Light cone ($|\mathbf{x}{|}^2=c^2{t}^2$)

The light cone of a spacetime event $O$ consists of $O$ together with the spacetime events having null separation from it. In the coordinates of an inertial frame centered on $O$, it obeysFor $t>0$ it traces an expanding spherical flash of light, with radius $ct$ at time $t$; for $t<0$ it describes possible incoming light rays. A spacetime diagram with one spatial coordinate shows the two lines $x=\pm ct$. The apparent $45^{\circ }$ angle uses equal scales for $x$ and $ct$ and is not a separate physical angle.

The light cone is the boundary, while its interior has timelike separation from $O$. An outgoing light ray follows the boundary; a future-directed massive observer's world line remains inside it. The exterior has spacelike separation: no signal travelling at or below the speed of light can connect its spacetime events directly to $O$. This separates possible causal influence from merely being assigned a later coordinate time.

In the diagram, $A=(4\mathrm{s},3\mathrm{ls})$ is inside the future boundary, $N=(3\mathrm{s},3\mathrm{ls})$ is on it, and $B=(2\mathrm{s},3\mathrm{ls})$ is outside. Their squared Minkowski intervals are respectively $-7,0,5$ in ${\mathrm{ls}}^2$. Thus $B$ occurs later than $O$ in the displayed inertial frame while still being unable to receive a light-speed-or-slower signal sent from $O$.

The usefulness is immediate for an experiment: a detector's reading can depend on a source event only if a causal influence can reach it. A Lorentz boost changes coordinates but preserves this classification. For a boost with $\beta =0.8$ and Lorentz factor $\gamma =5/3$, applying the Lorentz transformation gives, in light-second length units,The corresponding squared Minkowski intervals remain $-7,0,5$. In particular $B$ is now earlier than $O$, which is allowed for spacelike separation; $A$ remains later.

The past light cone of a spacetime event $O$ is the past-directed part of its light cone. In Minkowski spacetime, its spacetime events satisfy $|\Delta \mathbf{x}|=-c\Delta t$ with $\Delta t<0$. They are the possible emission events of light rays arriving at $O$. The interior also contains events that can send slower signals to $O$. The retarded potential of a moving point electric charge, given by the Liénard–Wiechert potential, uses the emission event where the source's world line intersects this boundary.

The future light cone is the future-directed part of the light cone of a spacetime event. Its boundary consists of the events reached by outgoing light rays. Events reached by slower causal signals lie inside the boundary; relativistic causality forbids a controllable signal from reaching the spacelike exterior.

The speed of light $c$ is the invariant speed of every inertial frame and the limiting speed of causal propagation in special relativity.

A one-way speed of light measurement assigns a travel time between spatially separated emission and reception events and therefore requires a distant-clock synchronization convention. In an inertial frame, Einstein synchronisation assigns the vacuum value $c$ in either direction. A completed-loop comparison such as the Sagnac effect is a different observable.

A light-second is a unit of length: the distance vacuum light travels in one second. It is useful in a spacetime diagram because a time of one second corresponds to a $ct$ coordinate of one light-second, making the slopes $x=\pm ct$ of the light cone particularly easy to read. The unit is a distance, rather than an elapsed time.

Null directions are tangent directions of the light cone; their Minkowski squared norm is zero.

A direction represented by a nonzero null vector has zero Minkowski norm. Such directions form the generators of the light cone.