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7.1.13. Spacetime Diagrams

Table of Contents

Seeing events in spacetime

A spacetime diagram is a picture that helps us represent where and when things happen. In ordinary graphs, we often plot position against time. In relativity, this idea becomes especially important, because time is not treated as completely separate from space. A spacetime diagram lets us place events, motion, and light signals on one picture.

An event is a single occurrence with a definite position and a definite time. For example, "a flash of light emitted at the origin at time zero" is one event. In a spacetime diagram, each event is shown as one point.

Usually, for simple motion in one spatial direction, we draw the horizontal axis as position $x$ and the vertical axis as time multiplied by the speed of light, $ct$. Using $ct$ instead of just $t$ makes both axes have units of length. This helps comparisons between space and time.

Axes and units

A common spacetime diagram has $x$ on the horizontal axis and $ct$ on the vertical axis. The origin represents the event $x = 0$, $t = 0$.

If an event happens at position $x$ and time $t$, its coordinates on the diagram are $(x, ct)$.

Important convention:
A spacetime diagram for one spatial dimension usually uses
$$
\text{horizontal axis} = x, \qquad \text{vertical axis} = ct
$$
so that space and time coordinates are expressed in compatible units.

This does not mean time and space are identical. It means they are displayed in a way that makes relativistic relationships easier to see.

Worldlines

A moving object does not appear as a single point on a spacetime diagram, because it exists at many times. Instead, its history is drawn as a curve called a worldline. Each point on the worldline is an event where the object is at some place at some time.

If an object is at rest in the chosen frame, its position does not change, so its worldline is a vertical line. If it moves with constant velocity, its worldline is a straight slanted line. Faster motion means a line that leans more toward the horizontal direction.

An object cannot move faster than light, so its worldline for a physical object always remains closer to the vertical axis than a light ray does.

Worldlines in a spacetime diagram

Slope and motion

The slope in an ordinary position versus time graph gives velocity. In a spacetime diagram with axes $x$ and $ct$, the meaning is slightly different. For a worldline,

$$
\text{slope} = \frac{ct}{x} = \frac{c}{v}
$$

for constant velocity motion.

So a slower object has a steeper worldline, and a faster object has a flatter worldline. Light has $v = c$, so its slope is

$$
\frac{ct}{x} = 1
$$

which means light travels along lines at $45^\circ$ if both axes use the same scale.

For a light signal in an $x$ versus $ct$ spacetime diagram,
$$
x = \pm ct
$$
These lines form the boundary between possible and impossible motion for material objects.

Light rays on the diagram

Light is special in relativity. Since every inertial observer measures the same speed of light, light rays always appear at the same angle in a correctly scaled spacetime diagram. In one space dimension, light emitted from the origin travels along two lines,

$$
x = ct \qquad \text{and} \qquad x = -ct
$$

One goes to the right, the other to the left. These lines divide the diagram into regions that have deep physical meaning.

Light rays from one event

Light cones

The two light lines from a chosen event form the edges of the light cone. In a full three dimensional space picture the structure is actually a cone, but in a two dimensional drawing with only one space coordinate, it appears as a V shape.

The region above the event and inside the light lines is the future light cone. Events there can be affected by something happening at the original event, because a signal traveling at or below the speed of light can reach them.

The region below the event and inside the light lines is the past light cone. Events there could have influenced the original event.

Events outside the light cone are separated in such a way that no signal traveling at or below light speed can connect them. These are sometimes called spacelike separated events.

A table helps summarize this.

Region relative to an eventMeaning
Inside future light coneCan be reached from the event by slower than light or light-speed signals
Inside past light coneCould have sent a signal to the event
On the light coneConnected exactly by light
Outside the light coneNo causal connection is possible at speeds $\leq c$

Causality rule:
If one event can cause another, the second event must lie in the future light cone of the first.
No physical influence can travel outside the light cone.

Comparing different motions

Spacetime diagrams make it easy to compare the motion of different objects. A stationary observer has a vertical worldline. A moving observer has a tilted straight worldline if the speed is constant. Two objects that meet correspond to worldlines crossing. The crossing point is the event of their meeting.

For example, if two spaceships move toward each other, each ship has its own worldline. Where the lines intersect, they occupy the same place at the same time.

Intersection of worldlines

Proper use of the diagram

A spacetime diagram is not a picture of what things look like visually. It is not like a photograph. It is a map of events. One axis gives position, and the other gives time. Because of this, distances drawn on the page are not ordinary physical distances in space.

Also, the diagram depends on the reference frame used. Different observers moving relative to each other use different axes for space and time. The detailed relation between those axes belongs to Lorentz transformations, but the important idea here is that the same physical events can be plotted by different observers in different ways.

Spacetime interval on the diagram

One of the most important quantities in special relativity is the spacetime interval between two nearby events. In one spatial dimension it is

$$
s^2 = c^2 \Delta t^2 - \Delta x^2
$$

This quantity has the same value for all inertial observers, even though $\Delta x$ and $\Delta t$ may differ from one observer to another.

The sign of $s^2$ tells us the kind of separation between the events.

Type of separationConditionPhysical meaning
Timelike$s^2 > 0$One event can influence the other with speed less than $c$
Lightlike$s^2 = 0$Connected by light
Spacelike$s^2 < 0$No causal signal can connect them

Invariant spacetime interval:
$$
s^2 = c^2 \Delta t^2 - \Delta x^2
$$
This value is the same in all inertial reference frames.

This is one of the main reasons spacetime diagrams are useful. They help us see whether events are timelike, lightlike, or spacelike separated.

Reading physical situations from the diagram

Suppose a flash occurs at the origin. Later, a detector at some position records the flash. If the detector event lies on the light line, the signal traveled at speed $c$. If the detector event lies inside the future light cone, the signal would have had to travel slower than light. If the event lies outside the cone, the flash could not have caused it.

This visual method is powerful because many questions about motion and causality can be answered by looking at the arrangement of worldlines and light cones.

Final idea

Spacetime diagrams are one of the clearest tools in special relativity. They show events as points, object histories as worldlines, and the universal role of light through $45^\circ$ lines in properly scaled axes. Most importantly, they make causal structure visible. By looking at which events lie inside, on, or outside a light cone, we can immediately see what kinds of physical influence are possible.

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