This presentation is driven from Jerrold Franklin of Temple university of Philadelphia’s paper “Lorentz contraction,Bell’s spaceships and rigid body motion in special relativity” and Dean and Bern’s paper: “Note on stress effects due to relativistic contraction”.

Bell’s spaceship paradox was formerly designed by Dewan and Beran and became widely known when Bell added a modified version…

This thought experiment was a demonstration of “relativistic contraction can introduce stress effects in a moving body”.  We procrastinated that in the rest frame S as everything of the two spaceships L & R are the same (with the exception of their initial place) so the distance of them should remain as constant at all times.
And we saw that if we analyze the distance in S’ which is a frame that its velocity is the same as our spaceships’ in S at a special instant,it gives us a distance which is getting greater than the rest distance d in S’ point of view. This actually is the Lorentz-Fitzgerald’s contraction as from S point of view the distance has contracted.

Dewan and Beran: everything for both spaceships is the same and they accelerate from rest so even in relativistic velocities the distance between them shouldn’t change(their velocities are the same all the time so the distance they cover in equal interval should be the same so the distance should remain constant).
But we can say that when we talk about length of something we mean the difference between the places of the beginning and the end of something, why can’t the distance between two spaceships be like that,It’s the difference between the top of L and rear of R. So as the length of a ruler in relativistic speeds will go through Lorentz’ contraction,the distance between L & R should be contracted too.

So the paradox here is if we add a silk thread between L & R,in rest frame S,will the distance between the spaceships remain constant as the silk thread contracts and snap,or will the distance get contracted too?

The resolution: The correct answer is the silk thread would snap due to constancy of the distance of the spaceships.
Let’s consider the distance of the spaceships would contract. This means in higher velocities,the relative velocity of the spaceships wouldn’t be zero. This would be a wrong thought cause of the inertial assumption of having the same velocity all the time. If this argument is not convincing enough,lets say the distance goes under Lorentz’ contraction,as shown in parts before,the distance between two spaceships in each S’ frame,is the new proper distance due to the spaceships being at rest in S’. If so each of these S’ can be like S and we can say that the spaceships are accelerating from rest in these frames. From   we see that the amount of contraction depends on the proper distance. This will bring up the questions like “How do the rockets obtain information about the distance between them(every proper distance in every S’)?”,”by what mechanism do they adjust their velocities in a way that depends upon d?”,”at very high accelerations,does R go backwards at first or does L catch up to it in order to reduce the distance between them?”,”if we had three spaceships,which ones have the larger and smaller velocities to give the contractions?”,etc.these difficulties and questions wouldn’t arise for a rod moving in relativistic velocities because the reason the rod contracts is the distance between its ends which is defined in its “proper Lorentz’ frame”(the rest frame which we can determine the proper length of the rod in it.there is only one proper length for it).