Summarizing: While the rest distance between the ships increases to γ γ L in S′, the relativity principle requires that the string (whose physical constitution is unaltered) maintains its rest length L in its new rest system S′. Therefore, it breaks in S′ due to the increasing distance between the ships. As explained above, the same is also obtained by only considering the start frame S using length contraction of the string (or the contraction of its moving molecular fields) while the distance between the ships stays the same due to equal acceleration.