Hey, guys. Now we're going to start talking about the second consequence of the second postulate of special relativity, which is length contraction. Alright? Let's get to it. Now, because time is measured differently in different inertial frames, this is actually not its own consequence technically. It is just a consequence of time dilation. Okay, because time is measured differently in different reference frames, length is also going to be measured differently in different reference frames, and this fact is known as length contraction. Okay. So we had time dilation, which said that if you measure time in the proper frame, time in the non-proper frame is going to be dilated. Right? Time is going to be longer. What length contraction says is if you measure the length in the proper frame, the length in the non-proper frame is going to be contracted. It's going to be shorter. Okay? So just be on the lookout for that—that your contracted lengths, your non-proper lengths, should always be less than the proper lengths. Okay? Now, in order to understand where length contraction comes from, we need to imagine measuring a rod in two different ways. Okay? First, we're going to imagine measuring it in its proper frame, which means at rest with respect to the rod. Okay? At rest with respect to the distance that we want to measure.
Now, because the frame that the rod is in is moving, we want to imagine a clock that is stationary in the lab frame moving past the rod. Okay? Because if the clock is stationary in the lab frame and the rod is moving past it, that's the same in the lab's frame, in the proper frame, as the clock, right, which I'm holding in my right hand, moving past that length. Okay? And basically, all we're going to do is we're just going to click the clock when we pass one end, let it pass the other end, and click it off. So it's like a stopwatch when it clears the other end. So we're just measuring how much time is elapsing as the clock passes. And given that time, we will get some measured length, okay, based on how quickly the rod is moving. Now, in the lab frame, instead of having a moving clock, the clock is stationary. Remember that the clock was always stationary in the lab frame. Only when we are in the proper frame of the moving rod does the clock appear to be moving. Right? Now, the clock is stationary, and the rod itself is moving past the clock. So, the same exact idea. The rod is moving at the same speed, \( u \), that the frame was moving, the proper frame. So, this rod is going to pass the clock, and we're going to click it on when the rod just approaches the clock, start measuring time, click it off just as the rod leaves. And we're going to measure a different time. Right? Because the time is different between the proper and the non-proper frame. Right? We have time dilation. So, those two times that we measure have to be different.
Now, if you actually work through the equations, you get that the length in the proper frame (remember the proper frame is the proper frame for the rod, which means that the rod is at rest)—the non-proper distance, right, the non-proper length, is the one measured in this case in the lab frame. And if you put them together, you're going to get something that looks like this: L'=L0γ. And remember that because the Lorentz factor, γ, is always going to be larger than 1, the contracted length \( L' \) is always going to be less than the proper length \( L_0 \). Okay? This is the opposite logic for time dilation: in dilation, you get this equation with γ in the numerator. Since γ is always greater than 1, dilated time is always larger than proper time. For length contraction, because γ is in the denominator and γ is always larger than 1, you always get a smaller non-proper length. Right? A contracted length. Okay? Very simple problem here to get us started in length contraction. A spaceship is measured to be 100 meters long while being built on Earth. That means that that is the proper length. While it's being built on Earth, we're assuming that the people who are building it and measuring it are at rest with respect to the spaceship. Why would they be building the spaceship as it flew by them? Right? That doesn't make any sense. So that 100 meters should be the proper length. Now, if the spaceship were flying past somebody on Earth, they would measure the contracted length, the non-proper length of that spaceship, because now that spaceship is moving past the observer at some speed. Okay?
First, let's just solve for γ: 11-u2/c2. And like most problems, \( u \), the speed, is given in terms of the speed of light. Right? 10% speed of light means that \( u \) is 0.1 times \( c \). So this is one over the square roots of 1 minus 0.1 squared, and this is going to be 1.005. Okay? And then this leads us to the conclusion that the contracted length, which is 100 meters over γ, is actually going to be 99.5 meters. Okay? So half a meter short, shorter than it was. Right? Basically half a percent shorter in length going 10% the speed of light, which is very, very, very fast. You only get a half a percent of drop in length. Okay? Alright, guys. That wraps up this video on length contraction. To the object. And then applying length contraction, super easy. Alright, guys. Thanks so much for watching, and I'll see you guys probably in the next video.