Bell’s spaceship paradox

Two ships begin at rest with separation L₀ in the laboratory. They start together in that frame and follow identical histories of constant proper acceleration a, measured by their own accelerometers. Both engines shut down simultaneously in the laboratory at target speed βc. The calculator compares the subsequent common inertial final rest frame with the laboratory. It does not animate the long acceleration period.

The laboratory separation remains Llab = L₀. After both engines have shut down, the simultaneous distance in their common rest frame is Lrest = γL₀, where γ = 1/√(1−β²). The geometric increase is (γ−1)L₀ and the required relative extension is 100(γ−1)%. This is a measurement in the common inertial end phase, not a universal definition of proper distance while the ships accelerate.

Event A is the rear ship’s cutoff and defines the coordinate origin. B is the front ship’s cutoff; A and B share the same laboratory time. In final-rest-frame coordinates normalized by L₀, A = (x′/L₀, ct′/L₀) = (0,0), B = (γ,−γβ). C = (γ,0) lies on the front ship’s subsequent inertial worldline and is simultaneous with A in the final rest frame. Thus A–B and A–C use different simultaneity conventions. For β = 0, B and C coincide. The diagram draws each inertial worldline upward only from its own cutoff, not as a straight extension into the acceleration phase.

For each ship, the laboratory burn duration is t = cγβ/a, elapsed onboard proper time is τ = c atanh(β)/a, and travel distance from launch is d = c²(γ−1)/a, with c = 299792458 m/s. The cutoff time difference in the final rest frame is t′B−t′A = −γβL₀/c. These durations and the full flight distance are listed separately; the event tables and diagram use coordinates relative to A.

The entered percentage is an assumed strain limit. The tool compares the required geometric extension with it; it does not calculate tension, stress, elastic response, material failure or a breaking time. “At the limit” includes only a rounding tolerance of 32 machine epsilons times the larger of the strain and limit percentages. The small positive quantity γ−1 is evaluated separately so tiny strains are not lost when γ itself rounds to one.

Compare calculates explicitly. Edits remove old results; Reset keeps the inputs. Valid inputs are retained across navigation. Reference: Lewis, Barnes and Sticka (2018), Bell’s Spaceships: The Views from Bow and Stern, §§2–3.