Why is the Wikipedia article about Bell's spaceship paradox disputed at all?

[nakurusil]

The proper length doesn't increase once the engines have been turned off, so it must increase during the acceleration phase. This means that the string would also break in c).

Correct. So if the rope breaks it breaks during the acceleration phase. This is why you must calculate the separation distance between the rockets during the acceleration phase. This is why you must not use Lorentz transforms, they do not apply to accelerated motion.

I pretty sure this line of reasoning is valid, but as it stands, I don't think it's rigorous enough to prove that the string breaks in c).

Correct. Except that your solution does not compute the separation distance between rockets correctly. You should not be using the Lorenz transforms, you schould be using hyperbolic motion. Using Lorentz transforms is akin to using the fact that the sum of the angles is 180 degrees in a planar triangle in order to calculate the third angle of a spherical triangle when you know the first two angles. In both cases there is no justification in blindly appliying a theory derived for one instance to a totally different instance.
By applying the correct theory (hyperbolic motion) you will get the correct answer. An it is not $$l=l_0\gamma$$. If you do the calculation correctly you will get a nonlinear expression that depends on acceleration and time.

 

[Fredrik]

I don't have to calculate the separation at all. I just have to show that it increases.

My calculation is exactly right for version b) of the problem, so stop denying that or at least try to prove that you're right. That hyperbolic motion stuff is specifically for version a).

 

[nakurusil]

I don't have to calculate the separation at all. I just have to show that it increases.

Yes, you do have to calculate the separation correctly, ESPECIALLY for "your case" (b) The question that was asked is: "will the rope snap". The rope has some elasticity, so it snaps only if the distance betwen the rockets increases beyond what the rope elasticity can accommodate DURING the acceleration phase. Without a correct calculation of the separation distance betwen the rockets, you cannot find out if the rope snaps. And in "your case" (b), the distance stops increasing after you shut off your engines.

My calculation is exactly right for version b) of the problem, so stop denying that or at least try to prove that you're right. That hyperbolic motion stuff is specifically for version a).

The hyperbolic motion is the rigurous solution for ALL cases.

 

[Fredrik]

Yes, you do have to calculate the separation correctly, ESPECIALLY for "your case" (b) The question that was asked is: "will the rope snap". The rope has some elasticity, so it snaps only if the distance betwen the rockets increases beyond what the rope elasticity can accommodate DURING the acceleration phase. Without a correct calculation of the separation distance betwen the rockets, you cannot find out if the rope snaps. And in "your case" (b), the distance stops increasing after you shut off your engines.

It's hard to tell if you're being serious. The rope will certainly snap if the proper length at any time exceeds the original proper length, and my calculation is more than sufficient to prove that it does in version b).

You're still insinuating that my solution of b) is incorrect. I suggest that you either stop doing that, or prove that you're right.

The hyperbolic motion is the rigurous solution for ALL cases.

Please explain yourself. Hyperbolic motion is constant proper acceleration. So how does a calculation that takes hyperbolic motion as a starting point solve the general case?

 

[ZapperZ]

This thread has gone long enough, and it is going nowhere long enough.

I will point out to everyone involved that to re-read the PF Guidelines that you have agreed to. If you do not think we meant everything we wrote in there, think again.

Consider this as your only warning before more drastic action is taken.

Zz.