Bell's spaceship paradox

Bell's spaceship paradox, posed by John Stewart Bell in 1976, is a sharper relativistic puzzle than the barn-pole. Two identical rockets, launched simultaneously in the lab frame, accelerate with identical proper-acceleration profiles. A thin string is tied between them at rest. In the lab frame, both rockets have the same world-line shape, just translated, so their separation never changes. Yet in the instantaneous rest frame of either rocket, the leading rocket is moving away from the trailing one — the proper distance between them grows monotonically. The string, whose unstressed length is its proper length, must stretch and eventually snap.

The resolution is that a "rigid" rod under acceleration cannot have all its points accelerate identically in the launch frame. Born rigidity (Max Born, 1909) requires that the trailing end of a rigid rod accelerate slightly faster than its leading end so that the rod's proper length stays constant — the difference exactly cancels length contraction. Bell's two-rocket setup explicitly enforces equal launch-frame acceleration, which is not Born-rigid, so the structure between the rockets is stressed beyond its breaking point. The puzzle was a CERN cafeteria argument that Bell turned into a published note; he reported that a non-trivial fraction of CERN theorists initially got it wrong, calling it "a relativity test that no working physicist should fail."