Does Bell's Paradox Suggest String Shouldn't Break Due to Length Contraction?

[Eli Botkin]

DaleSpam:
Your 86, as it now stands, is not the same as it stood when I made my 87 reply. You subsequently changed your mind; and I like that you rethought your statement. (However, I think it is not wise to alter records. Rather, one should make a new, corrective post, and thereby avoid this type of confusion.)

In your original 86 you clearly stated that you did not believe that there are inertial frames in which the observer would see a contracting separation. It seems that, maybe, now you agree that there are such frames, and if so, that is to the good.

Your new 86 has you saying “The string length contracts more than the distance between the ships decreases.” If only that were so for observers in any inertial frame, then I would be among the first to declare that SR alone is sufficient to answer the Bell Paradox and that the string breaks.

Your math in your post 43 does not address this question. Rather it just assumes that the string length is unaltered even as the ships’ separation is expanding.

But your declaration in the new 86 begs an SR mathematical proof. One needs a proof that the string length is less than the ships’ separation in all inertial frames after acceleration starts. Somehow, to me, this sounds like one would need to treat the string differently from the ships’ separation in applying the SR transformations, and that sounds like including theory which is non-SR.

I remind you that I never took a position on string-breakage (yes or no) in the Bell Paradox. I maintain only that it takes more than SR to determine that.

I would be happy to receive an SR proof (whatever source) of your new 86 declaration.

 

[Dale]

Your 86, as it now stands, is not the same as it stood when I made my 87 reply. You subsequently changed your mind;

Obviously. That is why I said what I said in post 88.

and I like that you rethought your statement. (However, I think it is not wise to alter records. Rather, one should make a new, corrective post, and thereby avoid this type of confusion.)

I make a lot of edits, and if I followed this suggestion I would be spamming the board, which irritates me when I see other people do it.

Instead, my personal policy is that I will make edits until someone responds to my post. Once someone has responded, I no longer edit. It took me only 9 minutes from the time of my post to work through the math, realize I made a mistake, check that you had not responded, and make a correct post. I feel that is fine behavior on my part. I am sorry that you were confused, but if you spend much time on the forum then it is guaranteed to happen again.

Your new 86 has you saying “The string length contracts more than the distance between the ships decreases.” If only that were so for observers in any inertial frame, then I would be among the first to declare that SR alone is sufficient

The principle of relativity requires only that the laws of physics be the same in all inertial reference frames. There is no requirement that every explanation of every scenario need be the same in all reference frames. E.g. cosmic ray muons reach the Earth following the same laws in all frames, but in some frames the explanation is time dilation and in other frames it is length contraction.

Your math in your post 43 does not address this question. Rather it just assumes that the string length is unaltered even as the ships’ separation is expanding.

Correct, I explcitly assumed that the string is stiff.

I maintain only that it takes more than SR to determine that.

I proved that SR is sufficient. And you admitted that the reasoning in the proof was valid. You did avoid answering the question about your opinion of the assumptions, so I will re-ask them:

1) Do you think the assumptions I made are the standard ones relevant to Bells spaceships?
2) Do you think they are correct assumptions?

I would be happy to receive an SR proof (whatever source) of your new 86 declaration.

And I once again encourage you to work it out.

 

[Eli Botkin]

DaleSpam:
I agree with you, "The principle of relativity requires only that the laws of physics be the same in all inertial reference frames." However, if those laws of physics predict that star A will collide with star B in anyone frame, then it better predict the same in all frames, otherwise there is something wrong with the prediction ;-)

You say ..."cosmic ray muons reach the Earth following the same laws in all frames, but in some frames the explanation is time dilation and in other frames it is length contraction." That is due to interpretation of equations, but the important physics is that for any observer they live long enough to reach the ground.

By assuming that he string length would be unchanging in post #43, you guaranteed for yourself that you would get the result you thought you should. Your proof was "valid" only in the sense that your assumptions mathematically lead to your conclusion. It's your assumption that's the problem. It needs to be mathematically shown that the string length doesn't obey the same physics as the separation length between ships.

You encourage me to work it out. But the reason we've had this lengthy go around is that I cannot show with SR alone that every observer (in any inertial frame) will see that the string's length is less than the ships' separation after acceleration starts. That condition, I would expect, is required for the string to break. If you've shown that (with or without invoking SR) please post your proof.

 

[Dale]

I agree with you, "The principle of relativity requires only that the laws of physics be the same in all inertial reference frames." However, if those laws of physics predict that star A will collide with star B in anyone frame, then it better predict the same in all frames, otherwise there is something wrong with the prediction ;-)

You say ..."cosmic ray muons reach the Earth following the same laws in all frames, but in some frames the explanation is time dilation and in other frames it is length contraction." That is due to interpretation of equations, but the important physics is that for any observer they live long enough to reach the ground.

Agreed.

Similarly, the important physics is that for any observer the string breaks. The disagreement about whether the explanation is the distance between the ships increasing or the length of the string decreasing is purely due to interpretation of the equations.

By assuming that he string length would be unchanging in post #43, you guaranteed for yourself that you would get the result you thought you should. Your proof was "valid" only in the sense that your assumptions mathematically lead to your conclusion.

Obviously. That is true of any proof.

It's your assumption that's the problem. It needs to be mathematically shown that the string length doesn't obey the same physics as the separation length between ships.

OK, so it sounds like you disagree with my assumption that the string is stiff. Is that because you believe that the assumption is not a standard one relevant to Bells spaceships or because you think that it is not correct?

Also, is the stiff string the only assumption that you disagree with? If not, then which others do you disagree with?

You encourage me to work it out. But the reason we've had this lengthy go around is that I cannot show with SR alone that every observer (in any inertial frame) will see that the string's length is less than the ships' separation after acceleration starts.

Then post your work and show me where you get stuck and I will help you from there, as I have offered already.

 

[Eli Botkin]

DaleSpam:
Of course “…the important physics is that for any observer the string breaks.” That was exactly the point I was making about colliding stars.

And agreement or disagreement about whether the explanation is an increase in ship separation or a decrease in string length seems to me to be irrelevant. Rather, it’s the increasing ratio of the former to the latter that is the physics that needs to be frame invariant, without assumptions that assure that outcome. If a correct mathematical proof from assumptions is all you require for satisfaction, then you may well be forgoing the correct physical outcome because you made an incorrect assumption. Deriving the physics from the equations must be more than just moving math symbols around and treating the process like a Rubics cube.

You keep saying that you assumed a stiff string as if that says any more than my saying that you assumed a string of constant length, thereby assuring the outcome of your analysis. You could have come to the same conclusion without the string length in the problem just by noting that the separation increases in the frame you selected ;-)

You need to forgo your constant string-length assumption and see what SR says about string-length just as you sought to see what SR says about the ships’ separation. And if SR treats them both the same way (as I think it does) and you have trouble believing that, then you will conclude that more than SR is needed for the problem’s resolution.

DaleSpam, as I’ve indicated at various points in our discussions, I do not have a solution to the question “does the string break.” Bell himself gave no mathematical proof of breakage though he was of strong opinion that it would break. However, if SR alone is the arbiter, then I don’t foresee breakage because I believe that both the string’s length and the ships’ separation are transformed between inertial frames equivalently.

There you have it. If you know of a solution that avoids your “fixed” string assumption, then bring it on. Otherwise I have nothing to add, except that I’ve enjoyed our time together ;-)

 

[Dale]

What do you think is wrong with assuming a stiff string? If we were to physically perform this experiment then we could choose to do it with a string of rubber or a string of steel. What is unreasonable about assuming steel instead of rubber? The stiff string assumption is simply an idealization of that, ie the limiting case of a string with a high Youngs modulus and a low breaking strength.

 

[Austin0]

DaleSpan:

You've taken me aback. I can hardly imagine that someone who exhibits such SR expertise would not be aware of the inertial frames within which the ships are approaching each other.

Say the ships accelerate to the right in the (rest) frame where they started at the same time. In that frame their separation remains constant.

An inertial observer moving (at some speed V) to the right will say that the lead ship started to accelerate earlier than the aft ship, leading to a continual increase in separation.

Conversely, an observer moving (at speed -V) to the left will say that the aft ship started to accelerate earlier than the lead ship, leading to a continual reduction in separation, without ever overtaking it.

I would encourage you to work this out for yourself ;-)

yes i noticed this myself also. And found it somewhat mysterious. The trailing ship by starting before the lead ship develops a velocity relative to that ship, and given equal proper acceleration there is no reason to assume that this velocity will diminish over time. Yet we also have to assume that it will never actually reach the lead ship.
The only explanation I could come up with is the diminishing coordinate acceleration, in the frame in which it starts first, results in the velocity differential asymptotically approaching zero.
SO it never reaches the lead ship.
What do you think, does this sound right?

 

[Eli Botkin]

DaleSpam:
Remember the Pole/Barn Paradox? Did you ponder whether or not the pole was stiff or elastic before applying the SR transformations ?

 

[Eli Botkin]

Austi0:
you say " Yet we also have to assume that it will never actually reach the lead ship."

That need not be an assumption. The Minkowski diagram shows that the two ship hyperbolic worldlines are the same shapes, laterally displaced from one another and, therefore, never intersecting.

 

[yuiop]

However, if SR alone is the arbiter, then I don’t foresee breakage because I believe that both the string’s length and the ships’ separation are transformed between inertial frames equivalently.

There you have it. If you know of a solution that avoids your “fixed” string assumption, then bring it on. Otherwise I have nothing to add, except that I’ve enjoyed our time together ;-)

Hi Eli, I will give two simple scenarios from which I think any reasonable person would conclude that the string must break (unless they can find fault with the scenarios).

First, the strength of the string is not particularly important, other than the assumption that there does not exist an infinitely strong string. We are basically trying to establish whether the string is physically stretched and under tension or not. For the purposes of the example I will assume a string that snaps when stretched to twice its rest length.

Scenario 1:

Two rockets are initially at rest on the ground 1 km apart. A string connects the rockets and is under negligible tension just sufficient to take up the slack. The rockets take off and are under instruction to stay 1 km apart at all times as measured in their own instantaneous rest frame. At 0.866c relative to the ground they are 0.5km apart as measured in the ground reference frame. At this velocity relative to the ground the ground based observers calculate that the length contracted length of the string is 1/2 km so they so there is no significant tension on the string because it spans a separation of 1/2 km. The rocket observers say the length of the string in their instantaneous co-moving reference frame is 1 km and it spans a separation of 1 km so they agree there is no significant tension on the string. Agree so far?

Scenario 2:

Same initial set up as Scenario 1, but this time the rocket pilots are instructed to stay 1 km apart as measured in the ground reference frame. When the rockets are moving at 0.866c relative to the ground, the un-tensioned length of the string should be 1/2 km (as calculated in the first scenario) but it is now stretched across a separation of 1 km as measured by the ground based observers (because that the distance the rocket pilots have been instructed to maintain) so the string is on the point of breaking.

From scenario 1 we know that if the rocket pilots measure the separation distance in their own reference frame to be 1 km, that the string is under no significant tension, but since they have been asked to maintain a separation distance of 1 km as measured in the ground based reference frame, then at 0.866c they must be separated by 2km as measured in the rocket based reference frame and the string must be stretched to twice its rest length and on the point of breaking.

Do you agree that all observers in Scenario 2 agree that the string is on the point of breaking and that the string will snap if they exceed 0.866c relative to the ground?

If not, what do you disagree with in the two scenarios?

 

[Eli Botkin]

yuiop:
First, you are not addressing (in your scenario 1) the issues that are being discussed in the Bell Paradox scenario. I’m certain that there are countless scenarios of two accelerating vehicles, connected by a string, wherein the string must break. Whether or not your selected scenarios do indeed make breakage certain, is something I would have to check mathematically, and that takes time.

At this point I’m not sure that it holds my interest since, as I said above, there are many scenarios that ensure that outcome.

Second, a note about your scenario 1:
Instructing the rockets “to stay 1 km apart at all times as measured in their own instantaneous rest frame” means that before they start they need to know what each of their accelerations, as function of time, needs to be. Those accelerations won’t be constants as in the Bell scenario. And there is more than one such set of acceleration histories that could suit the 1 km requirement. A calculation headache ;-)

Now your scenario 2:
“…the rocket pilots are instructed to stay 1 km apart as measured in the ground reference frame.” This is what happens in the ground frame when the acceleration histories are identically the same for both rockets. This is the Bell scenario.

But you need to tell me why the “…the un-tensioned length of the string should be 1/2 km…” in the ground frame. If you think it is because “…the rockets are moving at 0.866c relative to the ground,…”, then why is the rocket separation still 1 km, though the rocket frame (which is the string’s frame) is also moving at 0.866c relative to the ground?

Ultimately the question comes down to this:
1. Why is the string's length, as transformed between inertial frames, being treated differently than the rockets' separation length.
2. Arguments for breakage always seem to hinge on scenarios as viewed by observers that never see the rockets approaching each other, when in fact, there are such observers.

 

[Eli Botkin]

yuiop:
First, you are not addressing (in your scenario 1) the issues that are being discussed in the Bell Paradox scenario. I’m certain that there are countless scenarios of two accelerating vehicles, connected by a string, wherein the string must break. Whether or not your selected scenarios do indeed make breakage certain, is something I would have to check mathematically, and that takes time.

At this point I’m not sure that it holds my interest since, as I said above, there are many scenarios that ensure that outcome.

Second, a note about your scenario 1:
Instructing the rockets “to stay 1 km apart at all times as measured in their own instantaneous rest frame” means that before they start they need to know what each of their accelerations, as function of time, needs to be. Those accelerations won’t be constants as in the Bell scenario. And there is more than one such set of acceleration histories that could suit the 1 km requirement. A calculation headache ;-)

Now your scenario 2:
“…the rocket pilots are instructed to stay 1 km apart as measured in the ground reference frame.” This is what happens in the ground frame when the acceleration histories are identically the same for both rockets. This is the Bell scenario.

But you need to tell me why the “…the un-tensioned length of the string should be 1/2 km…” in the ground frame. If you think it is because “…the rockets are moving at 0.866c relative to the ground,…”, then why is the rocket separation still 1 km, though the rocket frame (which is the string’s frame) is also moving at 0.866c relative to the ground?

Ultimately the question comes down to this:
1. Why is the string's length, as transformed between inertial frames, being treated differently than the rockets' separation length.
2. Arguments for breakage always seem to hinge on scenarios as viewed by observers that never see the rockets approaching each other, when in fact, there are such observers.

 

[A.T.]

1. Why is the string's length, as transformed between inertial frames, being treated differently than the rockets' separation length.

What? The string's length and the rockets' separation are the same thing.

2. Arguments for breakage always seem to hinge on scenarios as viewed by observers that never see the rockets approaching each other

I gave you the reason for breakage in the frame where they don't approach each other.

 

[Austin0]

Austi0:
you say " Yet we also have to assume that it will never actually reach the lead ship."

That need not be an assumption. The Minkowski diagram shows that the two ship hyperbolic worldlines are the same shapes, laterally displaced from one another and, therefore, never intersecting.

Hi , you are of course quite right that the two worldlines would be identical in shape but they are not simply displaced laterally but also vertically (temporally).
So with a certain magnitude of lead time for the trailing ship, the lines could intersect even with identical curvature. So the assumption part is; that the maximum possible time difference due to relative simultaneity for the distance between them, in any frame, is always going to be less than this threshold magnitude.

 

[Dale]

Remember the Pole/Barn Paradox? Did you ponder whether or not the pole was stiff or elastic before applying the SR transformations ?

The stiffness of the pole is irrelevant in the barn/pole paradox. The pole is not under tension and is moving inertially, so whether it is made of rubber or steel the calculations are the same.

So again, what is wrong with assuming a stiff string in the Bell's spaceship scenario? Do you object because you think the assumption is non-standard or because you think it is wrong? Please answer these questions directly instead of with an evasion.

 

[Dale]

yes i noticed this myself also. And found it somewhat mysterious. The trailing ship by starting before the lead ship develops a velocity relative to that ship, and given equal proper acceleration there is no reason to assume that this velocity will diminish over time. Yet we also have to assume that it will never actually reach the lead ship.
The only explanation I could come up with is the diminishing coordinate acceleration, in the frame in which it starts first, results in the velocity differential asymptotically approaching zero.
SO it never reaches the lead ship.
What do you think, does this sound right?

This is correct. As they both asymptotically approach c in such a frame their relative coordinate velocity clearly goes to zero, meaning that the distance approaches some asymptotic value. It happens that this asymptotic value is non-zero, and so regardless of the length of the rope and the initial separation, eventually it breaks when it contracts below that finite distance.

 

[Nugatory]

2. Arguments for breakage always seem to hinge on scenarios as viewed by observers that never see the rockets approaching each other, when in fact, there are such observers.

Indeed there are such observers, and an observer moving, as viewed by the ground observer, in the opposite direction from the two spaceships is one. Let's call that observer the "left-mover", and then take on your question #1.

1. Why is the string's length, as transformed between inertial frames, being treated differently than the rockets' separation length.

They are being transformed from the frame in which they are constant and equal to L in the same way; the trick is that these frames are different. In one case we're transforming a distance that is constant in the front-ship frame (and therefore contracted by different amounts in all other frames including the ground frame). In the other case we are transforming a distance that is constant in the ground observer's frame (and therefore contracted by different amounts in all other frames including the front-ship frame). When I transform them both into the frame of your left-moving observer, I will get different contractions because the relative velocity between the "from" frame and the left-moving frame is different for the two transformations.

Thus, the left-mover sees the distance between the ships contract, but sees the length of the string contract even more (his speed is greater relative to the frame in which the string's length is L then it is relative to the frame in which the ship separation is L). Ground observer happens to be using the only frame in which the ship separation is constant, but in that frame the string contacts because it is moving. Lead-ship observer occupies a frame in which the length of the string is constant, but in that frame the distance between the ships is increasing (because the two ships are maneuvering to maintain a constant separation in ground observer's frame, so the separation distance is not constant in the front-ship frame).

Lemme know if this isn't clear enough... I have a space-time diagram that shows how the three frames (ground, front ship, and left mover) are related, along with the Lorentz transformation calculations.

(as an aside, I find that when time dilation or length contraction are giving me trouble, it often helps to go back to the Lorentz transformations themselves. Length contraction and time dilation can be derived by applying the Lorentz transforms to particular thought experiments, and if you're working with a scenario that doesn't match this those thought experiments, it's easy to misapply contraction and dilation).

 

[Eli Botkin]

A.T.:
You say "What? The string's length and the rockets' separation are the same thing."
Why the puzzlement? Isn't that what I'm implying when I'm asking (with tongue in cheek) "Why is the string's length, as transformed between inertial frames, being treated differently than the rockets' separation length? What I'm asking is why the string and the separation aren't being treated in the same SR way if, as you say, "they are the same" and so should be. Do we agree on this?

Please direct me to the reply number where you've answered my question about frames where the ships' separation reduces with time

 

[Eli Botkin]

Nugatory:
Indeed this isn't clear enough. To me it sounds like a comparison of apples and oranges. I would expect the comparison in frame A to be transformed to one in frame B , thereby assuring that the time coordinate is the same for all events in frame A and again in frame B. Why the need to transform one distance from frame A into frame B, then the other distance from frame C into frame A. Certainly both distances exist in frame A.

Maybe your spacetime diagram would clear this up for me. Thanks

 

[Eli Botkin]

DaleSpan:
Here is one thing that is wrong with assuming a stiff string.

Inertial observer A is moving opposite to the ships' acceleration, all of them with respect to the ground frame. As you now agree, for A the rear ship is closing in on the front ship. Or at least it should be except for its prevention by that stiff string whose compression forces acting on the ships plays havoc with the Bell Paradox scenario;-)

Unless, of course, there are no compression forces and the scenario continues to evolve without string compression (nor breakage as viewed in other frames).

 

[A.T.]

What I'm asking is why the string and the separation aren't being treated in the same SR way

No idea what "treated in the same SR way" means. And no idea why you even use two different terms for the same thing: string length = separation distance in any frame.

Please direct me to the reply number where you've answered my question about frames where the ships' separation reduces with time

The reason for breakage valid for all frames:

The atoms of the string cannot span the separation distance anymore

In some frames this is because the separation distance increased, in others because the atoms are contracted.

 

[Eli Botkin]

Nugatory:
I'm rereading your reply #107 and have some questions:

1. You say, "They are being transformed from the frame in which they are constant and equal to L in the same way;...". I read "the frame" to mean a single frame, and I presume it to be the ground frame in which they (both the string length and the separation) have, and maintain, the same length L. Is that what you meant?

But then you add, "...the trick is that these frames are different." You've switched to the plural "these," which I presume is more than one frame. Please clear that up for me.

2. I think you are saying that the string is of constant length L in the front-ship's co-moving frames. Would that then also be true for the rear-ship's co-moving frames? And if so, could that be a problem since there exists a relative velocity between the two ships in any co-moving frame of either ship? Of course, this constant string length in the front-ship co-moving frames is an assumption on your part which may not be assured of realism any more than would be an assumption of constant separation in all co-moving frames ;-)

 

[Nugatory]

Nugatory:
Indeed this isn't clear enough. To me it sounds like a comparison of apples and oranges. I would expect the comparison in frame A to be transformed to one in frame B , thereby assuring that the time coordinate is the same for all events in frame A and again in frame B. Why the need to transform one distance from frame A into frame B, then the other distance from frame C into frame A. Certainly both distances exist in frame A.

Maybe your spacetime diagram would clear this up for me. Thanks

[Edit - fixed a transposed A and B]
I'm busy cleaning up the picture now... But while I'm doing that, could I ask you to read the below, then read my previous post again more carefully?

You've mixed up the common destination frame when you ask "Why the need to transform one distance from frame A into frame B, then the other distance from frame C into frame A?"

We're trying to transform a distance known in frame A into C and another distance known in frame B into C, and we should expect that the A->C transformation is different than the B->C transformation because A and B are different.

Frame A: Ground observer.
Frame B: Front-ship observer
Frame C: Left-moving observer
We know the length of the string as measured in frame B.
We know the separation between the ships as measured in frame A.

What is the length of the string as measured in frame C? We use the B->C transformation on the known length in B to find what the frame C observer measures.

What is the separation between the ships as measured in frame C? We use the A->C transformation on the known separation in A to find what the frame C observer measures.

And once we know the distances as measured in frame C... We compare them to see if the string breaks.

 

[Dale]

DaleSpan:
Here is one thing that is wrong with assuming a stiff string.

Inertial observer A is moving opposite to the ships' acceleration, all of them with respect to the ground frame. As you now agree, for A the rear ship is closing in on the front ship. Or at least it should be except for its prevention by that stiff string whose compression forces acting on the ships plays havoc with the Bell Paradox scenario;-)

Unless, of course, there are no compression forces and the scenario continues to evolve without string compression (nor breakage as viewed in other frames).

What compression? Strings don't sustain compression. They are slack or in tension.

You seem to be unaware of the standard behavior of an idealized massless stiff string. This is basic first-semester Newtonian physics. An idealized massless stiff string is slack if the distance between its attachment points is less than its length and it cannot be stretched without breaking regardless of the tension applied. Do you understand that behavior of idealized massless stiff strings from Newtonian physics? In many first semester problems it is simply called an ideal string.

 

[Eli Botkin]

A.T.:
In this thread all other contributors have elected to distinguish between string length and the separation distance (length) between the ships. The string's ends have been fastened to the two ships. The differentiation is being made because the issue in the Bell Paradox is whether or not these two different lengths change in a way that requires the string to break.

By "treated in the same SR way" I mean only: dealt with through application of of the SR transformation equations.

Hope that clears up both points for you.

You say "The atoms of the string cannot span the separation distance anymore.
In some frames this is because the separation distance increased, in others because the atoms are contracted."

That sounds very nice, but I fear it needs SR's (or any other accepted theory like GR and QM) mathematical verification. If you feel up to that, I'ld appreciate that contribution.

 

[Eli Botkin]

DaleSpan:
Oops, of course you're right that " Strings don't sustain compression. They are loose or in tension."

Except, maybe, when there is only one space dimension, x, in the problem and there is no y or z to hang into ;-)

Better yet, would your solution have a problem if we replaced your "fixed" string with a "fixed" wooden pole so that we could deal with both tension and compression?

 

[Eli Botkin]

Nugatory:
I think that you've mixed up your frames:
"We know the length of the string as measured in frame A."
"We know the separation between the ships as measured in frame B."

"What is the length of the string as measured in frame C? We use the B->C transformation on the known length in B..."
"What is the separation between the ships as measured in frame C? We use the A->C transformation on the known separation in A..."

But I understand what you're doing. I'll keep A for string and B for separation.

But we also know the separation between the ships as measured in frame A? Why go to B?
Is there something special about the observer in B as opposed to the observer in A? or the observer D in the aft-ship? Will B and D agree on separation distance at the same time on their respective clocks? And if not then, then when? Oh..., so many questions ;-)

 

[Nugatory]

Nugatory:
I'm rereading your reply #107 and have some questions:

1. You say, "They are being transformed from the frame in which they are constant and equal to L in the same way;...". I read "the frame" to mean a single frame, and I presume it to be the ground frame in which they (both the string length and the separation) have, and maintain, the same length L. Is that what you meant?

But then you add, "...the trick is that these frames are different." You've switched to the plural "these," which I presume is more than one frame. Please clear that up for me.

The two frames I am referring to are:
1) The ground observer's frame, in which the separation of the ships is always L and the length of the string is something less than L as soon as the ships start moving.
2) The lead-ship frame in which the length of the string is always L and the separation between the ships is steadily increasing as long as the ships are accelerating.

The key to understanding this is to recognize that the path through spacetime of the trailing ship is not the same as the path of the trailing end of the string (except that we've tied the trailing end of the string to the trailing spaceship so the string breaks when its trailing end cannot follow its natural path).

2. I think you are saying that the string is of constant length L in the front-ship's co-moving frames. Would that then also be true for the rear-ship's co-moving frames?

Not the same while the two ships are accelerating, but the same once the engines are cut off and they're drifting at the same constant speed (zero relative to each other, so no relativistic effects). Note that they are not comoving while they're accelerating; there is relative velocity between them as they're accelerating because the acceleration has been specified to provide a constant separation in the ground observer's frame so the must be a non-constant separation in all other frames, including their own.

We're starting a red herring discussion below - read it if you want, but don't respond until you're SURE that you understand what I've said above.

And if so, could that be a problem since there exists a relative velocity between the two ships in any co-moving frame of either ship? Of course, this constant string length in the front-ship co-moving frames is an assumption on your part which may not be assured of realism any more than would be an assumption of constant separation in all co-moving frames ;-)

Strictly speaking, if the ships are comoving (that is, zero relative velocity) then they are in the same frame, except perhaps for a linear transformation of the origin. They will agree about all measured lengths, the rate of all observed clocks, and in general won't see any relativistic weirdness when they compare measurements.

The assumption of constant string length in the front-moving ship's frame comes the fact that the string is at rest relative to the front-moving ship... Always has been, always will be, as long as it is tied to the front-moving ship so moves with it. This is assuming that the string has the decency to break at the knot that attaches it to the trailing ship - but if you don't like that assumption, we can break the string anywhere else and what I said about the the constant length will still apply to whatever part of the string remains attached to the front-moving ship.

 

[Nugatory]

Nugatory:
I think that you've mixed up your frames:
"We know the length of the string as measured in frame A."
"We know the separation between the ships as measured in frame B."

"What is the length of the string as measured in frame C? We use the B->C transformation on the known length in B..."
"What is the separation between the ships as measured in frame C? We use the A->C transformation on the known separation in A..."

But I understand what you're doing. I'll keep A for string and B for separation.

You're right, I did. I just corrected it.

 

[Eli Botkin]

Nugatory:
I’ve been thinking some more about your solution. It occurs to me that there are many close similarities with DaleSpan’s solution.

DaleSpan assumed that the string was, what he termed, “fixed.” That is, the string was always of length L. He then correctly computed the ships’ separation distance in a ship’s co-moving frame and that, of course, was always > L. This, he believes allows him to claim that the string always breaks. It would, of course, have to break at T = 0.

In your own solution you haven’t assumed the string length to be “fixed” but instead selected frame A for its pre-transformation frame, and in frame A the string length is always L. Your co-moving frame B, as in DaleSpan’s case, yields a separation length > L. Then, transforming these two lengths from their respective frames to any frame C will result in the transposed string length being less than the transposed separation length. So you too can claim that the string will break.

Since the separation length is also available in A you could transform both lengths from any selected time in frame A to frame C and then compare them. That seems the more direct approach, avoiding what might be thought by some to be putting a thumb on the scale ;-)