Hello, and welcome back to MPC! Last week, we discussed some of the consequences of time dilation. Today, we will be discussing another effect of special relativity: length contraction.
Once again, I am going to return to the “spaceship model” that we used in the time dilation posts. We have already spoken about why time measurements for person 0 and person 1 will be different. However, we have ignored something. Let’s imagine that person 0 was flying to that star on the right side of the image, and let’s say that that star is a distance d away.
Figure 1: The scenario that we will be discussing (from person 1’s perspective)
Now, let’s think a little about velocity. Let’s say that the rocket ship is moving at a speed of 0.87c according to person 1 (that is 87% of the speed of light). In other words, the distance person 0 is traveling to the star (d) divided by the amount of time person 1 believes it will take for person 0 to reach the star (t1) is v1 (v according to person 1):
In this case, v1 is 0.87c. Let’s now consider the situation from person 0’s perspective. Person 0, of course, does not think that he is moving: instead of thinking that he is moving to the star, he thinks that the star is moving to him. To understand why this is, think about what happens when you are driving on the road and pass a speed limit sign. Our minds are trained to know that, if the entire world around us is moving past us (all of the trees, grass, etc.), then we are probably the ones moving. Nonetheless, regardless of how your mind interprets it, the world around you and the speed limit sign seem to be approaching you, and not the other way around.
Figure 2: The scenario that we are discussing (from person 0’s perspective)
We now ask ourselves the following question: at what speed does person 0 believe the star is approaching him at? Well, we can simply use the definition of speed! Person 0 just has to take the distance the star is traveling to reach him (which would still be d; remember, person 0 thinks that he is stationary and the star is rushing towards him, meaning that it still must travel a distance of d) and divide it by how long he believes it takes the star to reach him (t0):
Let’s take this one step further. We already know what t0 is, at least relative to t1. Recall our equation for time dilation:
In this case, where person 1 sees person 0 moving at a speed of 0.87c, the relativistic factor is:
Recall what this means. Because our relativistic factor is 2, if person 0 and person 1 were to measure the time it takes for an event to occur (such as a particle of light reaching the ceiling of the rocket or person 0‘s rocket reaching the star), person 1’s time measurement will always be twice as large. As a result, we can also say that person 0’s time measurement is half as large as person 1’s time measurement. So, we get:
Let’s plug this back into Equation 2…
**Notice how we also applied Equation 1**
…and we have a problem: person 0 believes that the star is moving towards him faster than the speed of light. As some of you may have heard, it is impossible for anything to exceed the speed of light (try plugging a v greater than the speed of light into the equation for the relativistic factor and you will see that it breaks apart; we will discuss why the equation breaks apart in the near future). Clearly, something is wrong! But what can it be? We had a similar problem when proving the existence of time dilation before (when we initially calculated the speed of light in a specific scenario, we were getting values that were not c), and the cause of that problem was that we made the invalid assumption that measurements of time for a specific event are always the same. Have we made a similar assumption? For now, be sure to post your ideas in the comments! Next week, we will discuss our mistake. See you then!
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