 Hi, I'm Zor. Welcome to a new Zor education. So we continue talking about Einstein's view on the universe, if I can say so. Well, actually it's about special theory of relativity. Now, we did discuss things related to time dilation. We presented Lorentz transformation and basically confirmed with the Lorentz transformation certain more I would say intuitive calculations about time dilation. Today we will talk about space, basically the lengths as you know the most important characteristic of space I guess is the lengths. So how lengths behaves, how the view onto lengths is different if you are looking at the same thing from two different inertial frames moving relative to each other. Now, this lecture is part of the course called Relativity for All presented in Unisor.com. The website contains not only the lectures, but also textual notes for each lecture, which basically is like a textbook, synchronized lecture, textbook, the chapter or whatever you call it. Also the same website Unisor.com contains prerequisite courses for this one, which is mass for teens and physics for teens. The knowledge which is presented in those courses is definitely mandatory to understand relativity. So you can have it from some other sources, if you don't you can use this website. The website is totally free, there are no advertisements, I don't want anything to distract you from learning. There are exams if you wish, you can take them as many times as you want until you will get perfect score. Also the website contains certain functionality related to supervised study. So let's say your parent or your supervisor or teacher wants to basically direct you as you are studying the material individually. So he can make an assignment and you will just go with supervisory attention. Signing in is not necessary if you are just doing everything by yourself, but if it's a supervised study then yes, you do need to sign in from a supervisor and from yourself to establish the connection, etc. Ok, so we are talking about length transformation. So I did mention that we discussed already the time transformation and we used Lorentz equation for this. I will use exactly the same kind of scenario, so you have two different inertial frames, the alpha frame and the beta frame. And there is an observer in alpha frame which is local to alpha and there is an observer in beta frame local to beta. Now in this particular case we will use, to analyze the length transformation, we will use a rigid rod which is positioned at rest in the beta system and the entire beta reference frame is moving relative to alpha frame with a constant speed across the x-axis. That's exactly the same as we considered before with time. So first of all, let me just remind you the Lorentz transformation. Now if you have alpha frame with x, y, z and time t coordinates and you have beta frame with lower case x, lower case y, lower case z and lower case t and there is a transformation between the coordinates in this particular case considering the beta is moving with speed v only across the x-axis of the alpha system the y and z coordinates are not changing but the time and the x-axis and x-coordinate they do change. So transformation from alpha to beta looks like this, t is equal, so that's the beta time. Alpha time minus v divided by c square times x divided by square root of 1 minus v square into c square. So that's how time is transformed. We have derived this formula in Lorentz transformation lecture. Now the x, again lower case x, the beta would be transformed as x minus v times t. v is the speed, t is time. Now obviously if beta is moving let's say to the right from relative to alpha system then you have to subtract that. But Lorentz transformation adds this already familiar factor which is actually characteristic of relativity. Now y is equal to y and z is equal to z, so these are not changing coordinates. So let's just have this in mind. Now reverse transformation that's actually easy because if beta is moving relative to alpha with the speed v to the right then alpha is moving relative to the beta with the same speed v to the left which is minus v. So basically you can use the same formula just change upper case and lower case and minus to plus. That's it. So capital T would be equal to lower case t plus v divided by c square times lower case x and divided by the same square root and capital X would be lower case x plus v lower case t divided by the same root. And again capital Y is equal to lower case y and capital Z is equal to lower case z. So these are Lorentz transformation from alpha to beta and this is from beta to alpha. And there is only difference here is minus square plus minus plus and these are the same. Now we will use these Lorentz transformation in the same way we were using for time transformation to basically demonstrate time dilation. This is for lengths to also demonstrate how lengths is changing. So I have basically two major cases. One is, I mean two major questions. One is does the length is perceived differently in alpha system relative to the true lengths, the proper lengths of this rigid rod which is located in beta system at rest in beta system and moving with the beta system relative to alpha. So first of all is there a change? Now the second question and we will answer positively. The second question is does it depend on how this particular rod is located in the beta system? You see the movement we have just decided it's stretching relative to the x-axis. So my question is, is rod supposed to be parallel to the x-axis, perpendicular to the axis, at the angle, at the axis, etc. How the change, perceived change of the length of this rod depends on its position relative to the x-axis. So we will consider two kind of opposite cases. One case would be when the rod is perpendicular to the x-axis, which means trajectory and the rod are perpendicular to each other, trajectory of the beta movement in the alpha system. And the second would be the parallel. What if the rod is positioned along the x-axis? How in this case the length will be changed? Okay, so we have two cases. Case number one, perpendicular. Now, in this particular case the coordinates in the beta system of the rod would be from the origin of coordinates, which is 0, 0, 0, x, y and z. Now, the rod is at rest in the beta system, which means position does not depend on the time. And it will stretch perpendicularly to x. And I will choose towards the z direction, if you don't mind, because it doesn't really matter. So it's stretched from 0, 0, 0 to 0, 0, r, where r is the rod's length. These are coordinates of the two edges, two ends of the rod. Now, I also assume that at time 0 for any system, alpha and beta, they completely coincide. And then, as the time increasing, the beta is moving relatively to the alpha with the constant speed v, preserving the parallelism of the axis. So it's just slowly moving uniformly with the constant speed v. Let's consider it to the right. So the v is positive, so it moves to the right along the x-axis of alpha. Okay, so these are coordinates in the beta system. They are independent of time. So for any time t, for any time t, I use this logical symbol for any time t, these are coordinates of the rod. All right, so let's just go and apply the Lorentz transformation straight forward and see what comes as coordinates in the alpha system. Okay, all right. So we will use these r from beta to alpha. And what do we see? t is equal to, so that's called t0. That's this point. In beta, it will coordinate which one? Time t for any time t, basically, right? Plus v over c-square times x-coordinate, which is 0, divided by square root of 1 minus v-square c-square. Okay. Now, x0. This is coordinate of the bottom of the rod, but in alpha system, right? So that's x, which is 0, plus v times t, divided by square root of 1 minus v-square c-square. Okay, y would be equal to y, which is 0, and z would be equal to z, which is 0. So these are coordinates of this point. Now, how about coordinates of the other end of the rod? All right. So the t would be t plus v divided by c-square, and this is x. x is still 0, so it's still 0 here. This is r, t-r. That's the second, the opposite end of the rod. x-r is equal to, again, x, which is 0, plus v-t, divided by square root of 1 minus v-square c-square. Oh, I didn't mention it, but c is obviously speed of life. I'm sure you know about this. Okay, y-r is equal to lowercase y, which is 0, and z-r, which is equal to lowercase z, which is r. So let's talk geometry. The beginning and the end, x-coordinate is the same, as you see. y-coordinate is the same, and z-coordinate is from 0 to r. So in the alpha system, what is the length of the rod? Well, basically it's z, which is length is equal to r. Now, what it means is that if the rod is perpendicular, okay, if this is x and this is z, and the rod is perpendicular, and the beta system, lowercase z, lowercase x, it moves here. Alpha observer is here in the alpha frame at the origin. So no matter what happens, as long as this rod goes along the x-axis, but its length is perpendicular, it does not change, basically. Okay, second case is perpendicular, I mean parallel. So let's consider the parallel case. Parallel case is much more complex, I would say. So first of all, we have to explain how exactly we can measure the length in the parallel case. So let's consider you have one frame, which is capital X, capital Z. It doesn't really matter, actually. And the rod is horizontally along the x-axis, and it's the same, lowercase z. This is the beginning of time when both alpha and beta systems coincide. So what I would do is the following. My alpha observer is in the origin. I suggest the following method. Let's just put this rod here in the beta system from minus r to zero. That's beta coordinates. And then the beta will move. And I will consider two events. Event number one is the beginning of time when t is equal to t is equal to zero, and x, y, z equals to lowercase equals to y equals to z equals to zero. So this is the point I observe my beginning, my right end of the rod. At this particular time, and I know it has these coordinates in all the systems. Now the beta system is moving now with the speed v. And I am watching the left end of the rod. As soon as the left end of the rod passes my alpha observer, which is in the sitting in the origin of the alpha system, that's the second event. So I will have two events. Each event has certain coordinates and timing, and I would like actually to compare the timing. Now whenever I have the time between these two events, if I will multiply it by the speed of the movement of the v, that would give me the length perceived by alpha observer. So this edge of the rod is moving with the speed v towards alpha. So all I know is, I would like to know, is the timing it takes for this edge to reach the origin of the alpha as the beta is moving to the rod. So these are two events, and the first event is obvious. It's all zeros. Timing equals zero, all coordinates equals zero. So that's quite obvious. So let's just think about the second thing. So what is the time from the beta perspective for this point to reach this point? Well, in the beta system, all it needs is with the speed v to cover the distance r. That's it, because the rod is at rest. So as long as we will move the whole beta system to the right by r, my left edge would be at alpha origin. So the timing from the beginning to this particular point reaching my original in the beta system, this tr is equal to r divided by v. That's the timing. Okay, that's what we know. Now what are coordinates of this point in the beta system? Well, the rod is at rest in the beta system. It used to be in the distance r to the left from the beginning. When it moves, it's still in the same position. It's still in the distance r from the origin of the beta system, which means lowercase xr is still minus r. As it used to be minus r, it remains minus r. So whenever this point will reach alpha origin, its coordinate in the beta system would still be minus r. Nothing's changed, right? Okay, I think that's basically sufficient to find tr. So what's the perceived time during which this point will reach my zero? I know the beta time and beta x coordinate. Now I will just use Lorentz transformation and I will get tr, which is r over v plus vc square times x and x is minus r divided by square root of 1 minus v square c square. What is it? Let's do r over v outside. That would be v square. This would be 1. So it would be 1 minus v square c square divided by square root of minus v square c square equals rv square root 1 minus v square c square. Okay. Now if I will multiply this perceived time, it takes for this h to reach the alpha's origin multiplied by the speed which the whole thing is moving times v. So the length, perceived length would be equal to r times square root of 1 minus v square c square. Now this is less than 1 as you understand because all the speeds are less than the speed of light. So the lengths, perceived lengths would be less than the proper lengths of the rod in the system where it is at rest, which means in the beta system. Well, obviously you know this abbreviation. So we'll have L is equal to r divided by gamma. Gamma is greater than 1. This is less than 1. It's reversed and this is greater than 1. So we are dividing by something so this is less. So the verdict is the proper lengths is always greater than the perceived lengths. Proper lengths is the lengths where this rod is at rest. And perceived lengths is how we view. So if I am an alpha observer and the rod is just moving across and it has the lengths of r, I see it in my perception is as smaller. And the faster the rod goes to the right, the smaller I will see it. I will perceive these lengths as smaller. But obviously we should be closer to see to get this r as really kind of significantly less than the proper lengths. So, well, that's what it is. This is the formula of contraction of the proper lengths. If you observe it from some kind of a point and the segment or rod, whatever, which is moving and stretched parallel to the trajectory, we are talking about, you see, it's a parallel case when the rod is parallel to the trajectory of the beta. Only then you will see this kind of contraction. Okay, that's it. I suggest you to read the notes for this lecture. So you have to go into relativity for all. Then the next menu item is Einstein view and then you will find Lang's transformation lecture. So read the notes. There are a couple of pictures there and I think it might actually provide a little bit better understanding. My purpose was to explain it in a relatively rigorous way. Through much less of the intuition but more towards just calculation based on Lorentz transformation which we have proven before in one of the previous lectures. That's it. Thank you very much and good luck.