One twin leaves, turns around, comes home younger. It is not an illusion and not a trick of clocks — proper time is a path, not a reading. Between two meetings in spacetime, the straight (inertial) worldline accrues the most time; the twin who turned around walked a bent path and lived less of it, by exactly 1/γ. Down the center the terms go in — a separation speed and a duration — the engine integrates dτ = dt/γ along each worldline, and the age gap comes out. The blue team builds it; the red team tries to break it.
source P. Langevin, L'évolution de l'espace et du temps, Scientia 10 (1911) 31–54; the effect follows from A. Einstein, Zur Elektrodynamik bewegter Körper, Ann. Phys. 17 (1905) 891. No stable open scan of Langevin's paper — cited by author/journal/year (wikisource search). Rendered, not quoted.
Along any worldline, the clock reads proper time τ = ∫ ds/c, where ds² = (c dt)² − dx². Split the journey into legs; on a leg of coordinate duration Δt and displacement Δx:
τ_leg = √((cΔt)² − Δx²)/c = Δt/γ, with β = v/c and γ = 1/√(1−β²) ≥ 1. Standing still (Δx=0) gives τ = Δt; moving costs time. The gap is real geometry, not a slow clock.
For the current journey, each leg's interval and proper time:
| leg | Δt | Δx | interval/c = τ |
|---|
Langevin (1911) drew the picture: the traveller ages less by 1/γ because a bent worldline is shorter in proper time than the straight one between the same two events. The straight, inertial path is the maximum — a real extremal.
This sphere is the-time-dilation integrated along the-spacetime-interval. Dilation is the rate; the paradox is that rate summed over a path. Each sphere is the next one's premise.
The blue team's live check: recompute both proper times, confirm τ_travel = τ_home/γ < τ_home, and confirm the straight path beats a sampled bent path. If red tampers, this badge is where it shows.
Two meeting events: A (departure) and B (reunion), at the same place in the home frame, separated by home-frame duration T. The traveller flies out at speed v = βc, turns around at the midpoint, and returns. That turnaround — a burst of acceleration — is what only the traveller feels.
| twin | worldline | accelerates? |
|---|---|---|
| home (S) | straight, v = 0 | no — inertial |
| traveller (T) | out, turn, back | yes — turnaround |
Feed a separation speed β and a duration T into the panel below — the engine integrates each worldline.
Vertical = home time ct↑ · horizontal = space x. Blue = home twin (straight). Red = traveller (bent). Dashed = light cone.
Move any control — τ_home = T, τ_travel = T/γ, and the age gap are integrated on the spot from dτ = dt/γ, never looked up.
What the machine proves, from geometry alone: the traveller returns younger by exactly τ_home − τ_home/γ. The stay-at-home logged the maximum possible proper time between A and B; every bent path — the traveller's included — logs strictly less. The asymmetry is real: only one twin turned around.
The blue team's witness (left) confirms τ_travel < τ_home live; the red team (right) tries to erase the gap.
General relativity is not required — flat Minkowski spacetime plus one turnaround settles it. The turnaround can be idealised as instantaneous; the missing proper time is charged to the leg-swap, not to any gravitational field. "Both frames are equal" is the assumption that fails.
"You need General Relativity / gravity to solve it." Cut. Special relativity in flat spacetime suffices — the asymmetry is the acceleration, handled without any curvature.
"The traveller's clock runs slow, so it's a clock defect." Cut. Every clock — biological, atomic, mechanical — reads the same τ. It is the path that is shorter, not the hardware that is faulty.
"By symmetry each sees the other younger, so it's a genuine paradox." Kept, corrected. The time dilation is symmetric while both fly inertially; the turnaround breaks the tie, and only the traveller's tally comes up short.
The red team's move: drop the γ on the moving legs so the traveller's clock ticks at the home rate — erasing the age gap and forcing τ_travel = τ_home. The blue team's witness (window 7) is watching.
Remove the 1/γ from the moving legs and the traveller no longer ages less — the τ_travel < τ_home check fails, the witness recomputes and turns red. Nothing is faked; the attack is real and it is caught.