Clocks Within Each Ship in Bell Spaceship Paradox

[harrylin]

Every reference frame is equally valid, but our intuitions about questions such as "will the string break" is formed based on nonrelativistic physics. From nonrelativistic physics, we would expect, if the front rocket takes off before the rear rocket, that that would put stress on the string connecting them. But the point of frame S_1 is that the rockets are moving nonrelativistically in that frame, so we would expect that nonrelativistic physics to give a good approximation to the answer.

Perhaps I was not clear enough. I made the OP aware of inertial reference frames going in the opposite direction, which is the group of frames that you did not consider. In that group of reference systems, RoS works against breaking the string as the front rocket takes off after the rear rocket.

 

[stevendaryl]

It is clear that in the ladder/barn paradox, contraction is not a "physical effect" and very much a frame dependent facet which involves RoS.

Well, it's a little ambiguous what it means for something to be "physical". Certainly an outcome that can be verified using any frame is physically meaningful. But the reasoning used to explain/predict the outcome is often frame-dependent. Different frames would explain the same result in different ways. But being a little more loose, you could consider frame-dependent (or more generally, coordinate-dependent) effects to be "physical" if they can reliably be used in predicting physical effects.

The difficulty in reasoning about thought experiments such as Bell's spaceships, or the barn and pole, or the twin paradox, or whatever is that we don't actually have any Lorentz-invariant equations of motion to apply. If we did, there would be no ambiguity: Just pick a frame, and apply the equations of motion. If the equations are Lorentz-invariant, then you get the same result, no matter what frame you pick. But if someone just tells you "I have a pole moving at close to the speed of light" or "I have a clock moving at close to the speed of light" or "I have a string connecting two rockets moving at a significant fraction of the speed of light", you don't have equations of motion to derive the result. Instead, you have to use rules of thumb and intuition about how poles, clocks, strings work. The rules of thumb that we are most comfortable with are Newtonian physics, which are only applicable when things are moving slowly relative to the speed of light. So you can try to analyze things locally, in a frame where things locally are moving non-relativistically, and hopefully piece together the local pictures into a global picture.

But Bell's argument with his spaceship paradox is that we can augment purely Newtonian reasoning with intuitions specially developed for SR. He claims that we should start to think of Lorentz contraction as a physical thing, to get an intuition about things moving relativistically. The rule of thumb is: If you take an extended object, and accelerate it to relativistic speed, then it will tend to contract. To prevent it from contracting requires applying stresses on the object. Whether you call such reasoning "physical" or not is a matter of definitions, but Bell's point was that, at least in many circumstances, such reasoning gives you a quick intuition about what the right answer is.

I don't think that the ladder/pole paradox contradicts Bell's contractionistic reasoning. From the point of view of the barn frame, the pole is contracted, and it is possible to close both doors simultaneously. And if the barn doors are really strong, compared to the pole, it IS possible to get the entire pole into the barn at once. Of course, using "contractionistic" reasoning, you would conclude that AFTER the barn doors are closed, the pole would expand to its normal length, and would get smashed to pieces by the strong barn doors.

You get the same conclusion from the point of view of the pole. From that point of view, it's the barn that is contracted, not the pole. But it's STILL possible to fit the pole into the barn: You smash the barn into the pole at relativistic speed, and the pole will be crushed to a size that fits inside the barn.

 

[stevendaryl]

Perhaps I was not clear enough. I made the OP aware of inertial reference frames going in the opposite direction, which is the group of frames that you did not consider. In that group of reference systems, RoS works against breaking the string the front rocket takes off after the rear rocket.

I understood your point, but I'm saying that that point of view doesn't really lead to the opposite conclusion, if you think through the details. Yes, in that frame, the rear rocket takes off before the front rocket, but the front rocket takes off before the information that the back rocket has taken off can propagate to the front. If the back rocket takes off at time t_1, and the distance between the rockets is L, then it will take at least until t_1 + L/c before the loosening of the string can propagate all the way to the front of the rocket. If the front rocket takes off at time t_2 < t_1 + L/c, then the stress on the string near the front rocket will be the same as if the front rocket took off first.

 

[harrylin]

[..] For an observer moving in the negative X direction, the back ship starts moving first, its clock set later than the clock on the front ship. In fact, the back of the back ship starts moving before the front of the back ship.

Exactly

For this viewer, the whole thing starts out contracted and out of synch, and so the intuition that the platform observer might have regarding the rope - that physics should work as expected in one's rest frame - is already subverted.

Physics must also work for that viewer. The point of Bell's exercise - as well of the point of this extension of it - is to enhance intuition of relativistic physics. No matter which frame you chose, length contraction in the sense of Bell (meaning no.2 in post #25 ) plays a role.

It is clear that in the ladder/barn paradox, contraction is not a "physical effect" and very much a frame dependent facet which involves RoS.

That is exactly the confusion that I addressed - here you talk about meaning no.1.

 

[harrylin]

I understood your point, but I'm saying that that point of view doesn't really lead to the opposite conclusion, if you think through the details. Yes, in that frame, the rear rocket takes off before the front rocket, but the front rocket takes off before the information that the back rocket has taken off can propagate to the front. If the back rocket takes off at time t_1, and the distance between the rockets is L, then it will take at least until t_1 + L/c before the loosening of the string can propagate all the way to the front of the rocket. If the front rocket takes off at time t_2 < t_1 + L/c, then the stress on the string near the front rocket will be the same as if the front rocket took off first.

I gave that additional example in order to prevent that the OP concludes that somehow RoS and not length contraction "explains" the breaking of the string. Instead, it's the departure times of the rockets as well as length contraction that must be taken in account, no matter what inertial reference system one chooses.

 

[harrylin]

[...] The rule of thumb is: If you take an extended object, and accelerate it to relativistic speed, then it will tend to contract. [..].

I wonder why you say "rule of thumb".
Length contraction as the result of motion is a physical prediction of SR (e.g. §4 of http://fourmilab.ch/etexts/einstein/specrel/www/ ). Next, in order to be sure what we can predict based on classical material science, we need to do a Lorentz transformation to the frame in which the string is in rest (in fact, you nicely clarified that point). And based on the calculated prediction (not a rule of thumb) we can develop correct intuition

 

[1977ub]

Well, it's a little ambiguous what it means for something to be "physical". Certainly an outcome that can be verified using any frame is physically meaningful. But the reasoning used to explain/predict the outcome is often frame-dependent. Different frames would explain the same result in different ways. But being a little more loose, you could consider frame-dependent (or more generally, coordinate-dependent) effects to be "physical" if they can reliably be used in predicting physical effects.

The difficulty in reasoning about thought experiments such as Bell's spaceships, or the barn and pole, or the twin paradox, or whatever is that we don't actually have any Lorentz-invariant equations of motion to apply. If we did, there would be no ambiguity: Just pick a frame, and apply the equations of motion. If the equations are Lorentz-invariant, then you get the same result, no matter what frame you pick. But if someone just tells you "I have a pole moving at close to the speed of light" or "I have a clock moving at close to the speed of light" or "I have a string connecting two rockets moving at a significant fraction of the speed of light", you don't have equations of motion to derive the result. Instead, you have to use rules of thumb and intuition about how poles, clocks, strings work. The rules of thumb that we are most comfortable with are Newtonian physics, which are only applicable when things are moving slowly relative to the speed of light. So you can try to analyze things locally, in a frame where things locally are moving non-relativistically, and hopefully piece together the local pictures into a global picture.

But Bell's argument with his spaceship paradox is that we can augment purely Newtonian reasoning with intuitions specially developed for SR. He claims that we should start to think of Lorentz contraction as a physical thing, to get an intuition about things moving relativistically. The rule of thumb is: If you take an extended object, and accelerate it to relativistic speed, then it will tend to contract. To prevent it from contracting requires applying stresses on the object. Whether you call such reasoning "physical" or not is a matter of definitions, but Bell's point was that, at least in many circumstances, such reasoning gives you a quick intuition about what the right answer is.

I don't think that the ladder/pole paradox contradicts Bell's contractionistic reasoning. From the point of view of the barn frame, the pole is contracted, and it is possible to close both doors simultaneously. And if the barn doors are really strong, compared to the pole, it IS possible to get the entire pole into the barn at once. Of course, using "contractionistic" reasoning, you would conclude that AFTER the barn doors are closed, the pole would expand to its normal length, and would get smashed to pieces by the strong barn doors.

You get the same conclusion from the point of view of the pole. From that point of view, it's the barn that is contracted, not the pole. But it's STILL possible to fit the pole into the barn: You smash the barn into the pole at relativistic speed, and the pole will be crushed to a size that fits inside the barn.

Basically with barn/pole, until the collision, there is no physical contact between objects in different frames. Either frame can be "correct" about the other's contraction. Bell's Spaceship is *entirely* about ongoing interaction of items in different frames. So that's a big difference. This also comes into play if we try to close the barn doors on the pole. From the barn frame, the doors come down simultaneously, then the front of the pole hits the door, then what happens thereafter is some combination of 2 effects... one effect (which could predominate) is that the ladder comes apart as it smashes into the 2nd door. In this version, observer moving with the pole never sees any paradox, just a plain old collision - and no "crushing" takes place. It would have done so basically the same way even if the 1st barn door never closed. The 2nd effect - if the pole is to strong enough to stay together - involves rapid deceleration - in which the frame of the ladder is caused to become identical with the barn frame - with deceleration taking place along the ladder at < c in all frames. I have to go back and read more about that collision.

 

[Mentz114]

Basically with barn/pole, until the collision, there is no physical contact between objects in different frames. Either frame can be "correct" about the other's contraction. Bell's Spaceship is *entirely* about ongoing interaction of items in different frames. So that's a big difference. This also comes into play if we try to close the barn doors on the pole. From the barn frame, the doors come down simultaneously, then the front of the pole hits the door, then what happens thereafter is some combination of 2 effects... one effect (which could predominate) is that the ladder comes apart as it smashes into the 2nd door. In this version, observer moving with the pole never sees any paradox, just a plain old collision - and no "crushing" takes place. It would have done so basically the same way even if the 1st barn door never closed. The 2nd effect - if the pole is to strong enough to stay together - involves rapid deceleration - in which the frame of the ladder is caused to become identical with the barn frame - with deceleration taking place along the ladder at < c in all frames. I have to go back and read more about that collision.

It seems to me that you are wrong in comparing the barn-pole paradox with Bells spaceships.

In the latter we have proper acceleration which is a frame-independent thing. The string breaks because in the rocket frame the ships are moving apart and the string stretches ( actually stretches) and thus breaks eventually. It is possible to find coordinates in which the ships appear not to move apart but that does not mean they are not, and physical reality is what happens locally.

In the barn-pole scenerio we assume two inertial frames and resolve the 'paradox' with a space-time diagram. What happens when the pole decelerates is irrelevant.

 

[1977ub]

Ok so this helps:
"At this point [soon after the ladder front colliding with the 2nd door] the ladder is actually shorter than the original contracted length, so the back end is well inside the garage. Calculations in both frames of reference will show this to be the case."
https://en.wikipedia.org/wiki/Ladder_paradox#Ladder_paradox_and_transmission_of_force

It helps to think of the ladder as two boulders with a spring between them. There will be a compression in all frames.

Now if Bell's "rope" is a 3 boulders with springs between, then once the two spaceships begin their acceleration, there should be some initial stress as the center boulder remains briefly in place while the end boulders start accelerating with the rockets (all in platform frame). Since the center boulder never has a direct acceleration from a rocket engine, I imagine it is always somewhat behind from the midpoint of the other two boulders.

 

[Mentz114]

Ok so this helps:
"At this point [soon after the ladder front colliding with the 2nd door] the ladder is actually shorter than the original contracted length, so the back end is well inside the garage. Calculations in both frames of reference will show this to be the case."
https://en.wikipedia.org/wiki/Ladder_paradox#Ladder_paradox_and_transmission_of_force

It helps to think of the ladder as two boulders with a spring between them. There will be a compression in all frames.

Now if Bell's "rope" is a 3 boulders with springs between, then once the two spaceships begin their acceleration, there should be some initial stress as the center boulder remains briefly in place while the end boulders start accelerating with the rockets (all in platform frame). Since the center boulder never has a direct acceleration from a rocket engine, I imagine it is always somewhat behind from the midpoint of the other two boulders.

I don't need help, thank you. I think most of what you've quoted is nonsense.

Well, I tried to help but clearly you have to find your own way with this.

 

[stevendaryl]

I wonder why you say "rule of thumb".

Because it's an assumption about the nature of the forces making up the object. For example, if you have a lump of clay, and accelerate it by pushing on one end, it will be compressed. If you accelerate it by pulling on the other end, it will stretch. So the length is not a predictable function of velocity.

 

[harrylin]

[...] Now if Bell's "rope" is a 3 boulders with springs between, then once the two spaceships begin their acceleration, there should be some initial stress as the center boulder remains briefly in place while the end boulders start accelerating with the rockets (all in platform frame). Since the center boulder never has a direct acceleration from a rocket engine, I imagine it is always somewhat behind from the midpoint of the other two boulders.

That's quite right.

In the standard Spaceship example that effect is neglected (it may be neglected if the rockets don't accelerate too fast). And assuming constant proper acceleration of the rockets then after take-off this situation will be stable.That detail does not affect the discussion in principle.

 

[harrylin]

Because it's an assumption about the nature of the forces making up the object. For example, if you have a lump of clay, and accelerate it by pushing on one end, it will be compressed. If you accelerate it by pulling on the other end, it will stretch. So the length is not a predictable function of velocity.

Of course it's predictable by means of known classical materials science just as you explained yourself. The forces on the string and the amount of dynamic stretching are only neglected in this example because they are not essential for the principle and enormously complicate the calculation.

 

[stevendaryl]

Of course it's predictable by means of known classical materials science just as you explained yourself. The forces on the string and the amount of dynamic stretching are only neglected in this example because they are not essential for the principle and enormously complicate the calculation.

You asked why I used the phrase "rule of thumb". It's a shorthand way of reasoning that doesn't require getting into complicated equations of motion, material properties, etc.

 

[harrylin]

You asked why I used the phrase "rule of thumb". It's a shorthand way of reasoning that doesn't require getting into complicated equations of motion, material properties, etc.

Ah yes, on that I agree.

 

[1977ub]

That's quite right.

In the standard Spaceship example that effect is neglected (it may be neglected if the rockets don't accelerate too fast). And assuming constant proper acceleration of the rockets then after take-off this situation will be stable.That detail does not affect the discussion in principle.

It depends what you mean by stabilize... It should never precisely stabilize, if the rockets truly follow identical programs, and there is no friction in space. Unless you mean a "stable" harmonic oscillation I suppose (in which I make it out that the midpoint of the oscillation will be somewhat to the rear of the midpoint of the "rope").

re "in Principle" : It depends upon the principle. If we accept going in that we are only asking about the integration of proper length of the rope vs the measured rope in S, and that's all we care about, using it as a proxy for "will the rope break," then I agree, that detail is not important.

I am accustomed to seeing the issue described as "will the rope break" if it is very fragile e.g. "silk" ?

If we look at that literally, then - whether or not we apply SR - the rope will break even if there is only the front ship! Acceleration pulling from the front causes a tension in the rope. Will the fact that there is a ship at the rear, pushing on the silk strand, stop that? No.

 

[harrylin]

It depends what you mean by stabilize... It should never precisely stabilize, if the rockets truly follow identical programs, and there is no friction in space. Unless you mean a "stable" harmonic oscillation I suppose (in which I make it out that the midpoint of the oscillation will be somewhat to the rear of the midpoint of the "rope").[..]

Away from Earth at "1g" acceleration, the inertial forces on the string are similar to the gravitational forces on a string on Earth that is held up vertically between two anchor points on a wall (this may sound like GR but it's still SR as it's just the same in classical mechanics). If next you move the top anchor point very slowly upwards, the string will undergo stretching exactly as in standard pull tests.

If we look at that literally, then - whether or not we apply SR - the rope will break even if there is only the front ship! Acceleration pulling from the front causes a tension in the rope. Will the fact that there is a ship at the rear, pushing on the silk strand, stop that? No.

I'm afraid that I can't follow that at all.
In Bell's example that effect is neglected, just as gravitation can usually be neglected in pull tests.

 

[Nugatory]

Ok. I see. I'm still trying to connect this to the relativity of simultaneity somehow, since length contraction and RoS seem to always be two sides of the same coin...

if the rockets truly follow identical programs,

The text that I have bolded above is where the relativity of simultaneity is hiding in Bell's paradox. If the rockets are following identical programs, then they are changing their speed by the same amount at the same time, and relativity of simultaneity warns us that this definition is frame-dependent - two programs that are identical in one frame will not be identical in other frames.

Pole-barn paradox: in all frames the door closes and in all frames the tip of the pole strikes the far wall of the barn (or, if the rest length of the pole is sufficiently short compared with the rest length of the barn and the pole-carrier slows down quickly enough, comes to a stop before it hits the wall). Because of the relativity of simultaneity, in some frames the door will close before the tip hits, and in other frames the door will close after the tip hits. Length contraction follows from the relativity of simultaneity because we define the length of the pole as the distance between the position of its front end and its back end at the same time and we define the length of the barn as the distance between the position of the door and the far wall at the same time; both of these quantities are frame dependent because they are defined using the frame-dependent notion at the same time.

Bell's spaceship paradox: In all frames, a tension meter in the string will register progressively increasing tension in the string, and if that tension reaches the breaking point of the string, it will of course break. This has nothing to do with length contraction of the ships themselves - we could model them as point particles (and generally do, for simplicity). In a frame in which both spaceships change their speed by the same amount at the same time, so that the distance between them remains constant, their speed is necessarily changing, so we explain the increasing reading on the tension meter reading by saying the length of the string is contracting more as the speed increases; and indeed if we were to untie the string from the trailing ship we would see an increasing gap opening up between the loose end of the string and the trailing ship. However, in a frame that is comoving with the string, the two spaceships are not changing their speeds by the same amount at the same time, and the increase in tension in the string is explained by saying that the distance between the two ships at the same time is increasing.

If you want to use the physical properties of the string to calculate exactly when it will break, you will find it easiest to work in a frame in which the string is at rest and the two ships are moving apart. To do that, you'll need to know how rapidly they are moving apart, and the easiest way to do that is start with their trajectories in the frame in which their acceleration are "truly uniform" (your words, not mine) and then transform into a frame in which one or the other end of the string is at rest. That way, length contraction never enters into the problem, and you can analyze the behavior of the string using only well-understood classical materials science.

 

[harrylin]

[..] However, in a frame that is comoving with the string, the two spaceships are not changing their speeds by the same amount at the same time, and the increase in tension in the string is explained by saying that the distance between the two ships at the same time is increasing. [...]

Note that such a frame is not an inertial frame (you know that certainly very well, but the OP may not be aware of the consequences!). The string's equilibrium length appears to remain unchanged at the cost of for example pretending that the distance between the two launchpad towers decreases during the flight for no good reason (how much depends on the angle of course). The action of the rocket engines results in zero motion of the rocket and instead, the Earth and distant stars are seen to magically accelerate. In other words, the laws of physics as assumed in SR do not hold. Nevertheless, the stresses in the string can thus conveniently and accurately be determined, as we both explained.

[where is the strike-through button? I made a correction in phrasing, adding "equilibrium"]

 

[1977ub]

Note that such a frame is not an inertial frame (you know that certainly very well, but the OP may not be aware of the consequences!). The string appears to remain the same length at the cost of pretending that the distance between the two launchpad towers decreases during the flight for no good reason. The action of the rocket engines results in zero motion of the rocket and instead, the Earth and distant stars are seen to magically accelerate. In other words, the laws of physics as assumed in SR do not hold. Nevertheless, the stresses in the string can thus conveniently and accurately be determined, as we both explained.

Can SR "handle acceleration" as I recently read? Can't we take "successive frames" that a particular ship passes through and say something meaningful. (I'm not sure how such an exercise would bear on this. )

 

[harrylin]

Can SR "handle acceleration" as I recently read? Can't we take "successive frames" that a particular ship passes through and say something meaningful. (I'm not sure how such an exercise would bear on this. )

Yes you can! In fact, taking "successive frames" is exactly what stevendaryl and I did in our discussions. And it's certainly possible to give a meaningful discussion of the forces that the string "sees" over time, as discussed in detail by stevendaryl and Nugatory.

 

[1977ub]

In other words, the laws of physics as assumed in SR do not hold.

Could you clarify? Which physical laws assumed in SR do not hold and why?

 

[harrylin]

Could you clarify? Which physical laws assumed in SR do not hold and why?

I answered that question in the sentences before the one you ask about...
Or maybe you overlooked that I was commenting about the use of accelerated frames?
The main physics law for this discussion is that lengths of static objects in inertial frames do not contract without a physical cause. The law of cause and effect is maintained in SR.

 

[Nugatory]

Note that such a frame is not an inertial frame (you know that certainly very well, but the OP may not be aware of the consequences!).

Yes, that's a good point... thanks. As far as I can tell, that frame is good for little other than showing that the string breaks because that's what strings do when you pull their ends apart. The physics of the string is easy to analyze in that frame, but as soon as you look beyond the string and the two ships it ceases to be helpful.

The non-inertial frame of Einstein's elevator/acceleration/gravity equivalence principle thought-experiment has the same problem that everything else in the universe seems to magically accelerate and length contract... That's why Einstein didn't put any windows in his elevator

 

[Nugatory]

Can SR "handle acceleration" as I recently read? Can't we take "successive frames" that a particular ship passes through and say something meaningful. (I'm not sure how such an exercise would bear on this. )

SR can indeed handle acceleration - google for "Rindler coordinates".

Although there is no inertial frame in which an end of the string remains at rest, there are "momentarily comoving inertial frames" (MCIFs) in which that end of the string is momentarily at rest. These are useful if we want to calculate the apparent positions and speeds of the ships, the ground, and everything else at a particular moment in time (from the point of view of an observer in the ship at that end). They are not, however, especially useful for calculating the increasing separation of the two ships as time passes because the string is only monetarily at rest in these frames.

Note that the earthbound observer in Bell's original statement of the paradox is using a frame that is momentarily comoving with the string - at time zero and only then is the string at rest in this observer's inertial frame.

 

[harrylin]

It suddenly strikes me that perhaps here the subtle difference should be explained between "MCIF"'s and accelerating frames in SR.
As a matter of fact, this was unclear to me some time ago, and it's probably still not understood by many.

So I'll re-hash what we discussed here above, focusing at that difference.

Using co-moving inertial frames is described in detail in post #22 by stevendaryl. He shut off the engines temporarily so as not to be hindered by acceleration, but that isn't really necessary for small acceleration as I discussed in post #47 . With co-moving inertial frames nothing magical is suggested: firing the rocket engines accelerates the rockets, and not the Earth and the stars. Also nothing happens to the launch pad. In each step a small incremental Lorentz transformation is made to a consecutive inertial frame in order to call that the new rest frame.
This allows to follow during the flight how the string stresses according to static mechanics; however, we never forget (or, we should never forget) that in this approach the rockets are accelerating from rest in the launch pad frame and that nothing happens with the launch pad frame's dimensions; apparent changes to the launch pad must be due to changes in the string's co-moving measurement system, caused by its change of velocity. In the end nothing different can be concluded as when making a single Lorentz transformation from the launch pad frame to the frame in which the string is at the point of breaking. It's essentially the same, but refined by splitting it up in a great number of steps.

Using accelerating frames is described in detail in post #48 by Nugatory. Although it looks very similar to stevendaryl's analysis, in the end Nugatory considered the string as being in rest all the time - the point of view of the accelerating frame. I similarly used the equivalence between gravitational force and inertial force in post #47. The forces on a string at 1g constant acceleration are just the same as those on a vertically held string in the Earth's gravitation, and the stress increase can be perfectly simulated by an increase in the distance between the attachment points in such a rest frame, as was found before by means of Lorentz transformations. Such a mechanical equivalence (according to classical mechanics and SR) is only meant to simulate the forces that act on the string, as experienced by the string; it's perfect for calculating the moment that the string will break.
In Bell's spaceship example the string is not in rest in a gravitational field but accelerates due to the rocket engine's propulsion, and momentum is conserved. If we forget that and pretend that the string was all the time in rest, then we find that the string has the same equilibrium length all the time and the string undergoes classical stretching. Consequently, length contraction plays no role in the related physical interpretation. However, that goes at the cost of a magical change of dimensions of the launch pad and the rest of the Earth - even a magical acceleration of the Earth and the distant stars. You must not look out of the window, and even not look at your fuel gauge!

 

[Stalin Beltran]

May I make a silly question? What are the initial conditions of this paradox? If I am right, the conditions were:

1. Two identical rockets, and if programmed the program is equal
2. Both rockets start at the same time, according to the initial reference frame (named S, for example)
3. Both rockets are tied by a inelastic string, which will break if it is elongated (that is, to the minimum elongation, it breaks)

Am I missing something? Something like "both rockets are some way forced to accelerate with respect to the initial reference frame (maybe)?

 

[harrylin]

May I make a silly question? What are the initial conditions of this paradox? If I am right, the conditions were:

1. Two identical rockets, and if programmed the program is equal
2. Both rockets start at the same time, according to the initial reference frame (named S, for example)
3. Both rockets are tied by a inelastic string, which will break if it is elongated (that is, to the minimum elongation, it breaks)

Am I missing something? Something like "both rockets are some way forced to accelerate with respect to the initial reference frame (maybe)?

3. Both rockets are tied by an elastic string, which will break if the strain as measured in its momentarily co-moving frame reaches the elongation to break.
4. The string will break at a certain moment, and as interpreted in S, this is due to length contraction because the distance between the rockets remains constant in this frame. Length contraction refers in this context to a change of the stress-free length of an object as observed from a single frame.

 

[Mentz114]

To do that, you'll need to know how rapidly they are moving apart, and the easiest way to do that is start with their trajectories in the frame in which their acceleration are "truly uniform" (your words, not mine) and then transform into a frame in which one or the other end of the string is at rest. That way, length contraction never enters into the problem, and you can analyze the behavior of the string using only well-understood classical materials science.

Surely inertial frames, comoving or not don't come into the problem. As Nugatory says - transform to the local frame basis of anyone of the ships and it becomes clear that the ships ahead are getting further away and the trailing ships are falling further behind. This stretches the string.

 

[harrylin]

Surely inertial frames, comoving or not don't come into the problem. As Nugatory says - transform to the local frame basis of anyone of the ships and it becomes clear that the ships ahead are getting further away and the trailing ships are falling further behind. This stretches the string.

In which sense that is correct and in which sense not, was the subject of the elaborated discussion here above - starting with Nugatory's post #48 .