Special Relativity IX: Spacetime Diagrams Part 1

Hello, and welcome back to MPC! Last week, we talked about the spacetime interval. We still have more to discuss when it comes to the spacetime interval. However, we are going to take a brief break and talk about something new today: spacetime diagrams.

In math and physics, we typically look for ways of representing concepts and phenomena visually. For example, we could describe the motion of a tennis ball thrown across a field with the following 2 equations:


These equations are great and make it very easy to calculate important values (e.g. where is the ball after 5 seconds?). That being said, by themselves, these equations are very limited. When we graph these equations, though, we get the following:


Figure 1: The trajectory of a tennis ball

With the graph in Figure 1 we can clearly see that the ball starts at a low position, reaches a high point, then returns to its starting position. This visual representation is very powerful and, when coupled with Equation 1 and Equation 2, allows us to thoroughly analyze the motion of the ball.

So far, we have discussed spacetime from a conceptual standpoint. We have also developed many equations to describe spacetime: the time dilation equation, the length contraction equation, the spacetime interval equation, etc. Today, we will start utilizing a visual representation of spacetime that will strengthen our understanding of it. This visual representation of spacetime is called a spacetime diagram.

In order to generate our first spacetime diagram, let’s imagine that we have just thrown a tennis ball up into the air. Its motion may look something like the following:


Figure 2: The motion of a tennis ball thrown into the sky

**Note: We are only analyzing the upward motion of the tennis ball. We will be ignoring the effects of gravity**

Clearly, the tennis ball is moving upwards. We can graph the tennis ball’s position at various parts of its upwards flight on a y-axis:


Figure 3: The motion of a tennis ball thrown into the sky plotted on an axis

The graph in Figure 3 depicts the ball’s motion through space. We’ve been talking about spacetime though! For this reason, let’s add another axis to Figure 3, the time axis (t-axis):


Figure 4: The motion of a tennis ball in a spacetime diagram

**It is very important to note that the tennis ball is not traveling diagonally — it is only traveling upwards. The x-coordinate (t-coordinate) of each point represents an instance of time, and not a position (in Figure 1, on the other hand, the x-coordinate represents a position). For example, the point highlighted in red on the graph does not represent the tennis ball at a position 1 meter above and 1 meter to the right of its starting position. Rather, the point highlighted in red on the graph represents the tennis ball at a position 1 meter above its starting position, and the tennis ball was at this position 1 second after it was thrown.**

In Figure 4, the points from Figure 3 were simply shifted to the right in order to accurately depict the tennis ball’s locations at certain instances in time. As we move from left to right on the graph in Figure 4, we can see that the tennis ball is moving upwards as time passes. And there we go, we have represented spacetime in a visual manner!

Well, almost. Yes, the graph seen in Figure 4 is one way of representing spacetime. However, it is more common for physicists to use a ct-axis instead of a simple time (or t-) axis.

What the heck is a ct-axis? Let’s imagine that right as we throw our ball up into the air, we also fire a particle of light into the atmosphere.


Figure 5: A particle of light launched alongside a tennis ball

(**Note: The arrow extending out of the particle of light is longer than the arrow extending out of the tennis ball in order to illustrate that the particle of light travels faster than the tennis ball**)

Let’s say that we also have a timer that we start as we release the tennis ball. After 1 second, we write down how far the tennis ball has traveled, but we also write down how far the particle of light traveled. We repeat this at 2 seconds, 3 seconds, etc.

Now, what we are about to do may seem a little strange but it will all make sense soon. In our previous drawing (Figure 4), the x-coordinate (t-coordinate) of a point represents a time measurement and the y-coordinate of that point represents the position of the ball at that instance of time. Instead of having the x-coordinate of each point represent a unit of time (1 second, 2 seconds, 2.5 seconds, etc.), let’s have the the x-coordinate represent the distance our particle of light has traveled.

That may sound confusing, but perhaps an example will help. Let’s say that we throw the tennis ball up and shoot the light particle up, and they both start at the same position (a position we will call 0). Let’s say after 1 second we measure the location of the tennis ball and particle of light. We find that the tennis ball has traveled 1.5 * 10^8 m (we threw it really hard) and the particle of light has traveled 3.0 * 10^8 m. On our graph then, the tennis ball’s y-coordinate would be the distance it has traveled after that one second, 1.5 * 10^8 m. Its x-coordinate, though, would not be 1 second; rather, its x-coordinate would be the distance the particle of light has traveled in that 1 second, 3.0 * 10^8 m.


Figure 6: A point plotted on the y-axis and the ct-axis

If we wanted to, we could also see how far the tennis ball has traveled after 1.5 seconds. Then, on our graph, the point representing the tennis ball would have a y-coordinate equal the distance the tennis ball has traveled in those 1.5 seconds and an x-coordinate equal to the distance the particle of light has traveled in those same 1.5 seconds. This new, strange x-coordinate is called our ct-coordinate.

Something that is very important to note is the fact that light has a constant speed. We spoke about how the speed of light, c is fixed in a previous post. Because c is always the same, light will always travel the same distance after 1 second, 2 seconds, etc. To be more specific, because the speed of light is 3.0 * 10^8 m/s, light will always travel 3.0 * 10^8 m after one second, 6.0 * 10^8 m after two seconds, etc. What this means is that, in our aforementioned situation with the tennis ball, we don’t even have to fire off a particle of light to create a spacetime diagram with a ct-axis. Why not? If we just use the timer and record the distance the tennis ball has traveled after 0.5 seconds, 1 second, 1.5 seconds, etc. we have the power to figure out how far a particle of light would have traveled (had we shot it off into the atmosphere) at each time interval. To do this, we just multiply a given time measurement, t, by the speed of light, c, which will give us how far the particle of light would have traveled after t seconds. For example, if knew that the tennis ball traveled 1.5 * 10^8 m in 1 second, we could just multiply that 1 second by c to determine that a particle of light would have traveled 3.0*10^8 m in that same second (had it been fired into the atmosphere). This is why the x-coordinate is our “ct-coordinate” — because each point on the ct-axis is just the distance light could/would have traveled in a unit of time, all points along the ct-axis just represent the product of c and some time measurement t, which is just ct.

Alright, so we now have a spacetime diagram with a y-axis and a ct-axis, but why? What was the point of replacing the t-axis with a ct-axis? Aren’t things just more confusing now? Yes, the ct-axis does make things more complicated. Nonetheless, using the ct-axis is actually very helpful. How so? That’s a discussion for next week! Next week, we will be looking at how we can use the ct-axis to our advantage. See you then!


If you want to get a headstart on some of next week’s materials, be sure to check out http://www.farmingdale.edu/faculty/peter-nolan/pdf/UPCh34.pdf.


(featured image: https://www.motionbackgroundsforfree.com/wp-content/uploads/2012/03/Screen-shot-2012-03-30-at-9.47.28-AM.png)


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