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Mathematics for Physics

175 Light Cone

A cone in spacetime that is used to mark the set of events reachable from a given event by light signals.

The locus of light trajectories. A light cone is the surface swept by every light ray through one event. An event is a happening at a definite place and time. This principle is used to draw all possible light paths from that event on a spacetime diagram.

Causal partitioning. The cone is a strict boundary between events that can be influenced from the vertex and events that cannot. Elsewhere is the region that cannot be reached from the vertex even at the speed of light. This principle is used to protect the order of cause and effect.

Invariance of the cone. Every inertial observer agrees on the same light cone at a given event. An invariant is a quantity that does not depend on which inertial frame is used. This principle is used to decide, in every frame, whether two events can be causally connected.

The null-interval boundary. On the cone the spacetime interval from the vertex is zero. A null trajectory is a path of zero interval. This principle is used to compute the worldlines of light.

Note: Also called the null cone. Also called the forward and backward light cones when future and past parts are distinguished.

175.1 References

  1. Emam, M. H. Covariant Physics. Oxford University Press, 2021. — lightlike world lines form the light cone of unit slope.
  2. Carroll, S. M. Spacetime and Geometry. Cambridge University Press. — light cone as points linked to one event by lightlike lines; null interval.
  3. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — forward and backward light cones; causal boundary.
  4. Shankar, R. Fundamentals of Physics I. Yale University Press, 2019. — past, future, and elsewhere.