Relativity and gravitational theory Definition

Light cone

Builds on Minkowski space Spacetime interval

Definition

Let $(V, \eta)$ be Minkowski space with signature $(-, +, +, +)$ and coordinates $(x^0, x^1, x^2, x^3)$ , and let $p$ be an event. The light cone of $p$ is the set $C(p) = \set{q : \eta(q - p, q - p) = 0},$ the events at spacetime interval zero from $p$ . Its future light cone is the part with $q^0 \ge p^0$ and its past light cone the part with $q^0 \le p^0$ .

Remarks

Example: with $c = 1$ , the event $q = (2, 1, 1, \sqrt{2})$ lies on the future light cone of the origin. Check: $-4 + 1 + 1 + 2 = 0$ and $q^0 = 2 > 0$ . Non-example: $q = (1, 2, 0, 0)$ is not on the light cone of the origin, since $-1 + 4 = 3 \ne 0$ ; it is spacelike separated from it. The events $q$ with $\eta(q - p, q - p) < 0$ and $q^0 > p^0$ form the interior of the future cone, the chronological future of $p$ . The light cone is preserved by every Lorentz transformation, and in a Lorentzian manifold the same construction is made in each tangent space.

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Minkowski space concept
Spacetime interval concept
Light cone definition

Tag	`7JFKFPHY4J`
URI	`m2://7JFKFPHY4J`
Kind	definition
Writing	W0 stub
MSC	83A05, 53B30
Defines	Light cone
Uses	Minkowski space, Spacetime interval
Changed	3 November 2026

Light cone

Definition

Remarks

Reading path

Further reading

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