Bell's Spaceship Paradox

[can't_think_of_a_name]

I have been trying to the Bell's Spaceship Paradox but I am getting confused. Can someone simply explain the solution to the paradox according to special relativity?

 

https://en.m.wikipedia.org/wiki/Bell's_spaceship_paradox

Quote

Bell's spaceship paradox is a thought experiment in special relativity. It was designed by E. Dewan and M. Beran in 1959[1] and became more widely known when J. S. Bell included a modified version.[2] A delicate string or thread hangs between two spaceships. Both spaceships start accelerating simultaneously and equally as measured in the inertial frame S, thus having the same velocity at all times in S. Therefore, they are all subject to the same Lorentz contraction, so the entire assembly seems to be equally contracted in the S frame with respect to the length at the start. Therefore, at first sight, it might appear that the thread will not break during acceleration.

 

This is what I think I understand but I am not sure. This could be wrong.
The ships starts off the same speed 0v. Then the ships accelerate. Due to the Lorentz transform the ships distance gets longer. The rope also stretches. How does this lead to the rope breaking?
In the ships frame I see the ship the same length. I just see a person running. How do I reconcile the 2 frames?

 

Here is a good space-time diagram.
K is stationary person and K' is the moving ships

https://imgur.com/a/renojPE

 

https://www.physicsforums.com/insights/what-is-the-bell-spaceship-paradox-and-how-is-it-resolved/ (The picture comes from this link) 

 

Edited October 4, 2020 by can't_think_of_a_name

[Halc]

5 hours ago, can't_think_of_a_name said:

This is what I think I understand but I am not sure. This could be wrong.
The ships starts off the same speed 0v. Then the ships accelerate. Due to the Lorentz transform the ships distance gets longer. The rope also stretches. How does this lead to the rope breaking?

The distance between the ships in frame S remains the same, but the length of the rope contracts in that frame and doesn't span the distance anymore. So it breaks. There is no paradox in this. One rigid ship of significant length does not accelerate at an equal rate along its length, so the pair of ships tied by a string does not behave as a rigid object would.

The ships in this example are assumed to be essentially small objects of negligible length.  Only the string has non-negligible length.

Quote

In the ships frame I see the ship the same length. I just see a person running. How do I reconcile the 2 frames?

I don't know what you mean by seeing a person running. In the frame of either ship during the acceleration phase, the rear one is going slower and the lead ship is going faster.  So again, the string must break. There is no disagreement between the frames, and thus no paradox at all.

Your spacetime diagram shows all of this.

Edited October 5, 2020 by Halc

[can't_think_of_a_name]

In S frame, why does the rope contract but not the distance between the ships?

There is mention of the ships and string not behaving as a rigid ship why?

[Halc]

The distance between the ships does contract in the S frame, but since the proper distance between the ships increases, it effectively stays the same in the S frame.

So say the ships (separated by 1 light hour) accelerate identically (same proper acceleration, commencing simultaneously in S) to 0.6c.  In the new inertial frame (T) where they both eventually come to rest, the ships are now 1.25 light hours apart (proper distance), which is length contracted to 1 light hours in frame S.  The 1 light hour string is length contracted to 0.8 LH in S, which isn't enough to connect them.  It is fully 1 LH in T, which also isn't enough to connect them.

[can't_think_of_a_name]

Followup question. This is a psuedo paradox but why was this even deemed a paradox at first? Also according to the online class from courseara I am taking why did this confuse scientist upon first hearing it?

 

Edited October 12, 2020 by can't_think_of_a_name

[Halc]

It is not a real paradox since the Theory (SR in this case) predicts nothing different than what happens in reality. It is only when the situation is assessed by a naive person who does not know the theory well that inconsistencies might first appear. If I am figuring out something and I get an inconsistency like that, my instinct is like that of a scientist: I assume I made a mistake somewhere, and do not assume the theory is wrong.

The wiki article expresses this: "Therefore, at first sight, it might appear that the thread will not break during acceleration."

This already implies somebody who isn't very familiar with SR, since anybody who knows it well knows that a rigid object (one that retains its proper length at all times) undergoes different magnitudes of proper acceleration along its length. The two spaceships undergo identical proper acceleration, therefore the assembly of the two connected ships does not constitute a rigid object and will not retain its proper length. So it is no surprise that the string breaks.

It is similar to the twins 'paradox', which seems to always want to use twins because it pushes an intuitive button in humans that twins are always the same age. But this naively makes the assumption that the twins have similar acceleration histories. Once this 'rule' is found to not be a valid rule, the paradox resting on the rule vanishes.

A true paradox will falsify a theory.  If a theory resting on a set of premises predicts that X is true and that X is not true, that is a true paradox, and it is a general indication that at least one of the premises leading to the paradox is false.  In the case of Bell's spaceships, all observers agree that the string should break given what they're doing, so there's no paradox.

Edited October 12, 2020 by Halc