Bell's Paradox Question

[Dale]

I've reviewed your math (your earlier reply #43) and find no error. What you've shown for certain is that for a sequence of instantaneously co-moving observers the ships' separation is some d' > d. I'm familiar with the math, having done this myself. And of course your conclusion favoring string breakage is pre-ordained since you've also stipulated that the string must remain L (= d) at any d'.

Good, so now you have seen it proven using SR.

But why does every co-moving observer "see" d expand to a value d' > d but not see L undergo a proportionate expansion to an L' > L? I think that would be an essential point to address so others couldn't claim that you have, in effect, proved what you assumed.

We did assume it. We assumed the string was stiff. If we had instead assumed it was elastic then it could have stretched, but then we would have been working a different problem, one requiring a relativistic version of Hookes law and the elasticity of the string material.

By selecting the frames of the co-moving observers (who always deal with a d' > d) you have avoided the problem of finding a solution for frames of observers for whom d' < d (and, as you know, there are many such frames.)

Yes, that is why I chose the MCIF.

 

[Eli Botkin]

DaleSpam:
Saying "Yes, that is why I chose the MCIF" sounds like an admission that you knew you couldn't get that SR conclusion in the other frames.

Calling the string "stiff" is just another way of saying "all L' = L, regardless of the frame.." I think that SR should decide that. You should be aware that SR tells us that even "proper lengths" of so-called stiff objects can be altered for a particular inertial observer if the object undergoes a history of acceleration. If this is unfamiliar territory, let me know so I would then describe how and why. But that would have to wait until next weekend since I will be sans computer until then.

 

[Dale]

Saying "Yes, that is why I chose the MCIF" sounds like an admission that you knew you couldn't get that SR conclusion in the other frames.

Wow! You sound paranoid.

There are an infinite number of equally valid ways of doing a problem, and we agree that we can choose to do it the easy way. There is also an obvious difficulty with some of those ways. So I pick a way which avoids the obvious difficulty precisely because it avoids the obvious difficulty (which you agree is valid to do).

To me the obvious conclusion is that I am lazy and don't want to do things the hard way, but what comes to your mind is instead that I am trying to hide the fact that it can't be done the other way. Sounds like you think I am some sinister agent of a cover up.

The problem can be worked in an infinite number of frames, and because of the principle of relativity we know that the answer must be the same. I welcome you to do it the hard way if that interests you, but I am lazy and will stick with the easy way. If you get stuck (as often happens when doing things the hard way) then post your work, and I have already offered to help get you unstuck.

 

[Eli Botkin]

QDaleSpam:
Since no paranoia has been detected in my family it's not likely that I carry that trait, nor do I think that you are lazy :-)

I think you should also address the issues in my 2nd paragraph, they are important for understanding what SR is saying.

 

[Dale]

Calling the string "stiff" is just another way of saying "all L' = L, regardless of the frame.."

No, stiff means that the proper length is always L, regardless of the forces on the string. In frames other than the MCIF it is certainly possible that L≠L'.

 

[Eli Botkin]

DaleSpam:
Returned from vacation, willing to continue discussion of our differences.

You say “… stiff means that the proper length is always L, regardless of the forces on the string.” I would modify that to: stiff means that the length is an unchanging value (regardless of forces applied) in any selected frame. In a different frame it will still be unchanging, but at a different value. SR’s transformation equations require that.

As for the term “proper length,” care must be taken not to think of it as a synonym for “true length.” “Proper length” is only shorthand for “the length measured in a frame wherein the body is at rest.” In SR that’s just another frame like any other.

So, if a log (or string) has “proper length” L in frame A, then, after acceleration to a velocity V, it will have a “proper length” >L in a frame B that has a velocity V relative to frame A.

Therefore it cannot be correct to assume that the stiff string maintains the same length L (or proper length L) throughout its acceleration interval.

 

[Dale]

I would modify that to: stiff means that the length is an unchanging value (regardless of forces applied) in any selected frame.

This is only true if the object and the frame are both inertial, which isn't the case here.

So, if a log (or string) has “proper length” L in frame A

Proper length is frame invariant. If it has proper length L then it has proper length L in frame A, B, C, D, ... Proper length only equals coordinate length in the rest frame.

 

[Eli Botkin]

DaleSpam:
You say "This is only true if the object and the frame are both inertial,..."

I presume that you would define a non-inertial object as an object under acceleration. But it is reasonable to consider such an object to be "jumping" from one instantaneously co-moving inertial frame to the next such frame. That's what acceleration is, its a transfer from one inertial frame to another. SR deals with coordinate transformations between inertial frames.

I don't know what you mean by "Proper length is frame invariant. If it has proper length L then it has proper length L in frame A, B, C, D, ... " Please tell me what SR says about proper length when it isn't related to the object's rest frame.

 

[yuiop]

EDIT: Oops, I had only read the first page of this thread when I posted this so completely missed 4 pages of the discussion. Hope it is still relevant.

This also makes sense-- but why doesn't the distance between the ships shrink along with the string? Shouldn't it all be length contracted?

It does and it doesn't :tongue: ...bear with me and I will try and explain. Imagine we have two spaceships parked on the ground that 1 kilometre apart. They are joined by a 1km string. This string is designed to snap when stretched to twice its rest length. In the ground frame both rockets take off simultaneously and accelerate equally, such that they maintain a spatial separation of 1 km at all times (as measured in the ground frame). When the rockets reach a velocity V, which corresponds to a gamma factor of 2, the length of the string in the ground frame should be 1/2km due to length contraction and is stretched over a space of 1km so it snaps. In the ground frame, there is no length contraction of the space between the rockets. Length contraction requires we have a velocity and we cannot assign a velocity to the space (vacuum) between the rockets.

Now let us have a look from the point of view of the rockets. Initially they see the separation as 1 km. As they accelerate the space between them appears to increase and as they arrive at the critical velocity V they measure the space between themselves as 2km. The string is approximately at rest in there reference frame because it is co-moving with the rockets so it should have a length of 1km but it is stretched over a distance of 2km so it also snaps from their point of view. (note that I am using a loose definition of rest frame for the rocket observers, because from their point of view they are are not exactly at rest with respect to each other, but it is a reasonable approximation for our purposes).

Note that according tot he rocket observers the distance between the rockets is 2km and according to the ground observers the distance is 1 km, so there is a sort of "length contraction" of the space between the rockets because different observers disagree on the length, but at no time does the spatial separation between the rockets contract according to any observer. In fact the distance expands according to the rocket observers and remains constant according to the ground observers.

The string on the other hand can be assigned a velocity relative to the ground frame and so it really does physically contract according to the ground observers.

In summary, according to the ground based observers the string contracts, but the space does not and according to the rocket based observers the space expands (because they consider themselves to be moving apart from each other) and the theoretical length of the string remains constant. The space between the rockets according to the ground based observers is 1/2 the distance measured by the rocket based observers, so the Lorentz transformation of space is still satisfied, even though there is no actual "contraction" of the separation space according to any observer.

You cannot accelerate space (vacuum) to make it contract, but your perception of the space between two markers can vary with your velocity relative to the markers.

If we do a variation of the paradox, whereby the rocket captains are instructed to maintain a constant distance of 1km between their rockets (by their own measurements) as they accelerate, then at the critical velocity V, the ground based observers will measure the space between the rockets to be 1/2km and so any string between the rockets will not snap, because this time the string and the separation distance, length contract at the same rate according to the ground based observers, while the rocket based observers say the string length and separation space remain constant so they also agree that the string does not snap.

 

[A.T.]

I don't know what you mean by "Proper length is frame invariant. If it has proper length L then it has proper length L in frame A, B, C, D, ... " Please tell me what SR says about proper length when it isn't related to the object's rest frame.

What do you mean by "proper length when it isn't related to the object's rest frame". Proper length is per definition the length measured in the the object's rest frame. From any other frame, the proper length of a moving object is measured by a co-moving ruler.

 

[Dale]

SR deals with coordinate transformations between inertial frames.

See http://math.ucr.edu/home/baez/physics/Relativity/SR/acceleration.html

I don't know what you mean by "Proper length is frame invariant. If it has proper length L then it has proper length L in frame A, B, C, D, ... " Please tell me what SR says about proper length when it isn't related to the object's rest frame.

The proper length is
$$\int_P \sqrt{g_{\mu \nu} dx^{\mu} dx^{\nu}}$$
where P is the space like path consisting of the intersection of the objects worldsheet with a hyperplane orthogonal to the tangent vector.

Eli, you seem to be grasping for straws now. As I said, the fact that the string breaks can be proven with SR. With this current line of questioning you are straining at very minor details that are already well established in the SR literature. How to handle acceleration and the definition of proper length are well known. The fact that you don't know about them is not a flaw in the proof I gave.

I am willing to continue the conversation in the context of improving your education, but not in the context of defending the proof above. It is a valid proof that the string breaks. Is that acceptable to you?

 

[Eli Botkin]

DaleSpam:
Thanks for your reply. I am always willing to continue my education, as we all should.

It seems to me that in your replies there has been an avoidance of a certain SR question, which is not a minor detail.

Why would an inertial observer, who is traveling in a direction opposed to the ships', predict that the string will break? In his frame the ships approach each other. Your own prediction of breakage is based on frames wherein the ships increase their separation while (you say) the string retains its "proper" length (reply #43). After all, a principal SR teaching is that the physics should be coordinate-free.

I would truly appreciate your view of this issue.

 

[Dale]

Do you accept the proof as valid?

 

[Eli Botkin]

What I accept is that under all your assumptions your mathematical deduction is correct.

You've correctly shown that for frames in which the ships are momentarily at rest, successive frames show an increasing ship separation. One of your assumptions is that the string length, L, stays constant from such frame to frame. Therefore you can correctly conclude that the string will break.

Now to my reply #82, would you be willing to address that? Thanks.

 

[Dale]

What I accept is that under all your assumptions your mathematical deduction is correct.

Fair enough.

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

 

[Dale]

Why would an inertial observer, who is traveling in a direction opposed to the ships', predict that the string will break? In his frame the ships approach each other.

The string length contracts more than the distance between the ships decreases. I would encourage you to work this out for yourself.

 

[Eli Botkin]

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 ;-)

 

[Dale]

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

I did, you apparently missed my update.

That is precisely why working things out for yourself is so important and why I encourage others to do so also. Did you think that I don't follow my own advice?

 

[Eli Botkin]

DaleSpan:
Your reply#88 puzzles me. What update did I miss? My reply #87 was an answer to your #86.

Are you still maintaining that there are no inertial frames wherein the ships' separation decreases? Can you show that?

 

[Dale]

Re-read 86.

 

[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.