Comparing Ropes Released from Ships A and B in Relative Motion

[hprog]

Suppose we have two ships A and B in uniform motion according to each other(A claims B to in motion and B claims A to be in motion).
Let each one of the objects have a roll of rope on it (let us call the ropes a-> for A's rope and b-> for B's rope).
Now when the ships were initially close to each other each of them throwed the end of his rope to the other ship where it was connected to a pole.
As such A has it's own roll of rope a-> on his ship (but the end of the rope a-> is tied to a pole on B and the farther they travel the more of the rope is being released into the space between them), and also the end of B's rope b-> (but not the roll itself) which is connected to a pole on A , and the opposite is true for B.
This naturally results in more and more of the rope being released into the space between the objects, and the farther they travel the longer the section of rope that has been rolled off from the roll becomes.
Now suppose that both A and B agreed that after a time t after they passed each other both of them will cut off the rope that has already been released off the roll, and we will then compare the the length of the cutted off section of both ropes (to avoid problems we might want to compare it in the same frame).

Now, since
1) The time of the object in motion (let us denote the moving object by Y and the resting object by X) is being dilated, which means that t' will happen later, which means that more rope will have been released for the Y's roll, which means that the Y's cutted rope will be longer than the one of X.
2) Since Y is the one in motion his roll is released as soon as he moves, while X's roll takes some time to be released since the tension of the rope has to travel the length of the entire released section before the tension arrives at the roll and only then more rope of X's roll is being released, and this delay is becoming greater the more the distance between A and B is becoming bigger.
As such it should be clear that the rope that has been cut off the moving object Y's will be larger than the rope cut off the resting object X's roll, and the difference is becoming bigger the higher the velocity involved and the longer the time t is.

But according to relativity each of the objects A and B can claim itself to be at rest while the other is claimed to be in motion, which means that each of them claims the rope that has been cut of its own roll to be smaller than the roll cut of the other object.
But of course only one of them can be true, so how does this fit with relativity?

 

[Dale]

Please derive the length of rope and the time of cutting in each frame mathematically. Also, you will need to specify the tension in the rope and the elastic modulus of the rope and the speed of sound in the rope. If you do that you will find there is no contradiction.

 

[Mueiz]

I think this is only a complicated version of the so called twin paradox .
The mistake in all forms of the twin paradox is in the last statement:

But of course only one of them can be true, so how does this fit with relativity?

Yes only one of them can be true but only in one frame!
Relativity never says that the true observation of two different frames shoud not contradict each other.
According to relativity two observers can contradict each other in the answer to question such as ..Which event happen first?Which distance is shorter? Which is older? and which rope is smaller?
What is common in all references is the laws of physics rather than the details of events

 

[Dale]

Good point Mueiz. Also, the symmetry of the situation does not require that each ship see the other as they see themselves, but only that each ship see the other as the other sees them.

 

[hprog]

It looks as you have misunderstood my question.
My question is what will be if after cutting the two ropes we then bring the ropes together in the same frame and compare them there, and in this case there might be no disagreement on which one is shorter.

(Well one might ask why should this be different than any length contraction problem in which bringing the object in the same frame will cause all objects to be no longer contracted, the answer is that all length contraction in relativity is due to the objects being in different frames and is a result of the relativity of simultaneity and as such it of course will no longer be if the objects will finally be in the same frame, however here the reason of one rope being shorter is because of time dilation which caused the rope to be cut in a later time or because of the time it takes the tension to travel and this should not be changed even if we bring the ropes together in the same frame.)

If you want to compare it to the twin paradox you can do so with a twin paradox that the twins actually meet after the trip, but unlike the typical answer to the twin paradox in which one of the twins had to accelerate this cannot be claimed here.

 

[PAllen]

I don't think this is like the twin paradox. There really is perfect symmetry between the two observers and their operations (if you set it up right). If you set it up symmetrically, you would end up with same length rope pieces when compared in the a common frame, after all of both rope pieces have come to rest.

To resolve the *apparent* contridiction, you need to analyze the situation in detail, as Dalespam noted in his first post. There are many details you have to resolve analyze this. Note that for each observer, rope end tied to a pole is stationary, while the rope being unrolled is moving rapidly. Thus length contraction affects one and not the other. Also, when they initially pass, tossing rope to each other, the end each one is grabbing and tieing it is creating great acceleration and stress on that rope, changing it from slow relative to one observer to slow relative to the other. Finally, each will perceive the events of cutting the rope as occurring in a different sequence. That is, if, by prior agreement, each cuts the rope after a minute on their clock, A thinks B cut his rope later, and B thinks A cut his rope later.

I can assure you that if you specify all the details plausibly, you will find that each sees a symmetric version of events (what A thinks happened with B's rope will be what B thinks happened with A's rope, etc.), and that the rope pieces will be the same length *when brought to common rest frame*, noting that each sees the ropes in opposite states of motion (until after cutting and bringing them to rest).

I am not willing to perform this rube-goldberg computation for you, but I do assure you there would be no contradiction if you carried it out correctly.

 

[PAllen]

Here is a qualitative description of one possible interpretation of this scenario.

A and B exchange and tie rope ends at mutual t=0 (ignore rapid acceleration of the rope ends and stress). The agree to cut their their rope from their roller at t of 1 minute, as each measures it.

At 1 minute for A, his rope is cut from roller. B continues to move rapidly away for some time, continuing to unroll rope (that is stationary with respect to A), while pulling the rapidly moving rope cut by A. Then B cuts his roller rope. A sees a long piece of stationary rope tied to his pole and a shorter piece comoving with B, tied to B's pole. B releases his pole rope, A 'grabs' it and slows it down. It now gets much longer, matching A's pole rope.

B, at one minute cuts his roller rope. A continues to move rapidly away for some time continuing to unroll rope (that is stationary with respect to B), while pulling the rapidly moving rope cut by B. Then A cuts his roller rope. B sees a long piece of stationary rope tied to his pole and a shorter piece comoving with A, tied to A's pole. [End of perfect symmetry]. B releases his pole rope, A grabs it and accelerates it to match A's pole rope. To B, it shrinks, to match the rapidly moving rope attached to A's pole.

Now A releases both ropes, B grabs and slows them down, they get longer and again match in length.

Perfect symmetry where expected, no contradiction, relativity is fine.

 

[PAllen]

But according to relativity each of the objects A and B can claim itself to be at rest while the other is claimed to be in motion, which means that each of them claims the rope that has been cut of its own roll to be smaller than the roll cut of the other object.
But of course only one of them can be true, so how does this fit with relativity?

This is the biggest of several misunderstandings. They can both be true, as I explain above.

 

[Dale]

I agree with PAllen, the situation appears to be completely symmetric (by design), so you immediately know that the lengths are the same by symmetry. However, if you feel suspicious of that answer then I would encourage you to work it out completely.

 

[hprog]

To resolve the *apparent* contridiction, you need to analyze the situation in detail, as Dalespam noted in his first post. There are many details you have to resolve analyze this. Note that for each observer, rope end tied to a pole is stationary, while the rope being unrolled is moving rapidly. Thus length contraction affects one and not the other. Also, when they initially pass, tossing rope to each other, the end each one is grabbing and tieing it is creating great acceleration and stress on that rope, changing it from slow relative to one observer to slow relative to the other. Finally, each will perceive the events of cutting the rope as occurring in a different sequence. That is, if, by prior agreement, each cuts the rope after a minute on their clock, A thinks B cut his rope later, and B thinks A cut his rope later.

Your solution is based on length contraction which is not applicable here for two reasons.

1) Since the length of the rope at any instant in time before cutting the ropes can clearly not be smaller than the space between the objects (think about that), and it can also not be larger than the space (since in this case the rope would curve down as a parabola and if this would be than both frames would agree on that) so if we have two ropes equally tied to both objects we must agree that both ropes are the same size (think about it).
(The mistake might be a result of the design of the experiment in which the rolls are on different ships, however this should not have any effect on length contraction.
But to show this clearer we might change the design of the experiment to having ropes that both ends are rolls, so each ship will have a roll for each rope.
But even if there would be length contraction we might even use this to make the paradox greater by designing the experiment that we do not know initially which rope to cut, but after analyzing the rope each of them cuts the rope that has not been contracted in his frame.)

2) Since after the ropes are brought together in the same frame the length contraction factor is rolled back while the time dilation is not.
To show that is easy, consider a clock that has been time dilated that we bring to rest, while the clock while no longer dilate, still the dilation that has been done is not rolled back and the clock will be set behind the other clocks in the rest frame.
However when you have a train of length L and you set it into motion and it is getting contracted then after you bring it back into rest it will be back into length L (this is due to the fact that length contraction is a result of the relativity of simultaneity).

So after all since length contraction and relativity of simultaneity does not apply here and the only factor here is time dilation, and since both frames disagree on which frame is time dilated, we clearly have a paradox.

 

[hprog]

I agree with PAllen, the situation appears to be completely symmetric (by design), so you immediately know that the lengths are the same by symmetry. However, if you feel suspicious of that answer then I would encourage you to work it out completely.

As I showed in the post above, the situation is symmetric but the result will not be ropes of equal length.

 

[yuiop]

As I showed in the post above, the situation is symmetric but the result will not be ropes of equal length.

They will be the same length. Look at it from the point of view of an observer that sees A and B going in opposite directions but at the same speed. The spooled portion of both ropes will also be going are equal speeds but in opposite directions so they will be length contracted by the same amount. When both ropes are brought to rest in the same rest frame they will have the same length. As others have said, the symmetry of the situation ensures the cut off portion of each rope will be the same when they are rest wrt each other.

 

[PAllen]

Your solution is based on length contraction which is not applicable here for two reasons.

1) Since the length of the rope at any instant in time before cutting the ropes can clearly not be smaller than the space between the objects (think about that),

It is true that the ropes are the same length according to each before either rope is cut. What you do not seem to see is that the ropes will not be cut simultaneously, and each sees the other cut their rope later. Please read my explanation carefully. I guarantee it is correct for the facts as you outlined them.

and it can also not be larger than the space (since in this case the rope would curve down as a parabola and if this would be than both frames would agree on that) so if we have two ropes equally tied to both objects we must agree that both ropes are the same size (think about it).

My explanation never had any rope longer than the space between them.

(The mistake might be a result of the design of the experiment in which the rolls are on different ships, however this should not have any effect on length contraction.

On the contrary, it is critical. It determines which rope is rapidly moving relative to which observer. If you want to propose another scenario, please describe it clearly (what you write below is too incomplete to make any sense out of). Include when clocks are set to match, what actions are done by each observer at what time on which clock, etc. Then try to work it out yourself first. If you don't know how, before proposing a paradox, you should actually study a good presentation of special relativity.

But to show this clearer we might change the design of the experiment to having ropes that both ends are rolls, so each ship will have a roll for each rope.
But even if there would be length contraction we might even use this to make the paradox greater by designing the experiment that we do not know initially which rope to cut, but after analyzing the rope each of them cuts the rope that has not been contracted in his frame.)

2) Since after the ropes are brought together in the same frame the length contraction factor is rolled back while the time dilation is not.

Again this shows elementary mis-understanding. As long as the rockets continue to move apart, the time dilation is symmetric - each sees the other's clock run slow. If each rocket decelerates symmetrically relative to the other, so they end up stationary relative to each other, their clocks will end up identical. There is no difference in this regard between length contraction and time dilation. Differential aging for the 'twin situation' results from objective differences in the space-time path between leave and meet points.

To show that is easy, consider a clock that has been time dilated that we bring to rest, while the clock while no longer dilate, still the dilation that has been done is not rolled back and the clock will be set behind the other clocks in the rest frame.
However when you have a train of length L and you set it into motion and it is getting contracted then after you bring it back into rest it will be back into length L (this is due to the fact that length contraction is a result of the relativity of simultaneity).

So after all since length contraction and relativity of simultaneity does not apply here and the only factor here is time dilation, and since both frames disagree on which frame is time dilated, we clearly have a paradox.

Advice: study a reputable book covering relativity. Think carefully about my explanation - it is correct.

 

[Dale]

As I showed in the post above, the situation is symmetric but the result will not be ropes of equal length.

As you showed where? I didn't see a single attempt anywhere by you to derive the length. A handwaving verbal summary is not the same thing as actually working it out which requires math.

If the situation is symmetric then it is mathematically impossible to get an asymmetric result. The symmetric situation defines symmetric boundary conditions and the differential equations are also symmetric, so the solution is guaranteed to be symmetric.

You must have made a mistake in your analysis (mostly by using handwaving instead of math).