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)