Hello, and welcome back to MPC! Last week, we ran into quite a conundrum when working with time dilation: we saw that time dilation allows an observer to see an object moving at a speed faster than the speed of light. This, of course, is not possible! Have we done something wrong?
Indeed, we have. As has happened to us too often in special relativity, we have once again made a seemingly reasonable assumption that is surprisingly invalid. Let’s try to walk through the calculations we carried out last week to see what assumptions we made.
Figure 1: A quick refresher of the scenario that we dealt with last week (person 1’s perspective)
Figure 2: A quick refresher of the scenario that we dealt with last week (person 0’s perspective)
The first thing that we discussed last week was the speed of the rocket ship according to person 1:
The equation above seems to be correct: it is just the definition of velocity (distance traveled divided by elapsed time). We then discussed the speed of the star (approaching the rocket ship) according to person 0:
What assumptions have we made here? You may be thinking that we have not made any assumptions: it’s just like equation 1, we are just using the definition of velocity. Yes, this is correct, but not completely correct. It is important that we don’t get too caught up in the letters. What could that possibly mean? Well, when you see this equation, you may just read it as distance divided by time — you may simply interpret the d as being a shorthand way of writing distance and the t0 as being a shorthand way of writing time. However, the problem is that d is not just distance — it is the distance that the rocket ship travels to reach the star according to person 1! In other words, if person 1 had a very long tape measure and were to measure the distance between the rocket ship’s starting point and the star, he would obtain a certain value which we are calling d. Remember that Equation 2 is supposed to represent the distance the star travels to reach the rocket ship divided by the elapsed time (as measured by person 0). By using d for the distance the star travels to reach the rocket ship in Equation 2, we are essentially saying that the distance the star travels to reach the rocket ship according to person 0 is the same as the distance the rocket ship travels to reach the star according to person 1.
So what? That should not invalidate anything! Isn’t the distance as measured by person 1 the same as the distance as measured by person 0.
Yes, it does make sense that the two distances would be the same. For example, imagine that we placed a giant, 1-mile long ruler in between two mile markers on a road:
Figure 3: Using a ruler to measure the distance between two mile markers
If you were just standing on the side of the street and decided to measure the distance between the two mile makers (by reading off of the giant ruler that has been conveniently placed on the mile markers), you would get 1 mile. Likewise, if you were in a car and tried to measure the distance between the two mile markers (once again reading off of the giant ruler; I would not recommend doing this), you would still get that the two mile markers are 1 mile apart. It does not matter how fast the person in the car is moving, the distances should always be the same (according to our everyday experiences)! To relate this to the scenario that we have been discussing with the rocket ship, imagine if the 1-mile long ruler were placed between the rocket ship’s starting point and the star (we will thus make d 1 mile, just because we have been using that measurement so much — I apologize for the use of imperial units):
Figure 4: Using a ruler to measure the distance between the rocket ship and the star (from person 1’s perspective)
As was the case with the mile markers, both person 0 and person 1 should measure the same distance between the rocket ship and the star when using the ruler (and this means that the distance the rocket ship must travel to reach the star according to person 1 should be the same as the distance that the star must travel to reach the rocket ship according to person 0). For this reason, it only makes sense that person 0 and person 1 would measure the same distances, meaning that we should be able to use d in both Equation 1 and Equation 2.
Whether or not this “makes sense” does not matter — it is still an assumption that we have made. Recall our discussion of time dilation, and how we discussed that 1 second to one person is not necessarily the same 1 second to somebody else. Similarly, how can we be certain that the 1 mile that person 1 reads off of the ruler is the same as the 1 mile that person 0 reads off of the ruler? We are assuming that person 0’s “1 mile measurement” is the same as person 1’s “1 mile measurement.”
And this is where all of our previous work falls apart. As it turns out, person 0’s “1 mile measurement” is different from person 1’s “1 mile measurement.” This may seem ridiculous and difficult to imagine, but it actually isn’t too complicated (at least compared to time dilation). Just think about that ruler stretching from the rocket ship’s starting position to the star (see Figure 4). To person 1, the ruler would look normal (see Figure 4). To person 0, though, the ruler would actually look squished:
Figure 5: Using a ruler to measure the distance between the rocket ship and the star (from person 1’s perspective)
**Notice how the divisions between the tick marks on the large ruler in Figure 5 are smaller than those on the large ruler in Figure 4**
See, although it may hard to believe, to person 0 (who is moving very quickly), the outside world will look squished. If person 0 had a ruler, it would not seem squished to him — only the outside world that he is traveling past will seem squished (notice how, in Figure 5, although the ruler inside of the spaceship is smaller — otherwise it would not fit in the rocket ship — the distance between the tick marks are the same as the distance between the tick marks of the ruler seen in Figure 4). So, although the ruler outside of the rocket ship may still read 1 mile, person 0 can use his own ruler which would tell him that measurements from the ruler outside are nonsense (because the ruler is squished; would you trust measurements from a squished/damaged ruler?) — “1 mile” on person 0’s large ruler is not the same as “1 mile” on person 1’s large ruler and the “true distance” that the star must travel (according to person 0) is some other value. Believe it or not, in this scenario, person 0’s ruler will actually tell him that the distance between him and the star (and the distance that the star must travel to reach him) is 0.5 miles! Amazing!
**Note: When I say “true distance” I mean the distance when we standardize units of measurement between person 0 and person 1. Person 0 may choose to use the ruler outside of the rocket ship as his main tool of measurement and call the distance between him and the star 1 mile, but according to person 1’s definition of 1 mile, the distance is really only 0.5 miles (in other words, if person 1 could somehow measure the distance between the rocket ship and the star from person 0’s perspective, he’d say that the distance is 0.5 miles because it is half as large as the distance he previously measured when standing on the surface of the Earth). By using the ruler inside of the rocket ship, person 0 is “standardizing the measurements” because that ruler is not “squished” in comparison to person 1’s ruler, meaning that his definition of 1 mile will become the same as person 1’s definition of 1 mile.**
Alright, perhaps I should explain what makes this so “amazing.” Recall in last week’s post how we deduced that person 0’s time measurement for how long it will take the star to reach the rocket ship is only half of person 1’s time measurement for how long it will take the rocket ship to reach the star:
**I apologize for skipping equation numbers — I am using the equation numbers that match up with those used in the previous post**
This caused a problem for us, for when we plugged Equation 4 into Equation 2 we got:
However, we are smarter now and we know that the distance that person 0 thinks the star must travel to reach his ship is 0.5 miles, or 0.5d (remember: d is the distance that person 1 thinks the rocket ship must travel to reach the star; person 1 still measures this distance to be 1 mile because the ruler is not “squished” for him). Therefore, Equation 2 is incorrect, and should really be:
Let’s now plug Equation 4 into Equation 5:
And voila! Using the fact that space itself can “squish,” we have found that the star no longer appears to be traveling faster than the speed of light. This “squishing,” better known as length contraction, has solved all of our problems. Amazing!
Just how we have an equation for time dilation, we can develop one for length contraction — the distance between two points when measured by person 0 will be the distance between those two points when measured by person 1 divided by the relativistic factor (instead of using d for distance, I have written this equation using L for length, which is how it is typically written):
**For those of you who are familiar with the concept of proper length, this form of the equation may be bothersome; I ask that you please accept the “formatting” of this equation for now, at least until we address the notion of proper length**
That’s all for this week. We have now covered the two major concepts of special relativity: time dilation and length contraction. Special relativity is a lot to digest and requires a deep understanding of “perspectives” and “frames of reference.” If you are having difficulty with time dilation and/or length contraction, I encourage you to dedicate some time to just sitting down and thinking about the concepts — they will make sense eventually. Feel free to post any questions you have in the comment section below! Next week, we will be summarizing what we have learned about time dilation and length contraction so we can solidify our understanding of them. See you then!
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