Bell's Spaceship Paradox Solved: The Thread Does Not Break

Bell's spaceship problem admits two internally consistent descriptions. In one, the ships hold a fixed launch-frame separation and their proper separation grows as γL; in the other, the proper separation is fixed and the launch-frame separation falls as L/γ. Both satisfy the Lorentz transformation exactly, and they give opposite answers for whether a connecting thread breaks. Neither takes the thread as an input, and length contraction cannot arbitrate between them, so which one describes the assembly is a question about its dynamics. We settle it by integrating the coupled relativistic elastodynamics — a covariant elastic thread of finite mass and sub-luminal signal speed, with massive endpoint ships and the control law an explicit input rather than a tacit stipulation. Under fixed throttle, with the thread's own weight set aside as Bell sets it aside, the ship–thread–ship system settles onto the Born-rigid branch — exactly when the material dissipates, and in the time-averaged configuration when it does not — and that branch is a stable fixed point for every finite servo gain. Bell's congruence survives only as the marginal case g → 1, in which a servo cancels the restoring stiffness exactly; reproducing his conclusion, a thread that parts, requires a DC loop gain above 7.5×10¹² against the thread's own 1.34 pN of back-reaction. The thread is what supplies the acceleration difference that puts the assembly on the Born congruence, and it does so at a cost of one part in 3.7×10¹⁵ of the engine thrust.

The two descriptions are also distinguishable by measurement, which makes the conclusion falsifiable rather than merely derived. The ratio of clock rates between two points of a body equals the inverse ratio of their proper accelerations, exactly; cutting the engines removes any question of simultaneity convention, since both clocks are then at rest in one frame and may be carried together and read side by side. Across a 30 m assembly coasting at 0.75c the Born-rigid congruence predicts 97.363 ns against Bell's 113.468 ns, a 16.5% discrimination. That measurement has been proposed but not performed; the relation it rests on is confirmed in two other instances of expansion-free motion, Mössbauer rotors and towers in a gravitational field to 10⁻¹⁸, which fix the relation without by themselves selecting the congruence. The thread does not break: the ships, thread and gap all remain 10 m in the comoving frame at every speed and all measure 6.614378278 m in the launch frame at 0.75c, with no strain anywhere. The criterion for a real thread is Π ≡ M_eff a₀²L / 2c²T_break < 1, which contains no γ, no v and no t: a thread breaks immediately or never, and there is no critical speed. For tonne ships and a cotton thread Π = 1.3×10⁻¹³. A real thread's own weight exceeds the coordination tension by 5×10⁸ and is ordinary Newtonian statics; borne by the front ship, as a tension-only member requires, it biases the assembly toward convergence rather than separation and leaves the thread slack, so on that account too it does not break.

This deposit contains the manuscript (44 pp., PDF and LaTeX source) and the verification suite it rests on: thirteen scripts reporting 369 checks, all passing, which recompute every number printed in the paper's tables and string-match each against the LaTeX source, together with the relativistic elastodynamics integrator, the figures, and the revision history. See README.txt in the archive.