Listen S is the new rest system of the rockets after the acceleration is finished Why is this so hard to understand So if you perform a Lorentz transformation of points A and B from system S where the ships are moving with v after acceleration to S where they are at rest after acceleration you are calculating the endpoints of resting objects in S thus you have the rest length of the objects and the distance between the endpoints And what do the Lorentz Trafos tell us Well they unambiguously show that the distance between A and B again those are the coordinates of the endpoints of in S resting objects has increased as compared to the coordinate length between A and B in S Of course this is in perfect agreement with relativity of simultaneity which shows that the spaceships started at different times in S thus they had different accelerations Summarizing 1 You have two endpoints A and B with different accelerations in S therefore they are moving away from each other in S and after the acceleration the distance of the now resting objects has increased and the string has been stretched 2 Of course one can also choose frame S in which the accelerations appeared to be the same due to relativity of simultaneity and therefore the increased distance in S apparently stays the same in S Actually one should abandon the word appear in relativity it serves no purpose and only distracts one from correctly applying the Lorentztransformation Now Dewan Beran and Bell unfortunately started with 2 and only afterwards at 1 so they had to focus their attention mostly on S and on length contraction which caused much confusion by some readers But the paradox is much easier to understand if one like Petkov and Franklin starts with 1 and then look at 2 BTW As explained above modern authors understand the paradox very well and some of them even try to confirm it experimentally Bell s spaceship paradox in electron interferometry