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Mathematics for Physics

210 Worldlines

A curve through spacetime that is used to represent the history of a particle as a one-dimensional set of events.

The spacetime trajectory. A worldline is the continuous path of a particle through spacetime. Spacetime is the four-dimensional set whose elements are events. This principle is used to record the whole history of an object as one curve.

The timelike constraint. The worldline of a particle with nonzero rest mass is always timelike. A timelike curve is a path whose speed is everywhere less than the speed of light. This principle is used to enforce that no material object reaches the speed of light.

Straight worldlines for uniform motion. Uniform motion traces a straight worldline. Accelerated motion traces a curved worldline. Constant velocity is a speed and direction that do not change with time. This principle is used to tell inertial motion from forced motion on a spacetime diagram.

Parameterization by proper time. Distance along a timelike worldline is the proper time on a clock that follows that path. Proper time is the time read by a clock carried with the traveler. This principle is used to compute how much a traveler ages between two events.

Null worldlines of light. Light travels on a null worldline on the light cone. A null worldline is a path of zero interval between neighboring events. This principle is used to draw the paths of light signals.

Note: Also called a world line. Also called the trajectory of a particle on a Minkowski diagram.

210.1 References

  1. Carroll, S. M. Spacetime and Geometry. Cambridge University Press. — worldline as a parameterized one-dimensional set of events; timelike and null paths; proper time.
  2. Emam, M. H. Covariant Physics. Oxford University Press, 2021. — world line as the path of a particle in a spacetime diagram; proper time.
  3. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — world line as trajectory on a Minkowski diagram; lightlike paths.
  4. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — continuous history of an object; straight worldlines for uniform motion.
  5. Shankar, R. Fundamentals of Physics I. Yale University Press, 2019. — timelike constraint and inertial versus accelerated worldlines.