Hello, and welcome back to MPC! Last week, we started discussing spacetime diagrams and how they can be used to represent the motion of objects through spacetime. Today, we will continue to analyze spacetime diagrams.
At the end of last week’s post, we spoke about the strange and seemingly useless ct-axis. Recall that we replaced our simple t-axis with this complex ct-axis. What was the point of that? Let’s start off by imagining that we are shooting a particle of light into the sky.
Figure 1: Our Particle of Light
**Note: Just like last week, we will be ignoring the effects of gravity.**
Let’s think about the particle of light’s motion. How far will the particle of light travel in one second? Well, we know the speed of light, c, is 3.0 * 10^8 m/s. Therefore, in one second the particle of light will travel 3.0 * 10^8 m.
Figure 2: Our Particle of Light After One Second
So we know that we will have a point on our spacetime diagram that has a value of 3.0 * 10^8 m on the vertical axis (y-axis). However, where will that point fall on the horizontal axis (ct-axis). Remember, the ct-value of a point represents how far a particle of light would have traveled in the same amount of time that the object in question has traveled. Last week, we said that we can get ct-values by imagining that we have shot a particle of light out into the sky at the same time that we shot out our original object (in this case, a particle of light). In order to make things less confusing, we will refer to the particle of light that we are drawing the spacetime diagram for (the particle of light illustrated in Figure 1) as Particle A and the particle of light that we have shot out for measuring ct-values as Particle B.
Figure 3: Particle A and Particle B
To recap, both Particle A and Particle B are starting off at position 0. Now, let’s shoot off both of the particles and after one second take a picture of the scenario:
Figure 4: Particle A and Particle B After One Second
Like we said before, Particle A travels 3.0 * 10^8 m in that one second. It should come as no surprise that Particle B also travels 3.0 * 10^8 m in that one second. Why does that make sense? Because they are both particles of light!
Alright, so we now use Particle A’s position (3.0 * 10^8 m) as our y-value and Particle B’s position (3.0 * 10^8 m) as our ct-value. On a spacetime diagram, Particle A’s motion so far looks like the following:
Figure 5: Particle A’s Motion on a Spacetime Diagram
**Note: The point in the bottom left corner represents Particle A right before it is launched into the sky**
So far so good. Let’s let Particle A and Particle B continue traveling and then take another picture after one second (from our previous picture; two seconds after they started moving originally).
Figure 6: Particle A and Particle B After Two Seconds
Using our formula for velocity, we discover that Particle A is now 6.0 * 10^8 m from its starting point. Additionally, Particle B is also 6.0 * 10^8 m from its starting point. We can add this “snapshot” (Figure 6) to our spacetime diagram:
Figure 7: Particle A’s Motion on a Spacetime Diagram
We can continue this process. If we were to do so, we would get a spacetime diagram that looks something like the following:
Figure 8: Particle A’s Motion on a Spacetime Diagram
Furthermore, we must note that we are taking our “snapshots” on one second intervals. If we had the technology to do so, we could take our snapshots at every half second, quarter second, or, ideally, at every instance of time. If we were to draw this last option (snapshot at every instance of time) on our spacetime diagram, we would actually get a line:
Figure 9: Particle A’s Motion on a Spacetime Diagram (with Small “Snapshot” Intervals)
The line in Figure 9 has a few interesting properties. The most important property is the fact that it forms a 45° angle with the ct-axis:
Figure 10: Particle A’s Motion on a Spacetime Diagram
Believe it or not, any object traveling at the speed of light will form a 45° angle with the ct-axis!
There we go, we already have one advantage of using a ct-axis over the t-axis: it is very easy to judge if an object is moving at the speed of light with a ct-axis. In other words, if you were ever given a spacetime diagram (with a ct-axis) and the motion of an object on the spacetime diagram forms a 45° angle with the ct-axis, you can immediately tell that that object is moving at the speed of light. Awesome!
Let’s now imagine an object that is moving faster than the speed of light — let’s say it is moving at twice the speed of light (6.0 * 10^8 m/s). We’ll call this particle Particle C and we will continue to use Particle B to measure our ct-values. Alright, so both particles start at position 0:
Figure 11: Particle C and Particle B
After one second, Particle C will have traveled 6.0 * 10^8 m and Particle B will have traveled 3.0 * 10^8 m:
Figure 12: Particle C and Particle B After One Second
On our spacetime diagram:
Figure 13: Particle C’s Motion on a Spacetime Diagram
After another second, Particle C will have traveled a total of 12.0 * 10^8 m and Particle B will have traveled 6.0 * 10^8 m:
Figure 14: Particle C and Particle B After Two Seconds
**Note: Particle C is so far ahead of Particle B that it cannot even be seen in the snapshot!**
On our spacetime diagram:
Figure 15: Particle C’s Motion on a Spacetime Diagram
And, once again, we can continue this process. If we were to continue this process and also take “snapshots” at every instance of time, we would get the following spacetime diagram for Particle C’s motion:
Figure 16: Particle C’s Motion on a Spacetime Diagram (with Small “Snapshot” Intervals)
Let’s look at the angle of this line with respect to the ct-axis — notice that it is greater than 45°:
Figure 17: Particle C’s Motion on a Spacetime Diagram
So, we can see that the motion of any particle traveling faster than the speed of light will have a spacetime diagram whose line forms an angle greater than 45° with the ct-axis. Although we will not show the proof of it here, the motion of any particle traveling slower than the speed of light will have a spacetime diagram whose line forms an angle less than 45° with the ct-axis.
However, let’s think for a moment: what is so special about the speed of light? It is the speed limit of the universe — nothing can travel faster than the speed of light! What this means is that we can never have lines in our spacetime diagram that form an angle greater than 45° with the ct-axis — they represent scenarios that are physically impossible!
Figure 18: Region of “Acceptable” Spacetime Lines
**Note: A spacetime line must be within the green region. Although not illustrated, the green region extends infinitely to the right, with its hypotenuse maintaining a 45° angle with respect to the ct-axis**
If we were to repeat what we did with Particle A and Particle B (Figure 9) but throwing the particles downwards instead of upwards, we would get the following spacetime diagram for Particle A’s motion:
Figure 19: Particle A Moving Downward on a Spacetime Diagram
This line also forms a 45° angle with the ct-axis:
Figure 20: Particle A Moving Downward on a Spacetime Diagram
Like before, all spacetime diagram lines pointing downwards must form an angle less than 45° with the ct-axis:
Figure 21: Region of “Acceptable” Spacetime Lines (for Downward Motion)
We can combine Figure 18 and Figure 20 to get the following:
Figure 22: Region of “Acceptable” Spacetime Lines (for Upward and Downward Motion)
The fun does not end there though! This may seem a little confusing, but we can actually make lines that point to the left of the y-axis (instead of the right). These lines actually represent events happening in reverse, events in the past. For similar reasons as before, these lines must form angles less than 45° with the ct-axis. We can combine this idea with the ideas represented in Figure 22 to get the following:
Figure 23: Region of “Acceptable” Spacetime Lines (for Upward and Downward Motion in the Reverse and Forward Direction)
The shaded regions in Figure 23 represent where all events (both past and future events) exist in spacetime. Any event outside of these regions cannot, by Einstein’s Theory of Special Relativity, possibly happen (assuming that all events start at the origin). The shaded regions in Figure 23 are collectively called the light cone (Why? If we were to added another dimension to our diagram — have objects move both horizontally and vertically instead of just vertically — the shaded regions would form two cones).
To summarize, using the ct-axis provides us with many “aesthetic” advantages. Using the ct-axis allows us to quickly judge if an object is traveling at the speed of light and allows us to visualize all possible events in spacetime in a clean manner (with the light cone). That is all for this week. Next week, we will resume our conversation on the spacetime interval. See you then!
For more information on spacetime diagrams, be sure to check out: http://www.farmingdale.edu/faculty/peter-nolan/pdf/UPCh34.pdf
(featured image: https://www.aspenmeadows.com/media/8595/ice-cream.jpg)