Hello, and welcome back to MPC! A few days ago, we discussed time dilation from a conceptual point of view. Today, we will look at a more mathematical approach to time dilation. If mathematics is not your best subject, I encourage you to at least try reading this post as it will provide you with insight into why and when time dilation occurs. Let’s get started!

Imagine a scenario in which a person, *person 1*, is standing on Earth, and another person, *person 0*, is flying through the sky on a rocket ship. For simplicity, let’s say that *person 0* is moving at a constant speed, *v*, in his rocket ship (note that this means that *person 0* is not accelerating). Let’s also pretend that *person 0* has a particle of light that is bouncing up and down inside of his rocket ship. Of course, this particle of light is moving at the speed of light, *c*. Here is a diagram of the situation:

**Figure 1**: A diagram of the scenario we will be discussing

Now, let’s think about how long it will take for that particle of light to travel from the bottom of the rocket ship to the top of the rocket ship once. First, let’s look at the particle of light from *person 0*’s perspective:

**Figure 2:** The movement of the particle of light from *person 0*’s perspective

Of course, *person 0* just sees the particle move from the bottom of the rocket ship to the top of the rocket ship. The particle of light is simply traveling a distance of *L* at a speed of *c*. Therefore, the time it takes for the particle to travel from the bottom of the rocket ship to the top of the rocket ship, from *person 0***’s perspective**, is:

It should be noted that this equation was derived using the definition of velocity (velocity is distance divided by time).

Now, let’s talk about the particle’s motion from the perspective of *person 1*. *It is the same right?* Well, not exactly. We have to remember that *person 1* is watching the rocket ship move at a speed of *v*. Because of this, to *person 1*, the particle’s motion looks like the following:

**Figure 3:** The movement of the particle of light from *person 1*’s perspective

It must be emphasized that, now, the vertical component of the particle of light’s motion (how quickly it is moving up if you were to completely ignore horizontal motion) is **not** *c* — the diagonal motion is *c*. *Why is this the case?* Recall that the speed of light is constant, so if the vertical component of the particle’s motion were *c* (as it currently is to *person 0*), the particle’s diagonal motion would be greater than *c *according to *person 1* (by the Pythagorean theorem) — a direct violation of the principles of relativity!

You may notice that this scenario is very similar to our billiards example in the previous post. Indeed, it is essentially the same! Let’s break it down! In the billiards example, one thing that we noticed is that the ball traveling diagonally traveled a greater distance than the ball traveling in a straight line. This is, once again, the case here. Let’s “zoom in” on this situation:

**Figure 4:** A magnified view of the movement of the particle of light from *person 1*’s perspective

It should be noted that in *Figure 4*, the letters represent *distances*, whereas in *Figure 3*, they represented *speeds*. In any case, this picture should make sense. The particle of light is traveling a vertical distance of *L*. Furthermore, it is traveling a distance of *vt**1* horizontally (by the definition of velocity), where *v* is the velocity of the rocket ship and *t**1* is how long it takes for the particle of light to move from the bottom of the rocket ship to the top of the rocket ship according to *person 1*.

It is very important that we have made a distinction between *t0* and *t*1*. *As discussed in last week’s post,* person 0* may claim that it takes 5 seconds for an event (such as the particle traveling to the top of the rocket ship) to happen while *person 1* may claim that it only takes 1 second for that same event to happen. Because *t0* and *t1* may be different, we must represent them with different variables.

So, according to *person 1*, what distance, *d*, is the particle of light traveling to reach the top of the rocket ship? By the Pythagorean theorem:

Using the definition of velocity, *t**1* is:

And there we go! Already, we can see that *person 1*’s measurement of the amount of time that it will take for the particle of light to reach the top of the rocket ship will be greater than *person 0’s* measurement (we know this because the numerator of *equation 3* will always be greater than the numerator of *equation 1*).

Now, let’s make things more interesting: how much longer will *person 1* say it takes for the particle of light to reach the top of the rocket ship than *person 0* says? We can calculate this. To see how, let’s rearrange *equation 3*:

Right now, we have two “problems.” First of all, we are trying to relate *t**0* and *t**1*, so it would certainly be useful if our equation contained both of those variables. Also, the *L* in the equation is rather bothersome. Could the relationship between *person 0* and *person 1*’s time measurements really depend on the height of the rocket ship? If we want to compare each person’s times, do we actually have to measure the height of the rocket ship? We may just have to. For now, though, it is in our best interest to see if we can get rid of that *L*. It may be possible, it may not be possible.

By rearranging *equation 1* we get:

If we substitute *equation 5* into *equation 4*, we get:

Look at that! We have killed two birds with one stone: our equation contains both *t**0* and *t**1* and no longer contains *L*. This was by no means guaranteed to happen. Nature has just been too kind to us!

Now, our next goal should be to either write *t**0* in terms of *t**1* or *t**1* in terms of *t*0. If we can achieve this, then it will be much simpler to calculate one of the times given the other. Let’s try using some algebra:

Viola! We have done it! We now have *t**1* in terms of *t**0*. So, if we were given *t**0*, *v*, and *c* (which is a well-known constant), we would be able to calculate *t**1*. We can also do this in the opposite direction — if we were given *t**1*, *v*, and *c*, it would not be too difficult to calculate *t**0*.

Well, we are almost done. Many physicists think that *equation 7* is too ugly, so they have done some extra work to make it neater:

*Are we done now?* Almost. Physicists also hate that ugly square root in the denominator, so they have defined the following variable:

That letter that looks like a *y* is actually the Greek letter *gamma*. When *gamma* is defined as it is in *equation 9*, it is known as the *relativistic factor*. Using the *relativistic factor *in *equation 8*, we get the following:

That looks pretty neat to me! With that, we are done. We have successfully (and neatly) written *t**1* in terms of *t**0*. Now, you may be wondering: *Who cares? So what if one person says that it only takes 1 seconds for the particle of light to reach the top of the rocket ship while someone else says that it takes 5 seconds?* You’re right, that information is pretty worthless. However, we can use it to uncover other, more interesting phenomena about our world. That will be the focus of next week’s post: the consequences of time dilation. See you then!

P.S.: This post may have been a lot to digest. I recommend that you think it over for the next few days. If you have any questions, feel free to post them in the comments! Also, if you would like this proof in video format, you can find a great version over here.

(featured image: https://cdn.shutterstock.com/shutterstock/videos/589834/thumb/1.jpg)

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