proper time

When observers moving differently disagree about how much time passed between two events, it is natural to ask: is there any time interval everyone can agree on? Proper time is that quantity. It is the time measured by a single clock that is actually present at both events — the time read straight off a wristwatch carried along for the whole trip. Because that one clock is right there for both the start and the finish, its reading is a real, physical, frame-independent fact.

Precisely: proper time, usually written with the Greek letter tau, is the time interval between two events measured in the frame where those two events happen at the same place — the rest frame of the clock that witnesses both. It is always the shortest time any inertial observer measures between the two events; every other frame, watching that clock move, records a longer, time-dilated interval delta-t = gamma times tau, where gamma is the Lorentz factor. So proper time is the 'true' ageing of the traveller, and coordinate time in a moving frame is the stretched-out version.

Proper time is the invariant heart of the theory: while individual clocks and rulers disagree, all observers compute the same tau for a given path through spacetime. It is the amount a clock actually ages along its route — the traveller's own lived time. In the twin paradox, the twin who ages less is simply the one whose path through spacetime carries the smaller total proper time.