KAHIBARO
Table of Contents
Causal Structure of Spacetime
In special relativity, a light cone is a picture that shows which events can influence each other. It is one of the clearest ways to understand causality in spacetime. An event means something that happens at a specific place and a specific time.
A light cone is built from the idea that light has the same speed, $c$, in all inertial reference frames. If a flash of light is emitted at one event, then after some time it spreads outward in all directions. The set of all points reached by that light forms the light cone.
The Event at the Cone's Tip
Imagine one event at the origin of spacetime, with coordinates $x = 0$ and $t = 0$. A pulse of light is emitted there. In one space dimension, the light moves to the right and left with speed $c$, so its motion satisfies
$$
x = ct \quad \text{and} \quad x = -ct
$$
These two lines form the boundaries of the cone on a spacetime diagram. In three space dimensions, the expanding light front is a sphere at each instant, and all these spheres together make a cone in spacetime.
The event where the light starts is called the apex, or tip, of the cone.
Future Light Cone and Past Light Cone
The light cone has two parts. The future light cone contains all events that can be reached by a signal moving at speed $c$ or less, starting from the original event. The past light cone contains all events that could have sent a signal to the original event, again moving at speed $c$ or less.
If an event lies in the future light cone, it can be affected by the original event. If an event lies in the past light cone, it could have affected the original event.
An event can causally influence another event only if the second event lies inside or on the future light cone of the first.
No physical signal can travel faster than light, so events outside the light cone cannot have a cause and effect relation with the event at the cone's tip.
Inside, On, and Outside the Cone
The position of an event relative to the light cone tells us what kind of separation it has from the origin event.
If an event is inside the cone, then enough time has passed for even a slower than light object to travel there. This is called timelike separation.
If an event is exactly on the cone, it can be reached only by something traveling at the speed of light. This is called lightlike, or null, separation.
If an event is outside the cone, it would require faster than light travel to connect it to the origin. This is called spacelike separation.
These ideas are closely connected to the spacetime interval. For motion along one spatial direction,
$$
s^2 = c^2 t^2 - x^2
$$
Then the classification is:
| Separation type | Condition | Meaning |
|---|---|---|
| Timelike | $s^2 > 0$ | One event can causally affect the other |
| Lightlike | $s^2 = 0$ | Connected by light |
| Spacelike | $s^2 < 0$ | No causal connection possible |
For light itself, the spacetime interval is zero:
$$
c^2 t^2 - x^2 = 0
$$
This gives the light cone boundary:
$$
x = \pm ct
$$
Light Cones on a Spacetime Diagram
A spacetime diagram usually has time on the vertical axis and space on the horizontal axis. Because light always satisfies $x = \pm ct$, its path appears as straight lines through the origin.
If we choose units where $c = 1$, these lines make $45^\circ$ angles with the axes. This makes the cone easy to draw.
The shaded upper region is the future light cone. The shaded lower region is the past light cone. The regions to the left and right are outside the cone and are spacelike separated from the origin event.
Worldlines and Causality
A worldline is the path of an object through spacetime. Light travels along the surface of the cone. Any massive object must move inside the cone, never on the outside. This is because massive objects always move slower than light.
A vertical worldline means the object stays at one position while time passes. A tilted worldline means the object moves through space as time passes. The tilt cannot be so large that it goes outside the light cone.
The dashed line represents a path outside the light cone. That would mean faster than light motion, which special relativity does not allow.
Worldlines of massive particles must remain inside the light cone.
Worldlines of light lie on the light cone.
Paths outside the light cone would require faster than light motion.
Why Different Observers Agree on the Cone
Different inertial observers may disagree about the time coordinate and space coordinate of an event. They may even disagree on the order of two spacelike separated events. But they all agree on whether one event is inside, on, or outside another event's light cone.
This is because the spacetime interval is invariant under Lorentz transformations. So the causal structure of spacetime is the same for all inertial observers.
That is why light cones are so important. They show what is absolutely possible and impossible in terms of cause and effect.
Spacelike Separated Events
Suppose two events are so far apart in space, and so close in time, that even light could not travel from one to the other. Then they are spacelike separated. No signal can connect them.
For such events, different observers can disagree about which one happened first. This does not create a contradiction, because neither event could have caused the other.
This is a central lesson of light cones. The relativity of time order is allowed only when causality is safe.
A Simple Example
Suppose event A happens at $x = 0$, $t = 0$. Event B happens at $x = 300 \text{ m}$ and $t = 2 \times 10^{-6} \text{ s}$.
Light would travel in that time a distance
$$
ct = (3.0 \times 10^8 \text{ m/s})(2 \times 10^{-6} \text{ s}) = 600 \text{ m}
$$
Since $300 \text{ m} < 600 \text{ m}$, event B lies inside the future light cone of A. So A could causally affect B.
Now suppose instead that event C happens at $x = 900 \text{ m}$ and the same time $t = 2 \times 10^{-6} \text{ s}$. Since $900 \text{ m} > 600 \text{ m}$, event C lies outside the light cone. A could not cause C.
Physical Meaning
Light cones divide spacetime into regions of possible influence and impossible influence. They tell us what events can communicate, what events can be causes of other events, and what events are forever outside causal contact.
In this way, the speed of light is not just the speed of a particular kind of wave. It is the fundamental limit that shapes the structure of spacetime itself.
The light cone is the geometric picture of causality in special relativity.
It separates spacetime into past, future, and elsewhere.
Only events within or on the cone can be physically connected by signals.