worldline

A worldline is a concept in physics that describes the trajectory of a pointlike object through spacetime. In a four‑dimensional spacetime, a worldline is the set of events that comprise the object's history. It is commonly parameterized by proper time for a massive particle, x^μ(τ), and the spacetime interval s^2 = c^2 Δt^2 − Δx^2 − Δy^2 − Δz^2 is invariant along the worldline. The four-velocity dx^μ/dτ has magnitude c.

Worldlines can be classified as timelike, lightlike (null), or spacelike. Timelike worldlines describe objects with mass and lie inside the light cone; lightlike worldlines describe massless particles such as photons; spacelike worldlines would imply superluminal motion and are not realized by known particles. In special relativity, a particle with constant velocity has a straight worldline in flat spacetime, while acceleration curves the worldline. In general relativity, worldlines are curves in curved spacetime, and free-fall trajectories are timelike geodesics which maximize or extremize the proper time between events.

Worldlines encode causality: an event on a given worldline can influence only events within its future light cone, and not events that are spacelike separated. In spacetime diagrams, the slope of a worldline reflects velocity, with vertical lines indicating stationary objects and steeper lines corresponding to higher speeds approaching the speed of light.

The term also appears in advanced contexts such as the worldline formalism in quantum field theory, where particle histories are represented as paths in spacetime.