We've talked a fair amount about length contractions. So, what we want to do in this video is think about how length contraction actually could occur. Notice that, interestingly enough, is a topic that physicists still debate. But what we want to do here is think about really what's going on a little bit more and get a little bit more understanding, hopefully, about how it happens. So here we have Bob, instead of on a space ship this time, we'll put him on a metal rod of length L sub b going at some velocity V. And that's the length is measured at rest in Bob's frame. Again in his frame of reference, he's just standing on the rod. He's not going any place. Alice, however, who's observing him, sees Bob and the metal rod going by at velocity V there. This is what Alice observes of course. The length contraction effect we derived several weeks ago. L sub a equals one over gamma L sub b where gamma is our usual factor. Greater than one if you have a non zero velocity. Therefore Alice will measure the length of the metal rod to be less than bobs so called proper length is sometimes called rest length, remember right? It's the length of the rod at rest. Now what we want to imagine lets say that. That the rod accelerates slightly. So instead of being at a velocity v, it moves up to a new velocity v + delta v, just a little bit more. Well we know just from our basic length contraction formula, that that means Alice will see the length contract a little bit. If the new velocity is higher than the old velocity, then gamma is a little bit higher as well, and therefore L sub a will get a little smaller that way. And so what we'd like to think about, you know, how does that occur? So let's think about what Bob has to do in terms of accelerating the rod so that its rest length, its proper length as he measures it stays the same. And you might think, well, let's just maybe attach to the rocket here to the end and turn it on so you get a little bit of acceleration going up to the new velocity. But if you turn on the rocket here it's not going to instantaneously affect the whole rod. It'll push the back a little bit before it gets up to the front. And so we have to imagine a situation here, again a thought experiment, of how Bob could accelerate the whole rod at once so the whole rod moves from velocity v up to velocity delta v. And one way to think about that is to divide the rod into little slices, so let's just diagram something like that, so here's the rod again, roughly speaking, and we'll just split it up into a bunch of slices like that. And then we could attach to the rocket perhaps to each slice. Or we could just have Bob if you really want to be so much fanciful about it. Say he just has a mallet and he needs to at a given instant in time, he hits each little slice with a mallet or at a given instant in time, simultaneously for all of them. He will turn on the little rockets he attaches to these slices so that each slice moves exactly at the same time up to the new velocity v + delta v. So each slice here just gets a little bit of push so it has that extra velocity. And then the whole thing moves as one. And if we can do that, again, a thought experiment here. Then Bob's, the length of the metal rod in Bob's frame would stay the same. It would not change at all. because again, if you're pushing the back first, before you push the front, then the length is going to change. So, the key thing here is that because the metal rod is at rest in Bob's frame of reference, it can't change when we accelerate it or to get it to a new velocity, it's always got to be the same length because these at risk as far as he's concerned nothing really has changed in terms of velocity. So, how might that affect though, what Alice sees so, let's think about this math and for the analysis here we don't have to worry about every single slice. Let's just think about the back slice here and the front slice here. And let's put our typical clocks here. In other words, Bob has his usual lattice of clocks. And so we'll put a clock on the back slice and one on the front slice And of course in Bob's frame those clocks are synchronized. And therefore, what he does if you want to go with a mallet example, where he. At the same instant in time where he hits the rear slice as well as the front slice, or has rockets attached to all of the slices so that right at the same time according to his layers of clocks. The rear slice gets a little pushed from its rocket and the front slice gets a little pushed from its internal rocket there if you want to think about it like that. Again, the idea is we're imagining a situation such that Bob can give a little impetus to it. But that the length is going to stay the same in his frame of reference. That can't change. Okay so that's sort of the set up. Now let's think about what does Alice observe from all this? And the key thing here it goes back to the relativity of simultaneity. Or another way to say that is synchronization is relative to that. If Bob has clocks synchronized in his frame his lattice of clocks we know that Alice observes Bob's clock compared to synchronized levels of the clock. Bob's clock would be out synchronization and vice versa of course going the other way. And another way, we see that of course is leaving clocks lag. So, if this is now so, now we're at V + A little bit more, V + delta-V to get up there. What does Alice observe? Well, she sees Bob's two clocks here which are synchronized to him and that synchronization enables him to give the little push to the back just at the same time as he gives the front a push. But Alice, of course, does not see it that way. She sees, remember the leading clock here? The leading clock lags the front clock. And so imagine that when Bob turns on his little rockets or his section with a little mallet or something just to give a little impetus of acceleration. We know that when that occurs, the two clocks are exactly the same. Now, if we could take a photograph and it would show the rockets turning on or the mallets hitting. And the two clocks identical. But, Alice of course sees this clock the leading clock lag behind the back clock. So in other words, this clock is ahead. In other words of course this clock is ahead of that clock. So Alice will see the little impetus from the rocket, the rocket turn on or the mallet hit, Alice will see this one happen first, and then, a little time later, the impetus given to the front. And so what happens? Impetus given to the back first pushes it in a little bit and therefore the length contracts. And that's a way to think about how length contraction actually works. Pretty much everything we've done in terms of deriving these results, even though we've come a long way since we started. We started with the relativity of time and that of course came from Einstein's principle that the speed of light is constant in all reference frames, inertial reference frames that we've been using. Hopefully that gives you an idea of how things actually undergo length contraction. That it has to do with the relativity of simultaneity. You can keep a proper length, a rest length, constant in Bob's reference frame. But when you need to move it up to a new velocity, then to do that, even though it's synchronized in Bob's frame, it's unsynchronized in Alice's frame. And that means the back gets a little push first and sort of compresses it a little bit, sort of the length contracts before the front. Gets it's little push there. We're actually going to see this concept appear and one of the paradoxes we'll be working on this week. And that is the paradox, we'll call paradox of two spaceships on a rope. So we'll be coming back to this in a little bit.