Proper time

Proper time τ is the time measured by an ideal clock carried along a particle's world-line. It is related to the coordinate time t of any inertial observer by dτ = dt/γ = √(1 − β²) dt, where β = v/c is the particle's instantaneous velocity in that observer's frame. Integrating along a timelike world-line yields the total proper time elapsed: τ = ∫ √(1 − β²) dt. This integral is the arc length of the world-line in the Minkowski metric ds² = c²dt² − dx² − dy² − dz², so proper time has the same geometric status in spacetime that Euclidean arc length has on a Riemannian manifold. It is invariant under Lorentz transformations — every inertial observer agrees on how much proper time elapsed on a given clock between two events on its world-line.

Proper time is the geometric content of time dilation. The lab-frame interpretation "the moving clock ticks slow" is observer-dependent; the invariant statement is that the proper time elapsed along a curved world-line is always less than the proper time elapsed along the straight (inertial) world-line connecting the same two events — the relativistic reverse of the Euclidean triangle inequality. The twin paradox is exactly this: the travelling twin's accelerated world-line has shorter proper-time arc length than the stay-at-home twin's straight one. Proper time is the parameter to use along world-lines because four-velocity, four-acceleration, and four-force are all derivatives with respect to it, and Lorentz-covariance is automatic.