We have talked of fair amount about length contraction so, what we want to do in this video is think about how length contraction actually could occur. Now this interestingly enough is a topic that physicist 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, how it happens. So here we have Bob, instead of on a spaceship this time, we'll put him on a metal rod of length elsa B going at some velocity, v. And that's, the length is measured at rest from Bob's frame. So a, again in his frame of reference he's just standing on the rod, he's not going any place. to Alice however, who's observing him, sees Bob and the metal rod going by at velocity v there. So, This is what Alice observes of course, the, the length contracture effect we've derived several weeks ago. L sub A equals one over gamma L sub B, where gamma is our usual lorentz factor greater than one if you have a non zero velocity. And therefore, Alice will measure that the length of the metal rod. To be less than Bob's so called proper length or sometimes called rest length, remember and that was the length of a rod at rest. Now what we're going 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 plus Delta V, just a little bit more, okay? Well, we know, just from our basic 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 like to think about is 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 a little rocket here to the end and turn it on so we get a little bit of acceleration going up to the new New velocity, but if you turn on the rocket

here, there's, 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. So, we have to imagine a situation here, again a thought experiment, of how Bob could. Accelerate the rod at once, so the whole rod moves from velocity V upto velocity Delta V. And one way to think about that is to divide the rod into little slices. So, let's just diagram soemthing like that, so, here's the road again, roughly speaking. And I'm going to just split it up. Into a bunch of slices like that. And then we could attach a little rocket perhaps, to each slice or we could just have Bob, of you really want to be [UNKNOWN] about it, say he just has a mallot and he needs to, at a given instant in time, he kicks each little slice with a mallet or at any given instance in time, simultaneously for all of them. You will turn on the little rockets that he attaches to each slice. So that each slice moves exactly at the same time. Up to the new velocity v plus Delta v. So each slice here gets just a little bit of push. So it has, that extra velocity. Then the whole thing moves as one. And if we can do that, again, a thought experiment here. Then Bob's, The length of, of the metal rod in Bob's frame would stay the same. It would not change at all if you're, you know? 'Cuz 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 get it to a new velocity. It's always go to be the same length because he's at rest. You know, as far as he's concerned nothing really has changed in terms of, of the velocity. So how might that affect though what Alice sees? So let's think about this a minute, and for, for the analysis here, we don't have to worry about every single slice.

Let's just think about the back, the back slice here and the front slice here and let's put our typical clocks here. In other words, Bob has his, 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 it does if you want to go with a [INAUDIBLE] example where you know, had the same instance time he hits the rear slice as well as the front slice has got rockets attached to all the slices so that right at the same time according to his laws of clock. The rear slice gets a little push from its rocket, and the front slice gets a little push from, 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, 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 of this. And the key thing here it goes back to relativity of simultaneity, or another way to say that is synchronization is relative. That if Bob has clocks synchronizing his frame, his lattice of clocks, we know that Alice, if Alice observes Bob's clocks, compared to her own synchronized lattice and clocks. Bob's clocks will be out of synchronization, and vice versa. Of course, going the other way. And another way we see that of course is leading clocks lag. So, if this is now, so now we're at V plus. A little bit more, v plus delta v, what does Alice observe. Well, she sees Bobs two clocks here which are synchronized to him and that synchronization enables him to give a little push at the back at the same time 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 rocket or hit, hit this section with a little mallet or something just to give a little impetus of

acceleration. He, we know that when that occurs, the 2 clocks are exactly the same, you know. We could take it 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. Another way to say it, of course, this clock is ahead of that clock. So Alice will see a little impetus from 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? The 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. Like pretty much everything that we've done in terms of deriving these results even though we've come a long way since we started. We started with a relativity of simultaneity and that of course came from Einstein's principle that the speed of light is constant in all- Of, reference frames. All inertial reference frames that we've been using. Okay, so hopefully that gives you an idea of how things actually go, undergo length contraction. That is 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. It's, you know, 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, so, it compresses it a little bit. Contr, so the length contracts before the front gets its little push there. We're actually going to see this concept appear and what the paradoxes were, we'll be working on this week. And that is the paradox, what we call the paradox of two spaceship on a rope. So we'll be coming back to this in a little bit.