Special Relativity II: Time Dilation Part 1

Hello, and welcome back to MPC! Last week, we covered a brief history of special relativity. We also spoke about the two main concepts on which relativity is based:

  • The laws of physics are the same in all inertial frames of reference.
  • The speed of light is constant in all inertial frames of reference.

Today, we will be discussing one of the strange consequences of these two statements: time dilation.

A few weeks ago, we spoke about how one second to you may not be the same thing as one second to me. This idea of time being able to “scale” in different scenarios is referred to as time dilation. Before we begin delving into the details of time dilation, I should note that this week’s blog post is going to be a little more mathematical. I am going to be dividing the post into two parts: a conceptual part and a theoretical part. For the conceptual part, the most challenging mathematics that you will need to understand is the Pythagorean theorem, which should not be too hard. Do not worry, the theoretical part of this week’s post will not require any “high-level mathematics” either — the hardest part will be solving algebraic equations. Let’s get started!

Part 1: Conceptual Understanding

Let’s pretend that we are playing billiards with “particles of light.” I have the first shot, and I hit the ball directly into one of the walls of the billiards table.


Figure 1: I shoot the ball directly into the wall

Remember, we are using particles of light as balls, so the speed of my ball must be the speed of light, or c. Let’s say that I am holding a stopwatch, and I measure the amount of time it takes for my ball to hit the wall as 1 second (this must be a very large billiards table).

Now, it is your turn. Like me, you shoot the ball at a distance of L from the wall. However, you do not shoot the ball directly into the wall; instead, you shoot the ball at an angle to the wall:


Figure 2: You shoot the ball at an angle to the wall

Just like my ball, your ball travels at a speed of c. Let’s say that you also have a stopwatch and that you measure the time it takes for your ball to hit the wall as 1 second.

Let’s look at the distance that my ball traveled compared to the distance that yours traveled. My ball, of course, just traveled a distance of L to reach the wall. Your ball, on the other hand, must have traveled a distance greater than L to reach the wall. How do we know? If we look at Figure 2, we will notice that L is only a component of the distance that your ball traveled– L is how far your ball traveled horizontally, but your ball also traveled vertically. By the Pythagorean Theorem, the distance your ball traveled is the square root of the sum of L squared and the vertical distance traveled squared. This will always be greater than L.

Do you see any problems? Remember, velocity is equal to distance per unit time. Even though our balls were traveling at the same speed, your ball was able to cover a greater distance than mine was able to in the same unit of time. Clearly, our balls are not going at the same speed.

Unless, the one second I measured on my stopwatch is different from the one second that you measured on your stopwatch. For example, what happens if every second on my clock was 2 seconds on your clock? Then, our scenario may make sense: your ball traveled a greater distance than mine, but it also took longer to travel that distance if I were to measure the time, meaning that both of our balls may be traveling at the speed of light (which they must be according to the laws of relativity). Note that your ball only took longer to hit the wall than my ball did from my perspective — even if you had the best stopwatch in the world, you would measure the amount of time that it took for your ball to hit the wall to be 1 second. This was not a result of the inaccuracy of your clock; rather, time itself was slower for you than it was for me when we ran the experiment. When time slows down in relativity, it is called time dilation

Hopefully, this explanation has provided you with a conceptual understanding of how the essence of time can change. Of course, the aforementioned scenario is not a realistic one: not only can you not play billiards with light, there is also no reason why time is traveling more slowly for you than it is for me. Why is your clock slower than mine and not vice versa? In fact, time dilation would technically not even occur in this scenario. So, when would it occur?  All of these questions will be answered in part 2! I do not want to make you wait too long, though, so part 2 will be a “bonus” post and will be posted at 11:30 am on this Sunday. See you then!

(featured image: http://www.villages-news.com/wp-content/uploads/2015/10/Billiards.jpg)


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