# Spacetime diagrams

Plotting world lines on spacetime diagrams can aid in the understanding of special relativity through visualization.

## Spacetime

Special relativity [1,2] starts out with the following two postulates:

1. The laws of physics are the same in all inertial frames.
2. The speed of light is the same in all inertial frames.1

With these postulates, one can derive how the coordinates $$(x,t)$$ of a particular event in one inertial frame are transformed to coordinates $$(x',t')$$ in another inertial frame as follows,2

\begin{aligned} t' & = \gamma \left( t - \frac{vx}{c^2} \right) \\ x' & = \gamma (x - vt) \end{aligned}

where $$v$$ is the speed the other inertial frame is moving relative to the original inertial frame, and the Lorentz factor $$\gamma$$ is

$\gamma = \frac{1}{\sqrt{1 - \left( \frac{v}{c} \right)^2}}$

In other words, the coordinates for space and time do not transform independently. Instead, they transform together as a set of spacetime coordinates from one inertial frame to another. Space and time should not be considered completely independent things, but should be considered intertwined as spacetime.

## A spacetime diagram

If you have a record of how a particular object moves across space as time passes by, you have a record of $$(x,t)$$ spacetime coordinatates for the object. You can plot these on a graph with an $$x$$ axis and a $$t$$ axis. This is called a spacetime diagram, and the plot of the $$(x,t)$$ coordinates is called a world line. These are useful for visualizing things in spacetime.

Below is a spacetime diagram with a world line of an object departing from the origin at half the speed of light, or $$v=0.5c$$. The horizontal axis is the $$x$$ axis, and the vertical axis is the $$t$$ axis with the future pointing up. The two dashed lines criss-crossing at the origin are the tracks of light passing through the origin, and these are often called the light cone in a spacetime diagram.

There is no reason why one cannot use spacetime diagrams with non-relativistic mechanics, but the different way coordinates transform between different inertial frames makes spacetime diagrams even more useful for visualizing things in relativity.

## From another inertial frame

In non-relativistic mechanics, transforming coordinates from one inertial frame to another does not change the $$t$$ coordinate or spatial lengths. If one were to transform the spacetime diagram in the previous section to the inertial frame for the moving object, it would merely slant the diagram. Horizontal and vertical distances would not change. In particular, the world line corresponding to the $$x$$ axis in the original diagram would remain the same in the other inertial frame. In addition, light emitted from a stationary source in the original frame would have a different speed.

However, this is not how coordinates actually transform in the real world given its relativistic nature. Unlike in non-relativistic mechanics, where space does not affect how time is transformed and there is no distortion in the spatial dimensions with relative velocities, space does affect how time is transformed and there is distortion according to the Lorentz factor for both temporal and spatial dimensions.

When the spacetime diagram from the previous section is tranformed into the inertial frame of the object moving at half the speed of light, it will transform as in the following diagram.

As in the diagram transformed non-relativistically, the world line of the observer is stationary in its own inertial frame, i.e., $$x=0$$ for the world line. But much else is different. Both the $$x$$ and $$t$$ axes are different from those in the original inertial frame, and events that had the same $$t$$ coordinates in the original inertial frame now have different $$t$$ coordinates. And the speed of light remains the same in both inertial frames.

You may or may not have also noticed that the world line now terminates at a $$t$$ coordinate slightly smaller than the $$t$$ coordinate it terminated at in the original diagram. If you did, you noticed time dilation, which will be discussed in the next section.

## Time dilation

The passage of time $$t$$ in one inertial frame is $$t' = \gamma t$$ in another inertial frame. This time dilation can be visualized using spacetime diagrams.

In the left diagram below is a world line stationary at $$x=0$$, starting at $$t=0$$ up to some time interval. On its right is the same diagram in another inertial frame moving at $$0.9c$$ towards the left, relative to the original inertial frame. You can see that the world line on the right terminates at a $$t$$ coordinate significantly higher than where it terminates on the left.

What does this mean? It means that time which elapses for the world line in the inertial frame on the right is significantly longer than how much it elapses in the inertial frame on the left. In other words, the time for a stationary observer is dilated for an observer moving at $$0.9c$$. The stationary observer does not notice anything funny with the flow of time, but the moving observer will notice that time is flowing much more slowly for the stationary observer.

## Length contraction

A length $$l$$ in one inertial frame is $$l' = \frac{1}{\gamma} l$$ in another inertial frame. Similarly to time dilation, length contraction can be visualized using spacetime diagrams, although it is a little more tricky. We can’t just compare how far along the $$x$$ axis a world line moves in two inertial frames, since that is measuring how far a point object travels, not its length. We have to think about what we are measuring and how we are measuring it.

Let’s say that we are going to measure the length of a stick which lies on the $$x$$ axis. Then what we want to measure is the distance between the two ends of the stick. If the two ends of the stick emit light towards each other at the same time, then we can measure the length of the stick by adding up the time for each beam of light to meet each other. Since the speed of light is always the same in any inertial frame, and the $$t$$ coordinate is the same for events happening at the same time, the length will be the distance parallel to the $$x$$ axis.

So what we will do with spacetime diagrams is to plot the world lines of both ends of the stick. For a stationary stick, the world lines will look like the diagram on the top below. The world lines are truncated so that they fit better in a spacetime diagram for another inertial frame, but you can imagine them extending forever into the past and future. Below this diagram is another spacetime diagram, this time for an inertial frame moving at $$0.9c$$ to the left, relative to the original inertial frame. The other diagram has transformed what is in the first diagram into its own inertial frame.

What you will notice from the diagrams above is that the length of the stationary stick as measured in the moving inertial frame is shorter than the length as measured in the stationary frame. In other words, the length of a stationary object contracts for a moving observer.

## Simultaneity

Another thing you could visualize using spacetime diagrams is how two events happening at the same time in one inertial frame do not happen at the same time in another inertial frame.

In the spacetime diagram on the left below, there are two world lines which change their color and how they are dashed at the same time. The spacetime diagram on the right shows the same world lines as observed from an inertial frame moving at $$0.9c$$ to the left, relative to the original inertial frame.

You should notice right away that what happened at the same time in the spacetime diagram on the left did not happen at the same time in the spacetime diagram on the right. In other words, simultaneity depends on the observer. Understanding this is crucial to understanding why seemingly paradoxical thought experiments such as the barn-pole paradox do not really give rise to a paradox.

## Conclusion

Spacetime diagrams do not replace the need for the equations specifying relativistic transformation of coordinates from one inertial frame to another. In particular, the equations are needed to transform spacetime diagrams from one inertial frame to another.

However, they are useful for visualizing how things may look differently from one inertial frame to another. This can aid the understanding of time dilation, length contraction, and the relativity of simultaneity. They can also aid in the understanding of various thought experiments with special relativity. In fact, I plan to use them to write up how one might understand apparent paradoxes such as the twin paradox.

• Proper time; how time passes for a moving object

• Twin paradox; how two twins could both observe time being slow for each other

• Time travel; what we could do if we could travel faster than light

## References

[1]
Sean Carroll. 2022. The biggest ideas in the universe: Space, time, and motion. Penguin Publishing Group.
[2]
Leonard Susskind and Art Friedman. 2017. Special relativity and classical field theory: The theoretical minimum. Basic Books.

1. With Maxwell’s equations for electromagnetism, one could argue that the second postulate is a consequence of the first postulate. This is because the equations predict a specific speed for light, and the equations would not be laws of physics if the speed of light was different for different inertial frames.↩︎

2. Space in our world is three-dimensional, so there should also be $$y$$ and $$z$$ coordinates, but we ignore them here, both because they remain unchanged when the movement is in the $$x$$ direction, and because it makes drawing spacetime diagrams on a two-dimensional surface much simpler.↩︎