= Light cone
{title2=$|\mathbf x|^2=c^2t^2$}
{wiki}
= Null cone
{synonym}
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 obeys
$$
|\mathbf x|^2=c^2t^2.
$$
For $t>0$ 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.
\Image[/minkowski-light-cone-events.png]
{title=Light-cone boundaries separate timelike, null and spacelike events relative to the origin}
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,
$$
\begin{aligned}
(ct,x)_A=(4,3)&\longmapsto (ct',x')_A=(8/3,-1/3),\\
(ct,x)_N=(3,3)&\longmapsto (ct',x')_N=(1,1),\\
(ct,x)_B=(2,3)&\longmapsto (ct',x')_B=(-2/3,7/3).
\end{aligned}
$$
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.
\Image[/lorentz-boost-preserves-intervals.png]
{title=A boost at 0.8c changes event coordinates and spacelike time order while preserving every Minkowski interval}
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