a worldline

On a spacetime diagram, a particle is not a dot — it is a curve. As time advances the particle occupies a succession of events, and threading them together traces a line through spacetime: its worldline. Where an ordinary trajectory shows where something is, a worldline shows where AND when — the full history of the object as a single geometric object.

A worldline is the curve x^mu(lambda) that an object sweeps out in Minkowski spacetime, most naturally parametrized by its own proper time tau, x^mu(tau). Its tangent is the four-velocity U^mu = dx^mu/dtau, and for a massive particle the worldline is everywhere time-like (its tangent lies inside the local light cone, U·U = c^2 > 0). A particle at rest has a vertical worldline (moving through time only); uniform motion is a straight tilted line; acceleration curves it. Light travels on null worldlines lying exactly on the light cone (ds^2 = 0), which cannot be parametrized by proper time since dtau = 0.

The worldline viewpoint reframes dynamics as geometry. Free (inertial) motion is a straight worldline — and, because the metric is indefinite, it is the one of LONGEST proper time between two events; a four-force bends the worldline (nonzero four-acceleration). This is the direct ancestor of general relativity, where free fall is motion along a geodesic — the straightest possible worldline in curved spacetime. Caveat: a worldline is not a path 'in space' that gets traversed over time; it is a fixed curve in spacetime, already containing the time dimension — the particle does not 'move along' it so much as it IS the curve.