[Fruityloop (OP)]

[MOD EDIT: if you have been led here by a link claiming this is an unresolvable paradox, you have been misled.
                  The truth is that the original poster is unable or unwilling to understand the answers given.]


I have read that there are no 'real' paradoxes in special relativity.
However, I disagree and I believe the following scenario
poses an unresolvable paradox.  The problem concerns two balls
than are connected by a chain and they are some distance apart
from each other and they are each blocking a light source.

So here's the situation:

Code: [Select]

O2(0.99999999999999999999c)-------->
  L1 B1--------O1-------B2 L2
     M1                 M2
L1 and L2 are light sources such as a lightbulbs or perhaps a small hole in a wall with light coming out.
B1 and B2 are lead balls blocking the light sources.
B1 and B2 are separated by a distance of 10 meters.
M1 and M2 are machines holding the lead balls in place and they are underneath balls B1 and B2 respectively, ready to pull them down.
There is an observer called O1 who is located at the mid-point of the
distance between the light sources L1 and L2.
The lead balls are connected by a chain which runs between them (that's the dashed line) and
has no slack, or perhaps just a little bit of wiggle room.
At a certain time, the machines pull down the lead balls simultaneously in the reference
frame of observer O1 and the light from both sources reach observer O1 simultaneously.
The chain remains intact because the distance between the lead balls doesn't change.

The is another oberver called O2 which is moving at 0.99999999999999999999c  relative to observer O1.
Gamma is 1/(1-v^2/c^2)^.5 = 7,071,067,811.865475244
The time between events for oberver O2 is given by (gamma times the velocity times the distance)/c^2.
So we have (7,071,067,811.865475244*(299,792,458 meters/second)*(10 meters))/(299,792,458 meters/second)^2.
Or (7,071,067,811.865475244* 10 meters)/(299,792,458 meters/second.) = 235.8654 seconds.
The time between events is about 235.8654 seconds or about 236 seconds.
So for oberver O2 ball B2 is pulled down about 4 minutes before ball B1 and the chain breaks.
So we have a paradox, for observer O1 the chain is intact and for observer O2 the chain is broken.
Both situations can't simultaneously be true.

Length contraction doesn't resolve the paradox.
I posted this on another message board and the response was that the chain
for observer O1 would break because the chain was pulled tight between the balls.
This is obviously silly nonsense.  If I have a chain that's pulled tight between two
balls and the balls move simultaneously, there's no reason for the chain to break.
There could be a little bit of slack in the chain and the paradox still stands.
I honestly believe that this paradox can't be resolved.

[PmbPhy]

So for oberver O2 ball B2 is pulled down about 4 minutes before ball B1 and the chain breaks.
So we have a paradox, for observer O1 the chain is intact and for observer O2 the chain is broken.
Both situations can't simultaneously be true.

Whenever you think that SR is wrong you can be guaranteed that you made a mistake. This happens every single day on the internet by different people. The result is that the person's understanding of SR is incomplete.

In this case you're making the false assumption that the chain can be treated as a rigid body. However its a well-established fact in SR a rigid body does not exist in the special theory of relativity. E.g. if you have a rod that's lying on the x-axis in frame S and you start accelerating every single part of the rod at the same time as observed in S then in frame S' which is in standard configuration with S and which is moving in the +x direction the rod will not remain rigid but different parts will accelerate at different rates.

In your case the chain stretches as observed by O2.

For more on this please see: https://einstein.stanford.edu/content/relativity/q2018.html

[Fruityloop (OP)]

Because information cannot travel faster than the speed of light. A rigid body is one in which the distances between every atom are fixed and do not change. But if we gave such a body a push from one end, information would have to travel through the entire body instantaneously to signal to all other parts of the body to 'move'. This means information has to travel faster than light which is precluded by special relativity. So, rigid bodies cannot be described by special relativity, and therefore do not exist in the physical world.

That's nice.  It's also completely irrelevant.  Nowhere in the opening post is it stated, implied, assumed, nor required that the transmission of movement along the chain from the movement of the balls must exceed the speed of light.

[PmbPhy]

That's nice.  It's also completely irrelevant.  Nowhere in the opening post is it stated, implied, assumed, nor required that the transmission of movement along the chain from the movement of the balls must exceed the speed of light.

So what? You missed my point. Besides, I didn't say otherwise. I merely showed you a site that describes why there is not such thing as a rigid body in SR. That it talks about other things is irrelevant. Anyway, my entire point is that the chain cannot be thought of as being rigid, i.e. it has some flexibility to it whereas you implied that it had to break implying that you're assuming that it's rigid, i.e. that the length doesn't change or if it tried it would break. That's a false assumption.

For there to not be a paradox the ends measured by O2 must not move simultaneously because if they did they'd communicate information faster than light. That's why that site mentions that.

[Fruityloop (OP)]

Anyway, my entire point is that the chain cannot be thought of as being rigid, i.e. it has some flexibility to it whereas you implied that it had to break implying that you're assuming that it's rigid, i.e. that the length doesn't change or if it tried it would break. That's a false assumption.

I don't know about you, but my experience with chains is that they're not very stretchy.
So I guess you're implying that the chain would remain intact for both observers even though the chain would need to increase in length for observer O2?

[PmbPhy]

I don't know about you, but my experience with chains is that they're not very stretchy.

The "stretchiness" of a material is determined by a quantity known as Young's modulus. You can see the definition for it at:
https://en.wikipedia.org/wiki/Youngs_modulus

Young's modulus for steel its 200 GPa where 1 GPa = 1 gigapascal  For comparison, Young's modulus for rubber is 0.01–0.1 GPa. So you see, even for steel there is a finite amount of stretchiness. But all this doesn't matter because if the chain moved as a rigid body (zero stretching) in frame O1 it will not have that property in frame O2.

So I guess you're implying that the chain would remain intact for both observers even though the chain would need to increase in length for observer O2?

Absolutely. You can try playing around with this. Merely consider what it'd be like to accelerate a rod rigidly in the inertial frame S and then see what happens to it in frame S' which is in standard configuration with S. To accelerate rigidly I mean that we accelerate all points on the rod in exactly the same way so that each end starts to move at exactly the same time. This means that in this frame the rod does not stretch. The differential form of the Lorentz transformation for time is

' =

Since the ends start to move at the same time we have . We'll set where L is the length of the rod. Then

' = -

Since the right hand is not zero we see from this that, as determined by observers in S', the ends of the rod do not start moving at the same time. So even though the rod is absolutely rigid in frame S it isn't as observed in frame S'. This is why it's said that there are no rigid bodies in special relativity.

So you see, you did not falsify relativity here. In fact this is a very well-known phenomena in SR. The year after Einstein published his theory of relativity he published another paper exploring the implications of it, this being one of them.

[Fruityloop (OP)]

... as determined by observers in S', the ends of the rod do not start moving at the same time. So even though the rod is absolutely rigid in frame S it isn't as observed in frame S'. This is why it's said that there are no rigid bodies in special relativity.

You're merely rehashing the opening post.  The balls move simultaneously for observer O1, but for observer O2 there is a difference of about 4 minutes.

So I guess you're implying that the chain would remain intact for both observers even though the chain would need to increase in length for observer O2?

Absolutely.

But all this doesn't matter because if the chain moved as a rigid body (zero stretching) in frame O1 it will not have that property in frame O2.

So your resolution to the paradox is that the chain doesn't break for observer O2 even though it grows in length.  This is because the chain is in a different frame of reference.  Now putting aside the ridiculous absurdity of this for a moment, this can't be true because this leads to another unresolvable paradox.  Imagine for a moment that the movement of the balls is not simultaneous for observer O1 and also not simultaneous for observer O2.  Now the chain will break for observer O1, but it won't break for observer O2.  So I don't see this as a solution.

[PmbPhy]

You're merely rehashing the opening post.  The balls move simultaneously for observer O1, but for observer O2 there is a difference of about 4 minutes.

All that I'm doing is sketching out the reasons that there is no such thing as a rigid body in relativity and I was merely "rehashing" it for purposes of clarity. I modified your opening post to demonstrate why there's no such thing as a rigid body in relativity. Therefore, if a body is non-rigid it has so "stretchiness" to it and that is why the chain doesn't break. Do you understand now?

So your resolution to the paradox is that the chain doesn't break for observer O2 even though it grows in length.  This is because the chain is in a different frame of reference.

That's 100% correct.

  Now putting aside the ridiculous absurdity of this for a moment, ...

I keep explaining to you that "There are no such things as rigid bodies in SR"? Clearly you must be able to understand this by now. So if a body is not rigid then what does that imply? It implies that its non-rigid of course, i.e. elastic. That means, of course, that the length of object changes even though in one frame such as O1 it remains rigid. What is it about that which you have such a difficult time understanding? This is very well-known physics. So well know that all relativists know it like the back of their hand. Any one of them, even someone completing a relativity college course for the first time would be able to understand this. Merely look in the text where you learned SR (if you actually did learn it properly) to see this.

Anyway, please stop with the "ridiculous absurdity" talk. It's not very polite and I've been very polite to you throughout this entire discussion. To be honest, when someone says things like that all it tells me is that they have a poor grasp of the physics involved.

Those people who know SR solid have long ago learned that what appeared to me true in non-relativistic mechanics is no longer true in SR. While you've been able to grasp the fact that simultaneity of events is not an invariant property you have yet to be able to get over the logical "hump" that there's no such thing as a rigid body in SR and as a result you're not taking that into account leads to contradictions.

Or do you actually not believe that there is no such thing as a rigid body in reality?

this can't be true because this leads to another unresolvable paradox.

That's quite wrong as I'll show you!

Imagine for a moment that the movement of the balls is not simultaneous for observer O1 and also not simultaneous for observer O2.  Now the chain will break for observer O1, but it won't break for observer O2.  So I don't see this as a solution.

The flaw in your assertion is that you're automatically making statements which are not true in general. For example: If it's true that the movement of the balls is not simultaneous for observer O1 then you have to say how their motion is not simultaneous. In the present case if the balls don't move at the same time in O1 then you'd actually be trying to stretch the chain because the distance between the balls is no longer constant. If the stretch is not enough to break the chain, or you left in some slack, then you'll find that the chain doesn't break in either frame. The chain breaking is an invariant event so that if it doesn't break in O1 then it won't break in any other frame and therefore the stretchiness is enough. However if you didn't leave enough slack in the chain the chain will break because you're trying to stretch it. Of course that all depends on whether you're stretching it beyond its elastic limit to its breaking point.

As I suggested above (and for which you totally missed the point) try working this out with various scenarios and you will invariably find that what I said is true. But its dependent on the fact that there are no rigid bodies in SR.

What you're essentially claiming is that the Lorentz transformation is wrong. However those transformations have been tested over and over again in particle accelerator labs and in other experiments and observations and in all of them they've been consistent with observation. It's your interpretation and false assumptions that are wrong here.

[Fruityloop (OP)]

Because information cannot travel faster than the speed of light. A rigid body is one in which the distances between every atom are fixed and do not change. But if we gave such a body a push from one end, information would have to travel through the entire body instantaneously to signal to all other parts of the body to 'move'. This means information has to travel faster than light which is precluded by special relativity. So, rigid bodies cannot be described by special relativity, and therefore do not exist in the physical world.

So how does this give us chains that magically stretch?

When you make silly ridiculous statements I need to point it out.

[PmbPhy]

So how does this give us chains that magically stretch?

Wow! You have no problem with a body shrinking from Lorentz contraction but insult me when I explain that a similar thing happens for stretching regarding rigid bodies.

How? It's a property of spacetime, not a property of matter, of course. Clearly you don't really understand SR and here you are claiming its wrong. You're a real piece of work. It "magically" stretches in the same way that a rod "magically" contracts when in motion due to Lorentz contraction. Gee! You never thought of that, have you?

When you make silly ridiculous statements I need to point it out.

This is just like someone who thinks that he's proved special relativity to have a problem with it. I.e. they start posting rude comments like that. I asked you very politely to not post such comments.

When you started a thread in this subforum you're required to follow the rules.  The top thread in this subforum is a "sticky" which is labeled Forum acceptable usage policy. It requires you to follow the rules listed at:
http://www.thenakedscientists.com/forum/index.php?topic=8535.0      which states in particular

2.  Keep it friendly
 
Do not use insulting, aggressive, or provocative language.

If you feel another forum user is using insulting language, seek to calm things down, or if that fails, report the matter to the moderators.  Under no circumstances should you seek to trade insults, or make accusatory remarks to that, or any other, forum user.

Show respect to other forum users.  In particular, there are times when forum users might post about delicate personal issues.  Please refrain from trivialising or making inappropriate remarks, or remarks that might embarrass the poster.

You violated that rule when, in Reply #7 you wrote Now putting aside the ridiculous absurdity of this for a moment, ... I clearly and politely responded asking you to stop it, i.e. I wrote

Anyway, please stop with the "ridiculous absurdity" talk. It's not very polite and I've been very polite to you throughout this entire discussion. To be honest, when someone says things like that all it tells me is that they have a poor grasp of the physics involved.

But you ignored that, didn't you? What's worse is how wrong you are about the physics. This fact about elasticity and there being no such thing as a rigid body, i.e. nothing exists that can't stretch, is a well-known fact and is found in almost all SR textbooks.

By posting comments such as When you make silly ridiculous statements I need to point it out. you're breaking this rule. This is grounds for being suspended.

By the way. Why haven't you answered the following questions that I asked you? If you don't tell me what's going on in your thoughts then how can I correct you when its a matter of fact that you need to be corrected because you're going against well-established results of derivations and experiment. I.e. I asked

(1) Therefore, if a body is non-rigid it has so "stretchiness" to it and that is why the chain doesn't break. Do you understand now?
- Clearly you didn't and don't understand the physics. It seems to me that if you did then you'd have responded to this very simplequestion.

(2) So if a body is not rigid then what does that imply? - That was partly a rhetorical question but I wanted to know how you'd respond to it so that I know what is going on with you. The fact is that you clearly don't understand what it implies. It's not even clear whether you know and understand why there are no rigid bodies in SR. Do you?

Why is it that you don't appear to have a problem with Lorentz contraction but have a problem with stretching? After all the Lorentz contraction applies to an infinitely rigid body.

(3) So if a body is not rigid then what does that imply? - You gave no answer to that question either.

(4) What is it about that which you have such a difficult time understanding? - Again, no answer.

(5) Or do you actually not believe that there is no such thing as a rigid body in reality? - And again, no answer.

Notice how I've answered ALL of your questions and you've answered none of mine.

Regarding contracting and stretching, as Ohanian explains in Einstein's Mistakes on pages 282-283

When the equations for relativistic quantum mechanics were finally formulated by Dirac, Lorentz's attempt to calculate length contraction from atomic physics had nearly been forgotten, and Dirac nor any of his contemporaries thought to apply his new equations of relativistic quantum mechanics to the problem of length contraction. It wasn't until 1941 when the American physicist W.F.G Swann revisited Lorentz's arguments in the context of relativistic quantum mechanics and showed that, indeed, the length contraction emerges from a quantum-theoretical calculation of the length of a solid body when the length of a moving body is compared with the length of a similar body at rest. Swann's argument also applies to time dilation, and they can be used to show how the internal dynamics of an atom or of any kind of clock lead to a time dilation, that is, a reduction of the frequency of vibration or ticking of a moving atom or a moving clock. But Swann's argument was as quickly forgotten as Lorentz's.

I'm certain that Swann's argument can be applied to the stretching of a body.

[PmbPhy]

Out of all the special relativity textbooks that I have almost all of them explain that rigid bodies don't exist and SR can be used to show why.

I'll list six of the occurrences of this in the SR texts that I have, the last of which I'll quote:

Relativity; Special, General and Cosmological by Wolfgang Rindler.
Basic Relativity by Richard A. Mould.
Special Relativity by A.P. French.
Introducing Special Relativity by W.S.C Williams.
Theory of Relativity by W. Pauli.

From Special Relativity: A Modern Introduction by Hans C. Ohanian. From the section entitled The synchronization of clocks and the Relativity of Simultaneity. On page 49 the author writes[/b]

[Fruityloop (OP)]

So how does this give us chains that magically stretch?

Wow! You have no problem with a body shrinking from Lorentz contraction but insult me when I explain that a similar thing happens for stretching regarding rigid bodies. It "magically" stretches in the same way that a rod "magically" contracts when in motion due to Lorentz contraction. Gee! You never thought of that, have you?  This fact about elasticity and there being no such thing as a rigid body, i.e. nothing exists that can't stretch, is a well-known fact and is found in almost all SR textbooks.

Length contraction is part of Special Relativity.  But I've never read anything about objects that magically stretch.  Do you have any sources or is this something you just made up?  We need to stick to the theory.  You seem to be implying that because rigid bodies don't exist, we can have things like chains that magically stretch.  The conclusion doesn't follow from the premises.  How can a chain stretch for 4 minutes?

Out of all the special relativity textbooks that I have almost all of them explain that rigid bodies don't exist and SR can be used to show why.

I'll list six of the occurrences of this in the SR texts that I have, the last of which I'll quote:

Relativity; Special, General and Cosmological by Wolfgang Rindler.
Basic Relativity by Richard A. Mould.
Special Relativity by A.P. French.
Introducing Special Relativity by W.S.C Williams.
Theory of Relativity by W. Pauli.

From Special Relativity: A Modern Introduction by Hans C. Ohanian. From the section entitled The synchronization of clocks and the Relativity of Simultaneity. On page 49 the author writes[/b]


Quote from: Hans C. Ohanian

I guess I have to repeat myself.

Because information cannot travel faster than the speed of light. A rigid body is one in which the distances between every atom are fixed and do not change. But if we gave such a body a push from one end, information would have to travel through the entire body instantaneously to signal to all other parts of the body to 'move'. This means information has to travel faster than light which is precluded by special relativity. So, rigid bodies cannot be described by special relativity, and therefore do not exist in the physical world.

That's nice.  It's also completely irrelevant.  Nowhere in the opening post is it stated, implied, assumed, nor required that the transmission of movement along the chain from the movement of the balls must exceed the speed of light.

[PmbPhy]

Length contraction is part of Special Relativity.

So isn't the non-rigidity of bodies. Where on Earth did you learn SR from? I.e. what textbook on the subject do you have?

That this is part of SR see: https://en.wikipedia.org/wiki/History_of_special_relativity#Rigid_bodies_and_Ehrenfest_paradox

Einstein (1907b) discussed the question of whether, in rigid bodies, as well as in all other cases, the velocity of information can exceed the speed of light, and explained that information could be transmitted under these circumstances into the past, thus causality would be violated.

All you really need to do is pick up a text on the subject and read it. I gave you a list of six well-known/very popular SR textbooks. The text Special Relativity by A.P. French was written by a physicist at MIT and is used to teach SR at MIT even today and it explains this fact.

  But I've never read anything about objects that magically stretch.

You claim to know SR and you use terms  like "magically stretch"? The two don't go together. I showed you in a derivation and you keep ignoring it. I'll do it again:

Let there be a rod having zero elasticity (i.e. infinite Young's modulus) lying at rest along on the x-axis in the inertial frame S. Now accelerate the rod in the +x direction. If the rod is perfectly rigid then all parts will move at the same time. Define the following events:

Event #1 (E1): Left end of rod starts to accelerate at t = 0
Event #2 (E2): Right end of rod starts to accelerate at t = 0

What do observers in the inertial frame S' which is in standard configuration observe? Using the Lorentz contraction you determine that the ends don't start moving at the same time because while E1 and E2 are simultaneous in S they are not simultaneous in S'. Since the ends don't move at the same time the length changes and therefore the body is not rigid.

When you keep using the term "magic" all you're doing is showing your ignorance in SR. Why would you want to give everyone the impression that you're ignorant in SR? If others chose to chime in they'd explain how wrong you are.

  Do you have any sources or is this something you just made up? 

Oy vey!!! I just gave you a list of six references in my last post. Why did you ignore it? I'll repeat it:

Relativity; Special, General and Cosmological by Wolfgang Rindler.
Basic Relativity by Richard A. Mould.
Special Relativity by A.P. French.
Introducing Special Relativity by W.S.C Williams.
Theory of Relativity by W. Pauli.
Special Relativity: A Modern Introduction by Hans C. Ohanian.

  We need to stick to the theory.

That's the only thing that I ever do. The fact is that I know this subject matter a great deal better than you've shown that you do so I know that I'm sticking quite closely to the theory. This was published by Einstein in 1907 in fact.

You've got me curious. Why do you insist of ignoring most of what I'm telling you? I've repeatedly said that it's in most SR textbooks. I.e. in Reply #9 I wrote

This fact about elasticity and there being no such thing as a rigid body, i.e. nothing exists that can't stretch, is a well-known fact and is found in almost all SR textbooks.

In Reply #9 I wrote

Out of all the special relativity textbooks that I have almost all of them explain that rigid bodies don't exist and SR can be used to show why.

I already gave you a list of textbooks that explain it. You can download any and/or all of them from http://bookos-z1.org/ to verify that for yourself.

I'll quote one of them to make this certain. From Special Relativity: A Modern Introduction by Hans C. Ohanian,  (2001).

More generally, a physical signal of any kind cannot have a speed exceeding the speed of light. A direct consequence of this limitation is that an absolutely rigid body cannot exist, because such a rigid body could be used to transmit signals with infinite speed. For instance, a sudden push exerted against one end of an absolutely rigid rod would cause an immediate displacement of the other end of the rod, which would constitute a signal with infinite speed. Physical rods made of solid materials are always somewhat elastic. They are stiff but not absolutely rigid, and the speed of a compression signal propagating along the rod depends on the speed of sound in the solid material; this speed is always much lower than the speed of light.

I guess I have to repeat myself...
....
That's nice.  It's also completely irrelevant.  Nowhere in the opening post is it stated, implied, assumed, nor required that the transmission of movement along the chain from the movement of the balls must exceed the speed of light.

Your problem here is that you're focusing on the premise rather than the conclusion. The non-rigidity of bodies in SR is a consequence of the physical speed limit of c. I already explained that to you but I can figure out what you're unable to understand it. I'm sorry to say, and this is not meant as an insult, just an observation, that you have a problem focusing on what is being explained to you.

For example, in Reply #3 I clearly wrote

So what? You missed my point. Besides, I didn't say otherwise. I merely showed you a site that describes why there is not such thing as a rigid body in SR. That it talks about other things is irrelevant.

That means that for some reason that I don't currently understand. You clearly ignored the most important part of that quote, i.e. the part that says Why can't a rigid body exist? The part of the quote that says [Because information cannot travel faster than the speed of light. is the reason that rigid bodies don't exist. Your problem was that, for some crazy reason, you only focused on that part and ignored what was deduced.

The only reason I posted that quote was to show you where someone said that rigid bodies don't exist. That website is a relativity website at Stanford University so its trustworthy. However you only focused on the first part and ignored the relevant part and you did it twice. Why?

Since you're having some problems understanding what I've explained to you I'll explain it again:

The premise of an argument is the proposition that the argument is based on and is used to lead to the conclusion. In this case the premise is Information cannot travel faster than the speed of light. The conclusion of the argument is the proposition Rigid bodies cannot be described by special relativity, and therefore do not exist in the physical world. That's why Ohanian wrote Physical rods made of solid materials are always somewhat elastic.

[PmbPhy]

I'm going to post all the quotes from all six textbooks that I mentioned above so as to drive home this point. Keep an eye on this post because I'm going to keep editing it to keep adding parts from different texts as well as a link to the original paper by Einstein on the subject. That paper is entitled On the Inertia of Energy Required by the Relativity Principle by A. Einstein, Annalen der Physik 23 (1907). The section which it appears in, i.e. section #3, is entitled Remarks concerning the dynamics of the rigid body[/i]. In it Einstein wrote

We will now show that not only the assumption of an instantaneous spread of some effect, but also, more generally, any assumption of the spreading of an effect with a velocity greater than the velocity of light is incompatible with the theory of relativity.

That means that rigid bodies don't exist.

From Special Relativity by A.P. French, page 27:

One particular consequence of the physical speed limit equal to c is that the classical concept of an ideal rigid body finds no place in relativity. (And strictly speaking, it cannot be justified in classical mechanics either.) For by a rigid body we mean an object which physical information can be transmitted in an arbitrarily short time, so that the object is set in motion instantaneously, as a single unit, when a force is applied to any point in it. For an ordinary box, the information that one end has been struck is transmitted as an elastic wave, which we know is many orders of magnitude slower than a light signal.

[Fruityloop (OP)]

Let there be a rod having zero elasticity (i.e. infinite Young's modulus) lying at rest along on the x-axis in the inertial frame S. Now accelerate the rod in the +x direction. If the rod is perfectly rigid then all parts will move at the same time. Define the following events:

Event #1 (E1): Left end of rod starts to accelerate at t = 0
Event #2 (E2): Right end of rod starts to accelerate at t = 0

What do observers in the inertial frame S' which is in standard configuration observe? Using the Lorentz contraction you determine that the ends don't start moving at the same time because while E1 and E2 are simultaneous in S they are not simultaneous in S'. Since the ends don't move at the same time the length changes and therefore the body is not rigid.

I don't understand the relevance of this to the opening post.  We aren't dealing with a rod of zero elasticity which I thought you already pointed out couldn't exist anyway.

Let me see if I can summarize your solution to the opening post...

A rigid body doesn't exist because that would require propagation speeds of movement along a material body to be faster than light, which isn't possible.  Therefore materials have some kind of 'stretchiness' to them.  Therefore a chain can stretch for nearly 4 minutes without breaking.  Um, yeah right.  But as I pointed out this would just result in another paradox if the movement of the balls for observer O1 weren't simultaneous and also the movement of the balls for observer O2 weren't simultaneous.  The chain would break for observer O1 but wouldn't break for observer O2.
I think we're done here.  Next person please...

[PmbPhy]

I don't understand the relevance of this to the opening post.

You've got to be kidding me. The conclusion of any derivation follows the term therefore. In this case the conclusion was the body is not rigid. Clearly that was a simply way to illustrate how a body can be elastic in one frame even when its rigid another frame. It's all based on two facts (1) lack of simultaneity and (2) the finite speed of light.

  We aren't dealing with a rod of zero elasticity which I thought you already pointed out couldn't exist anyway.

You just can't follow a simple argument, can you? The whole idea was to prove that assertion to be wrong. I shouldn't have used the term "zero elasticity" though. That was an error on my part. The appropriate phrasing would have been something like

Let there be a rod lying at rest along on the x-axis in the inertial frame S. Now accelerate the rod rigidly in the +x direction. This means that If the we accelerate all parts of the rod with the same acceleration curve. This means that the rod will behave rigidly in frame S, i.e. all parts of the rod will start to move at the same time and will have the same velocity at the same time as observed in S.

That way the rod acts like its rigid in frame S. Notice that all parts will start to move simultaneously in S. That means that they won't to move as observed in frame S' which is in standard configuration with S.

Let me see if I can summarize your solution to the opening post...

A rigid body doesn't exist because that would require propagation speeds of movement along a material body to be faster than light, which isn't possible. 

There you go!  This should be clear to you by now. I just posted two examples of this from two different authors. I have more which I'll quote later after I type them all up later this afternoon or tonight. There will be five quotes. The sixth one by Pauli is too long to quote.

Therefore materials have some kind of 'stretchiness' to them.  Therefore a chain can stretch for nearly 4 minutes without breaking.

Precisely. If you have trouble understanding this just remember that you have no experience observing things moving past you at near light speed (this is actually extremely close to the speed of light). However, what you failed to do is to analyze the motion of all the parts of the chain, i.e. "how" the chain stretches. In frame S each link of the chain "drops" down from the upper position to the lower position. If you actually chose to analyze this in S' then what you'd observe is a series of events, one link dropping at a time in a manner of speaking. Not exactly though. What actually happens is hard to visualize because one part of the chain is doing nothing while other parts are dropping down. By my calculation that series of events takes 4 seconds, not 4 minutes. However I could very well be wrong on that.

Um, yeah right.

And here we go with the sarcasm again. This is so typical of you. Once you show that don't understand something or can't grasp it you turn into a wiseass. My, my. Aren't you the professional one!

But as I pointed out this would just result in another paradox ...

You don't appear to know the meaning of the term "paradox." Please look it up before you use it again. I'll do the work for you just to make sure that you learn what it means. From: https://en.wikipedia.org/wiki/Paradox

A paradox is a statement that apparently contradicts itself and yet might be true (or wrong at the same time).

So being a paradox doesn't necessarily mean that there's a contradiction. But in this case there's no contradiction whatsoever.

But as I pointed out this would just result in another paradox if the movement of the balls for observer O1 weren't simultaneous and also the movement of the balls for observer O2 weren't simultaneous.

In what post did you discuss this? In every post I've read of yours I've only seen you mention the events being simultaneous in O1 and thus the events were not simultaneous in O2. In that scenario special relativity predicts that the chain doesn't break for either observer.

I think we're done here.

I think that's best. You don't have a solid grasp of relativity to be able to understand the scenario you outlined. So its best that you stop trying, especially since you're unable to understand me or the textbooks that I've both quoted and showed you where to download it and read it for yourself.

Next person please...

Don't bother. Since you can't understand what I've described to you then you simply are unable to understand the physics. Until you go out and pick up a good text on SR and start learning it properly you shouldn't try to claim its wrong because you're just making yourself look ignorant.

[PmbPhy]

By the way, this paper talks about the subject of non-rigid bodies in SR: http://arxiv.org/ftp/arxiv/papers/1205/1205.0221.pdf

A Discussion of Special Relativity by Galina Weinstein

Five topics: A rigid body does not exist in the special theory of relativity; distant simultaneity defined with respect to a given frame of reference without any reference to synchronized clocks; challenges on Einstein's connection of synchronization and contraction; a theory of relativity without light, composition of relative velocities and space of relative velocities.

See 1. Kinematics of the "Rigid Body" – No such Thing

[Fruityloop (OP)]

None of what you're posting is relevant to the opening post.
How does the non-existence of rigid bodies in SR enable such things as chains that stretch for nearly
4 minutes without breaking?
Can you give a quote from any text that enables one to draw such a conclusion?
Or is this something that you just made up?  Because I have never read anything that would
lead me to believe such a thing.

[PmbPhy]

None of what you're posting is relevant to the opening post.

It most certainly is!

How does the non-existence of rigid bodies in SR enable such things as chains that stretch for nearly
4 minutes without breaking?

I've already told you that so many times I've lost count. Your problem is that you keep missing the point. All bodies have elasticity to them, period!! Even a steel chain. 

If frame O2 was moving at a speed of v = 0.9c then it would happen 9 nanoseconds to happen rather than 4 minutes. But because you've chosen a frame to be moving so fast all you've done is to chose a frame to watch these events unfold extremely slow. Don't you understand the effects of time dilation?

Can you give a quote from any text that enables one to draw such a conclusion?
Or is this something that you just made up?  Because I have never read anything that would
lead me to believe such a thing.

No text explains every single scenario that newbies with their own little scenarios can dream up. Texts are only designed to give the reader the skills to solve analyze problems like this. The only problem here is that you can't understand relativity that well. You're terrible at this and that's all there is to it.

If you don't believe me then all you have to do is contact one of the authors of an SR textbook and ask them about the resolution to your little problem. They'll all tell you the same thing. Go ahead, I challenge you to do so. Are you confident enough in your beliefs to accept that challenge or not?

[Ethos_]

So for oberver O2 ball B2 is pulled down about 4 minutes before ball B1 and the chain breaks.
So we have a paradox, for observer O1 the chain is intact and for observer O2 the chain is broken.
Both situations can't simultaneously be true.

Whenever you think that SR is wrong you can be guaranteed that you made a mistake. This happens every single day on the internet by different people. The result is that the person's understanding of SR is incomplete.

In this case you're making the false assumption that the chain can be treated as a rigid body. However its a well-established fact in SR a rigid body does not exist in the special theory of relativity. E.g. if you have a rod that's lying on the x-axis in frame S and you start accelerating every single part of the rod at the same time as observed in S then in frame S' which is in standard configuration with S and which is moving in the +x direction the rod will not remain rigid but different parts will accelerate at different rates.

In your case the chain stretches as observed by O2.

For more on this please see: https://einstein.stanford.edu/content/relativity/q2018.html

I agree with you Pete, and the link you've provided explains it perfectly.

The OP is assuming that rigid bodies exist in both frames.

"Why can't a rigid body exist?"

"Because information can't travel faster than the speed of light."

"A rigid body is one where the distance between every atom is fixed and does not change."

The issue here is; Defining the body as rigid to which reference frame?

If one wants to get really technical, there are no rigid bodies even when observed in the same frame. When in separate moving frames of reference, the rigidity becomes even more suspect.