Hey, everyone. Welcome to this next episode of your daily equation. In the last episode, we spoke about the impact of motion on the passage of time. And remember, it all came from the constant nature of the speed of light. If speed, according to Einstein, has strange properties at high speeds, namely near the speed of light, then since speed is nothing but space per time, then we learned that space and time have weird properties, and we worked out the weird properties of time in the last episode. Today as the counterpart to time dilation, what we did previously, we’re going to talk about the weirdness of space, which yields the equation as we will see, that is called length contraction, or Lorentz contraction. Lorentz after a famous physicist, who actually, strangely enough, even though we’re focusing on Einstein here, he actually came up with this equation first.
He didn’t completely interpret it correctly, and that’s really why these ideas are deeply associated with Einstein, but other people were thinking about these ideas as well. Okay, so let’s get into it, and I’m going to describe length contraction by using a concrete example first. But before I show you that little animation, let me just give you the basic idea, and then we’ll try to derive it first, intuitively, through animation, and then I’ll write down some equations that will capture this rigorously, mathematically. Okay. What’s the basic idea? The basic idea is, if I am watching an object race by me, and the canonical example that we will use is a train. If I watch a train race by me, and say, you are on that train, you will measure the length of the train, say, and get a particular value. If I then measure the length of the train that rushing by me, I will get a smaller value.
A shorter length only in the direction of motion. Lengths are contracted along the direction of motion according to an observer, in this case, me, watching that object in motion. That’s the basic idea. And how are we going to understand this? Where does it come from? Let’s get into a concrete example, and in fact, I’m going to use that example of the train. Let me bring up some animations, I think that will help make it clear. So, imagine that the train is rushing by me, but let’s focus upon you first. Imagine that you are on the train. That is you, generic you, right there. And how would you go about measuring the length of the train? Well, you pull out a tape measure, and you simply go from one end of the train all the way to the other end of the train, and you’d read off.
In this particular case, these numbers are completely made up. It’s 210 meters, according to your tape measure. How would I go about measuring the length of the train as it rushes by me? Well, I can’t really use a tape measure, at least not in any conventional way, because the train’s rushing by me. So, as I bring the tape measure up to the train, that’s going to rush away and I won’t be able to do the usual approach to measuring the length of an object with a ruler, with a measuring tape. Instead, there’s something clever that I can do, which is this. If I have a stopwatch, and if I know the speed, the velocity of the train along the track, here’s what I can do. As the train approaches me, right when the front of the train passes me, I turn on the stopwatch.
Okay? I let the walk go until the caboose, the very end of the train, goes by me, and then I click, I stop the watch. So, I get the elapsed time, from my perspective, that it took the train to rush by me, and then I simply use distance is velocity times time. I know the velocity of the train. I know the amount of time that elapsed between the front of the train passing me and the rear of the train passing me. I simply multiply those two together to get the length of the train that I would measure. Show that in a little visual here. So, there’s me, and there is where I’m going to stand. And when the front of the train passes me, I start the watch, I let it tick along. And then finally, when the back of the train passes, click. I stop the watch.
In this case, I got, say, 5.9 seconds. If the speed of the train was 30 meters per second, I would simply multiply those two numbers together. And the claim is that when I carry out that arithmetic, I will get a smaller number for the length of the train then you got using the tape measure approach. Again, these numbers, completely made up. This is not the amount of contraction at a slow speed of 30 meters per second, I should say. So, it’s really just illustrative of the qualitative effect that the length of an object in motion will be shrunken. Okay, so that’s the basic idea. Now, how do we argue for it?
And there are many ways that we can go about this, but the simplest is to make use of what we already derived, time dilation. And simply by using our earlier understanding of time dilation, we can get this result that I will measure a shorter length of the train. So, let’s do that. Again, I’ve got my handy iPad here to do that, and this should come up on your screen if the, yeah, the technology seems to be working. So, what did we learn about time dilation? Well, we learned that when someone is looking at a clock in motion from their perspective, they will say that that clock is ticking off time slowly compared to their clock. Now, I’m going to do something a little bit strange right now. I’m going to take your perspective on the train and consider Delta T according to you, versus the Delta T, the amount of time that you will claim elapses on my watch.
The reason why I’m doing this perspective, I’m looking at things from your perspective first, is a little bit subtle. Let’s do the calculation and then it’ll indicate why I had to do it this way for this particular derivation. But Delta T, right, the amount of time that will elapse on your watch compared to Delta T on my watch. We know the answer to that. You will say that more time elapses and you know the factor by which it is going to be greater. It’s one over the square root of one minus V squared over C squared from last time. In other words, the amount of time that elapses on my stop watch, compared to the amount of time that would elapse on your watch measuring the same events, would be given by squared of one minus V squared over C squared, times Delta T you.
So, less time on my clock compared to your clock. Why is that relevant? Well, if I consider the length of your train according to me, that’s my measurement of the length of your train. What am I doing? Well, as we described in that little animation, I’m taking the velocity of the train times the amount of time that goes by on my stopwatch. But now using the relationship between time according you, time according to me, I can write this as V times square root of one minus V squared over C squared times Delta T you. And then we know that if we write this essay, just move this guy over. One minus V squared over C squared, V Delta T you. This combination over here is just the length according to you, right? And therefore, length according to me is square root of one minus V squared over C squared times length according to you.
And so, there you have it, right? Because this factor over here, let me actually give it a little color to distinguish it. This guy over here is a number that will always be less than one, because it’s the reciprocal of gamma. In fact, I can write this if I would like as, equal to L U, divided by gamma. Gamma is always bigger than one, now that I’ve put it upside down there. And therefore, the length according to me will be less than the length according to you, who measures the length of the train while being on the train itself, being stationary with respect to the train. So, that’s the little derivation that the length of the train according to me will be less than the length of the train according to you.
Why do I have to play this funny game of going to your perspective, watching my clock, you might wonder. Well, couldn’t the person on the platform, namely me, say that the clock on the train is running slow and wouldn’t that give us the reverse result? If you think about it, if we tried to play this same game by using clocks on the train, as opposed to a clock on the platform, we’d have to make use of two such clocks. Because as your train is rushing by me, you could start your watch as you pass me, but you wouldn’t then pass me again to stop the watch.
Instead you’d need someone situated at the back of the train to click off when that person passes by me. There’s an asymmetry there, so you need to have two clocks in the train, and that yields a subtlety that we will come back to in one of the subsequent discussions. And that’s why I didn’t do it that way. So, this slightly circuitous approach, where I go from your view of my clock to my view of your length, is actually the shortest way to get to the result that we just derived. Now again, as with all things in special relativity, the effects are small in everyday life, because the factor of V over C is usually incredibly tiny, and therefore this gamma is often very, very close to one. It is very close to one at small speeds, but large speeds, it can make a really big difference.
So, let me just show you an example. Imagine that you have a taxi cab that is streaking down 5th Avenue in Manhattan at a speed very near the speed of light, and you’re watching this very fast moving taxi cab. What would that look like? Well, let me just show you a little animation of it. Of course, we’re imagining that the speed is close to the speed of light. That’s a little hard in everyday life, but we can do it in animation. And look at that taxi cab, isn’t that strange, right? The taxi cab is shrunken in the direction of motion only. The height of the taxi cab is unchanged. It’s that its length has been squeezed down by this factor of gamma. Now, you note something else. If you look at that picture a little bit more carefully, it’s not only that the taxi cab is squeezed along the direction of motion, it’s also twisted a bit, right?
We’re seeing the back bumper at a funny angle, relative to what you might expect. And the reason for that is that we are in a situation with relativity where there’s a difference between what’s actually happening out there in the world and what we perceive when we consider the rays of light bouncing off of an object. And if you consider the rays of light bouncing off of the taxi cab, you’re actually seeing the taxi cab at different moments in time, different points on it, because the light from different locations on the taxi cab have to travel different distances to your eyeball, and therefore you’re not seeing the taxi cab, the whole thing at one instant of time. You’re seeing different points on the taxi cab at different moments in time, dependent on how far away those points in the taxi cab are from your eyeball.
And when you take that complexity into account, you get that interesting twisting effect that you’re seeing in the animation. But the bottom line of what’s actually happening to the taxi cab from our perspective is what we derive mathematically, it’s length in the direction of motion is being shrunken by a factor of gamma. Now, imagine that you were inside of that taxi cab. How would things look from your perspective? Well, from your perspective, the taxi cab is not moving relative to you. In fact, as we’ve emphasized, if you’re moving at a fixed speed in a fixed direction, you can claim to be at rest, and it’s everything else that’s rushing by you in the opposite direction. So, from your perspective, it’s life as normal inside of the taxi cab. And if you look out the window, it’ll be the outside world that has all this weird stuff happening with lengths being contract, and again, based upon the light travel time, interesting twisting and curving from your perspective.
So, let me show you that alternative perspective. Here it is. So, there you are inside the taxi cab. Everything appears normal inside, but look at what things look like on the outside. Things are shrunken, they’re twisted, because of the weirdness of the rate at which different clocks are ticking and the different distances that the light has to travel, all folded into this length contraction in the direction of motion. So, that’s the bottom line of how motion affects space, shrunken in the direction of motion. The other perpendicular directions are not influenced at all. And as we’ve seen, we actually were able to derive it from our understanding of how clocks that are in relative motion will tick with respect to one another. Okay, so that’s today’s daily equation. Keep in mind that the length, me, being equal to length of you, divided by gamma, you have to interpret what these symbols mean.
It’s the length according to me of your length, as measured with respect to a stationary object. You are on the train itself, but if you keep the symbols in your mind straight, we now understand the relationship between time for you, time for me, length for you, length for me. I think next time we’re going to take up, I think I’m going to look at maybe relativistic mass, or the relativistic velocity combination formula. See, as I go forward, again, love to hear more of your suggestions, which I’m keeping a list of, and as we go forward, I’ll try to incorporate your suggestions into the equations that we discuss. Okay. But that’s it for today. That is your daily equation. Look forward to seeing you at the next episode. Take care.