Two identical rockets float at rest, one ahead of the other, joined by a taut, fragile thread. At in your frame both fire identical engines and follow identical velocity profiles, so in your frame their separation never changes. Does the thread break?
Solution
Yes.
In your frame the separation is constant by construction — but the thread is now moving, so its unstressed length is Lorentz-contracted. A thread forced to span an uncontracted gap while its equilibrium length shrinks is a thread under growing strain, and it snaps. In the rockets’ instantaneous rest frames the same story reads differently: there the separation between the ships grows (the leading ship, by relativity of simultaneity, “started earlier”), and the thread is stretched outright.
To keep the thread relaxed, the ships would have to maintain constant proper separation — Born-rigid motion — and that requires the trailing ship to accelerate harder than the leader, the two following different Rindler hyperbolae with . Identical accelerations and rigid formation are incompatible demands; Bell’s spaceships are the cleanest demonstration that length contraction has dynamical teeth.
Deeper in the notebook: 03. Bell’s Spaceship Paradox and Born Rigidity · 01. Uniform Acceleration; the Rindler Wedge and Horizon