The Twin Paradox
Two twins synchronise their clocks. One boards a rocket, flies out at high speed, turns around, and comes home; the other stays put. When they reunite, the traveller is measurably younger. The "paradox" is the objection: motion is relative, so couldn't each twin call the other the one who moved, and expect them to be younger? They can't both be right.[1]
Both twins start and end at the same two points, so both agree on the total elapsed coordinate time — but not on how much they aged. Ageing is the length of your own worldline (its proper time), and in spacetime the straight path is the longest one. The homebody takes it and ages most; the traveller's bent path is shorter, so she returns younger by the factor γ. It isn't symmetric because only the traveller turns around — she switches frames, and that break is what singles her out. Crank β toward 1 and the age gap yawns open.
The resolution is written in the diagram. What a clock measures is the length of its own worldline — its proper time — and in spacetime the counter-intuitive rule is that the straight worldline is the longest. The stay-at-home twin travels straight up the time axis and ages the most; the traveller's path bends out and back, a shorter total proper time, so she ages less by exactly the factor .[2]
The situation is not symmetric, and the spacetime diagram shows why: only the traveller turns around. At that turnaround she switches from an outbound inertial frame to an inbound one, and her notion of "now" back on Earth jumps forward — the missing years the homebody experiences while she is between frames. The stay-at-home twin never changes frames, which is what breaks the tie. Whoever feels the engines fire is the one who comes back young.
It's tempting to conclude that the acceleration is the aging difference — that the engines somehow "do" the time loss. That's not quite right, and the distinction matters.
What acceleration settles: which twin is the traveller. Only she feels proper acceleration — an accelerometer on her ship reads nonzero at the turnaround, one on Earth never does. That is a frame-independent fact, and it is what dissolves the paradox: the two twins are not interchangeable, because only one of them ever changes inertial frames.
What acceleration does not settle: how much younger she returns. The size of the gap is geometry — the proper time
integrated along each worldline — and it depends on the speed and duration of the inertial cruise, not on how hard or how briefly she turns around. Make the turnaround arbitrarily violent and brief (enormous acceleration, negligible duration) and the age difference barely changes; lengthen the coasting legs and it grows without any change in the acceleration. The straight worldline simply has the longest proper time, and the traveller's bent one is shorter — acceleration is necessary to bend the path at all, but the years accrue on the straight segments, not "during" the burn.
The two viewpoints agree exactly. In the simultaneity picture, the turnaround is where the traveller's line of "now" on Earth swings forward and skips over a swath of Earth-time; shrink the turnaround to an instant and that skip becomes a sudden jump, but its size is fixed by the geometry of the two inertial legs. Acceleration marks the traveller; the trip's shape sets the toll.