelevant I do not know. Acceleration at "rocket launch" can be as gentle as one likes - that's why Bell refers to "spaceships" instead because many people can't get the idea of a Saturn 5 taking off out of their heads in order to focus on the relevant issue - the relativistic effects. The two sorts of effect are quite independant which means one can always arrange the thought conditions to make the undesirable one negligable.

Yes, there are two different effects, I outlined them in my post to Jorrie. BOTH need to be taken into consideration if you want to solve the problem correctly.

The unconnected rod ends are equivalent to the two rockets or the two spaceships for the purpose in hand. The point that the ends alone give the same effects as the whole rod is saying that the two spaceships give the same results as the string ends, without any tension.
It's called analogy.

Why don't you try to write down some math , instead of waving your hands?
You can start by quantifying the "gentle" acceleration, I've just asked you to quantify.
You can proceed with producing your own version of the mathematical solution to Bell's paradox. Enough of your armwaving, let's see some math.

nakurusil said:
The rod is connected between two rockets in Bell's paradox, have you forgotten? Let's not start this nonsense again, I gave you a very detailed answer, please leave it at that.

Narkurasil, please read what was asked. The OP asked a question about proper acceleration at the ends of a rigid rod being uniformly accelerated at one end only. I clearly referred to that as well in my reply. No Bell's paradox (where identical proper accelerations at both ends are an assumption) involved.

Jorrie said:
Narkurasil, please read what was asked. The OP asked a question about proper acceleration at the ends of a rigid rod being uniformly accelerated at one end only. I clearly referred to that as well in my reply. No Bell's paradox (where identical proper accelerations at both ends are an assumption) involved.

How do you accelerate a rod at one end? By applying a force at that end. This results into having the end where the force is applied accelerating ahead of the other end.Because the force will propagate from end to end in the rod with finite speed. Are you trying to create trouble or are you that slow in understanding?

Others be the judge

nakurusil said:
Are you trying to create trouble or are you that slow in understanding?

Neither. I rest my case.

Boustrophedon's original post says that some books and articles are using the term "proper" incorrectly. The people who replied to that seem to think that B has either read some bad articles, or simply misunderstood them. I have a different suggestion. I think the books and articles are using the term "proper" correctly, and that B is wrong about what would be correct usage.

The "proper" quantities can always be determined by the coordinates of co-moving inertial observers.

The proper time along a curved world line is the sum of time coordinate increases of an infinite sequence of inertial frames.

(A curved world line can be approximated by N straight line segments of equal length. Each straight line segment corresponds to an inertial frame. The time coordinate of this inertial frame increases from one end of the straight line segment to the other. The sum of these increases approximates the proper time. The approximation is exact in the limit N --> infinity.)

The proper length of an accelerated object is the sum of spatial coordinate increases of an infinite sequence of inertial frames. (Imagine N points on the object. The tangent to the world line of each point is the time axis of an inertial frame. The space axis of this inertial frame assigns a spatial distance between one point and the next. The sum of all these distances approximates the proper length. The approximation is exact in the limit N --> infinity).

The proper acceleration is the second derivative of the space-time coordinates of a co-moving inertial frame with respect to proper time.

Boustrophedon said:
I frequently come across the statement that "the "proper acceleration" of the rear end of an accelerated rigid rod must be greater than that of the front end."
...
Since co-moving observers verify a constant "proper" length for the rod, and an observer at each end will record the same "proper" time, then by definition the "proper" acceleration must surely in fact be identical at each end of the rod, whether they are measured by inertial accelerometers or by reference to the background inertial frame.
You're making a mistake here. You're assuming that the world lines of each part of the rod is identical except for starting position. But the statement that you "frequently come across" is a statement about Born rigid motion, and that contradicts your assumption.

Boustrophedon said:
So although the platform clocks are perfectly synchronised to the guy on the platform, the train rider's judgement of "simultaneity" has tilted "upward" or forward in time ahead of him while dipping backward in time behind, so that "now" ahead of him is later in time than for the platform in that direction, and he therefore regards the clocks as set forward in time.

Now what happens as the train accelerates from rest ? The "lines of simultaneity" at every point of the train start parallel and smoothly rotate so as to increasingly tilt forward in time towards the front and backwards in time towards the rear.
You seem quite focused on the idea of using accelerating frames, but you also seem to underestimate the problems with accelerating frames. I believe that what you're thinking is something like this:

If an inertial observer draws the world line and the lines of simulaneity of another inertial observer, the slope of the world line is 1/v and the slope of the lines of simultaneity is 1/v. This seems to imply that the "line" of simultaneity of an accelerating observer must bend upwards, as the observer's world line bends to the right. These "lines" (actually curved space-like hypersurfaces of Minkowski space) seem to be good candidates for what the accelerating observer should call "space" at different times.

If it really makes sense to say that these space-like hypersurfaces are "space" at different times, for an accelerating observer, the length of the object as measured by him/her must be sqrt(ds^2) integrated along a curve in "space", from one end of the object to the other.

There are plenty of reasons why this doesn't make sense. The strongest one is that the length of the object would depend on how its velocity is going to change in the future.

Boustrophedon said:
Since Pervect has introduced the so-called "Bell spaceship problem" I should point out that the falsity of Bell's claim follows inevitably from...
I'm not sure if it's appropriate to answer a post about something that was the topic of a thread that's now locked, but it seems that this thread has become a discussion about the same thing, and what you say here makes me curious.

In the other thread, I solved one version of the spaceship problem explicitly. (In this version, the acceleration is arbitrary for a certain proper time, and then the engines shut off). You seemed to agree with what I said except for one detail. I made a new post after that and proved that I was right about that detail. You didn't answer that. Perhaps you didn't see it, or perhaps you didn't have time to answer before the thread got locked.

Anyway, the solution is still there if you're interested, and the result is that the string breaks. If you have any questions or comments, I'm willing to answer them, either here or in private messages if the moderators think we're getting too far off topic.

Boustrophedon said:
It is easy to see that the observers disagree about the length of a rod simply and solely because they disagree about whether each other recorded the ends simultaneously. This is a non-physical, purely kinematical effect and cannot possibly give rise to any "stresses", "forces", "tensions" or "string-breaking". The same causes ( differing simultaneity ) would give the same effects ( differing length estimates ) if the ends of the rod were not even connected.
Everyone who disagrees with your conclusions agree with you about these details, so there's no need to explain them to us.

Boustrophedon said:
Since Pervect has introduced the so-called "Bell spaceship problem" I should point out that the falsity of Bell's claim follows inevitably from the demonstration that both time dilation and length "contraction" effects follow in a elementary way from the divergence in simultaneity without any "extra" assumptions that a moving rod or clock is any different from a "stationary" one.
It is easy to see that the observers disagree about the length of a rod simply and solely because they disagree about whether each other recorded the ends simultaneously. This is a non-physical, purely kinematical effect and cannot possibly give rise to any "stresses", "forces", "tensions" or "string-breaking". The same causes ( differing simultaneity ) would give the same effects ( differing length estimates ) if the ends of the rod were not even connected.
Boustrophedon, it's clear that you have some basic misunderstandings of special relativity. Your interpretation of the Bell spaceship paradox is just one manifestation of this.

While the Lorentz contraction is surely a "kinematical" (or, better, a geometrical) effect, you must realize that forcing both ships to have the same acceleration (with respect to an inertial frame) does introduce stresses in the string. Of course the string breaks.

Several knowledgeable members have patiently described how proper acceleration is treated in special relativity; I suggest you study their posts and the papers they have cited.

One of the main goals of PF is to help students learn the current status of physics as practiced by the scientific community. Please reread the general posting guidelines that cover all of PF and the particular guidelines that appear in the stickies at the top of this forum. Personal interpretations and theories that contradict established, mainstream physics are not permitted in this forum.

On that note, I am going to close this thread. Please do not bring up the same topic in yet another one. If you seriously think that you have been misrepresented, you are welcome to submit your ideas to our Independent Research Forum, provided that they meet certain https://www.physicsforums.com/showthread.php?t=82301.