Mentor

 Quote by Eli Botkin 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.
 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 ;-)

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 Quote by Eli Botkin 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?
 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?

Mentor
 Quote by Eli Botkin 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.

 Quote by Eli Botkin 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.

 Quote by Eli Botkin 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.

 Quote by Eli Botkin 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.

 Quote by Eli Botkin 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?

 Quote by Eli Botkin 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.
 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 any one 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.

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 Quote by Eli Botkin 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 any one 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.

 Quote by Eli Botkin 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.

 Quote by Eli Botkin 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?

 Quote by Eli Botkin 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.
 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 ;-)
 Mentor 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.

 Quote by 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 ;-)
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?
 DaleSpam: Remember the Pole/Barn Paradox? Did you ponder whether or not the pole was stiff or elastic before applying the SR transformations ?
 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.

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 Quote by Eli Botkin 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?
 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.
 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.

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