Hello, and welcome back to MPC! Last week, we discussed time dilation from a conceptual and mathematical standpoint. Today, we will discuss the importance of time dilation and the consequences it has on our world.
In our mathematical analysis of time dilation, we talked about a person standing in a rocket ship with a particle of light bouncing up and down inside of it. Specifically, we showed that a person outside of the rocket ship would measure the amount of time it takes for the particle of light to reach the top of the rocket ship as greater than what the person inside of the rocket ship would measure. This is interesting and all, but who cares? So what if two people disagree about a simple and useless time measurement? Believe it or not, this discrepancy in time measurements does not simply reflect two clocks that have become unsynchronized — it reflects the transformation of time itself!
Let’s conduct a little “thought experiment.” Let’s imagine that the person inside of the rocket ship, person 0, is not moving at all. (see Figure 1) Also, let’s imagine that the person inside of the rocket ship is raising his hand. To make things simple, let’s say that he starts raising his hand when the particle of light is at the bottom of the rocket ship and finishes raising his hand when the particle of light is at the top of the rocket ship. Once again, let’s have another person, person 1, standing below the rocket ship watching person 0. Right now, everything is normal: person 1 just sees person 0 raising his hand.
Figure 1: Our first scenario — the rocket ship is stationary
Now, though, let’s reimagine this scenario with the rocket ship moving at a high speed. From person 0’s perspective, everything feels the same. How do we know this? According to the principles of relativity, all inertial frames of reference are the same (specifically, the laws of physics are the same in all inertial frames of reference). Because the stationary rocket ship and the rocket ship moving at a constant velocity are both inertial frames of reference, they must be the same and person 0 should not be able to tell the difference between them (if he could, then clearly the laws of physics have changed in some way, violating the principles of relativity). Therefore, both situations must feel the same to him. Person 1, though, sees something very different. Remember, because of time dilation, person 1 thinks that the particle of light takes longer to reach the top of the rocket ship now than it did when the rocket ship was stationary. Let’s just say that it took the particle of light 1 second to reach the top of the rocket ship when the rocket ship was stationary (this is a very big rocket ship), and it now takes the particle of light 2 seconds to reach the top of the rocket ship. This means that, while person 1 believes that it takes person 0 1 second to raise his hand in the stationary scenario 1, person 1 now believes that it takes person 0 2 seconds to raise his hand. In other words, person 0 sees person 1’s actions in slow motion (when compared to how he would have seen them if the rocket ship were not moving)! Not only that, everything inside of the moving rocket ship will appear in slow motion to person 1 (a clock ticking, a ball bouncing, etc.).
Figure 2: Our second scenario — the rocket ship is moving
**Note: I used the following phrase to describe the “slow motion” actions: “(when compared to how he would have seen the [actions] if the rocket ship were not moving).” It is interesting to note that, to person 1, everything also appears in slow motion when compared to what it would like if he/she were actually on the rocket ship. We know this because, when person 1 is on the rocket ship, everything seems stationary, meaning that it is equivalent to our first scenario in which the rocket ship was not moving relative to person 1.**
This may seem crazy, but it is true! When something is moving relative to you, everything occurring inside of that something is happening in slow motion! Now, I anticipate a counterargument: What about a car? When I am standing on the sidewalk and I see a car drive past me with people talking inside of it, the people’s mouths do not seem to be moving in slow motion. You’re right, but we have a reasonable explanation for that. Recall equation 9 and equation 10 from last week’s post:
Based on equation 10, we can consider the relativistic factor (the gamma) as being how many times slower actions in the moving frame of reference appear to the person in the stationary frame of reference. For example, if t0 were 1 second and the relativistic factor were 2, then t1 would be 2 seconds and person 1 would be measuring all of person 1’s actions as being twice as slow. So we can say that person 1’s actions appear γ times slower to person 0. Now, when the car is passing you, say at 27 m/s (or about 60 mph), how many times slower do the actions of the person in the car appear to be? If we plug in 3.8 * 10^8 m/s for c (a huge number) and 27 m/s for v (a relatively small number), the relativistic factor is…1! Well, technically, it is greater than one, but it is by such a small amount that a calculator (at least my calculator) just says that it is 1. In other words, the people inside of the car would appear to be moving in slow motion to you, but only marginally; in fact, so marginally that you would not notice it!
Now is also a good time to think about what the values of the relativistic factor can be. For those of you who are mathematically inclined, you may notice that gamma is always greater than or equal to 1. What does this mean? Essentially, it means that special relativity only allows things to appear to be moving in “slow motion” — time cannot appear to be speeding up, it can only appear to be slowing down.
To summarize, whenever an object is moving relative to a person, that person will see all actions that occur within that object in “slow motion.” This is a fascinating revelation, but you still may be wondering why this matters. We discussed that the effects of time dilation are negligible in everyday scenarios (i.e. the car example), so how does it affect us directly. In a few weeks, we will be discussing the effects of special relativity on our world. Before we do that, though, we must discuss another mind-bending concept, length contraction. We will discuss this next week. See you then!
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