 Hi, I'm Zor. Welcome to Unisor Education. Today we will talk about Laurence's transformation. This lecture is part of the course called Relativity for All presented on Unisor.com. I suggest you to watch this lecture from the website from the Unisor.com. You just have to go to Relativity for All course and then it will be part of the Einstein's view. Primarily because every lecture has textual part which is basically like a piece of a textbook exactly related to the lecture. So you always have some kind of a presentation as a video and the textbook where everything is more accurately than what I'm doing on the board written. Secondly, the website has certain functionality like exams for instance. You can take any number of times you want. There are some other aspects of functionality like supervised studying etc. on the website. And the site is completely free. No advertisement, no strings attached. And one more thing the same site contains two prerequisite courses. Maths for teens and physics for teens. Now you cannot really do any kind of a relativity related things without mathematics or knowledge about classic classical physics. Okay, back to business. Now before as part of this course we were talking about Galilean transformation. And we were talking about certain inadequacy related to usage of Galilean transformation everywhere. And in particular, Maxwell equations which describes the electromagnetic field are not really dealing well with Galilean transformation. They're not invariant so to speak. So we were talking about this before. Then another thing, the constant of the speed of light. In every inertial system the speed of light is the same. It's a constant and it cannot be explained within the Galilean transformation framework. So we need something else. Now there were attempts related to ether as the media where light actually is going through. Didn't really agree with experiments. So we need some new approach. And this approach was actually quite well was explained by Einstein in his work in 1905. So the most important thing is that if you have two inertial systems, let's say you have inertial system alpha with coordinates x, y, z and the time t and system beta which I will use lower case as coordinates. And again the Galilean transformation when this system is moving relative to this did not adequately explain whatever was going on, especially with electromagnetic fields. So we need some new way to transform coordinates from one system into another. So we assume that beta system is moving relative to alpha with some kind of a speed v. And for simplicity we think that let's consider only the movement along the x-axis. So this is x, y and z for alpha system. It will be the same. This will be lower case x, lower case y and lower case z. It's moving along the x-system. And at moment t equals to t equals to zero. So whenever we start counting, whenever we start the clocks in both systems, everything basically coincides. So origin of beta is coinciding with origin of alpha. Axes are overlapping each other. So this is one of our beginning point. And then this vector v, if you will explain, if you will put this vector in the vector notation in the alpha system, it will have speed v along the x-axis and zero and zero along the y and z. So vector v which explains basically the movement of the beta system relative to alpha is just directed along the x-axis. So this is what we are having in the very beginning as two systems, two reference frames, and we would like to convert coordinates from one into another. Now, what kind of rules we have, which we have to basically preserve. There are two main postulates, if you wish. One is the principle of relativity, which means all the laws of physics have to be expressed in a quantitative way in a way of equations, exactly the same in all inertial systems. Which means if we have some kind of an equation which is true in alpha system, it should be exactly the same equation in the beta system, but instead of capital X, Y and Z and T we will use lower case. So the movement should obey in exactly the same way. The laws are supposed to be the same way. Another postulate is constancy of the speed of light. So the speed of light is the same in these two systems. These are two axioms, postulates, if you wish, which we have to preserve when we are describing how these coordinates are transformed into these coordinates if beta is moving along the x-axis of alpha. So these are the rules. Now let's try to find some kind of transformation of coordinates from one system into another, which basically obey these rules. And this is, by the way, a purely mathematical task as far as I am explaining. Einstein explained it differently with clocks and some people might not actually find it relatively well understood or something like this. If we follow purely mathematical rule, which I am suggesting right now, I think it would be just, you know, mass is mass, nothing you can do about it. As long as you preserve these two principles, principle of relativity and constancy of the speed of light, well, whatever comes up comes up. You can't really argue with mass, right? Okay, so let's try to do this. Now I did not mention it, but it's kind of silently assumed that empty space, I mean really empty, like no forces, no fields, no gravitation, no air, nothing, is actually homogeneous, which means one piece of this space, if you wish, empty space is exactly the same as another one. Now the same thing I should say with the time. It's kind of an assumption that both time and space are, empty space are homogeneous. They are completely the same in all different directions. So it's isotropy and whatever. There are some terminology about this, I forgot. But in any case, they are the same in all directions and at all places in the universe. As long as it's completely empty, there are no fields around it, no matter, nothing. Now why I am mentioning this? Because I would like actually to concentrate on linear transformation of these coordinates into these. Because if it's not linear, if it's something like quadratic, if x supposed to be squared to get a regular x, it disturbs the uniformity of the space. So linear transformation always preserves this uniformity, homogeneousness. So let's just assume that we are looking for linear equations which transform these coordinates into these. And from the very beginning, as I said, we are moving only along the x-axis. So we can disregard y and z. These y and z is exactly the same as those y and z. And let's just not talk about this. So we will talk only about x and time. Okay? All right. So we have two coordinates which we have to somehow transform from one into another. Well, let's just write down the most general linear transformation, x plus q times t. And t is equal to pq r times x plus s times t. These are the most general, general linear transformations where p, q, r and s are completely unknown coefficients. We are looking for these coefficients and we will try to find them based on whatever the postulates we have, whatever the rules of engagement we have. Now, in theory, there might be constant c and d or something like this. Now, I did not put them y because in the very beginning I said that when time is equal to zero in both systems, the coordinates are equal to zero. So if t equal to zero, x is equal to zero and this x is supposed to be equal to zero. And this t and this x and this t, which means if we will put any constant, non-equal to zero, it will disturb our initial condition. So the initial condition is that alpha and beta reference frames are coinciding in the beginning of time and the beginning of time is the same in both cases. Okay, so this is the most general kind of thing. So let's just think about how to find these pq, r and s coefficients and we will find them eventually. Okay, first of all, let's talk about our x. And this is lowercase x. It's moving with speed v, which means this point, which is origin of beta system. So v is beta and this is alpha. So the origin of the beta system is moving with a speed v along the x axis of alpha system. So what does it mean? Well, it means that if x is equal to v times t, then lowercase x is equal to zero. Lowercase x is equal to zero, that's the origin. So if I will move with a speed v during some time t, I should really find the origin of the beta system. So this is one of the rules, right? How can we satisfy it? Well, let's substitute t in the first equation and we will have p times v times t is equal to q times t. And we know that x in this particular case should be equal to zero. Right? So obviously it's for every kind of time. t doesn't really matter what t is. So we have the first equation. Sorry, it's plus here. Plus. So we have from here q is equal to minus p times v. Okay? We got the first equation. We need four. Right? Okay, next. Next we will do something related to light. So let's just think again. You have this system and this system. And let's say at the very beginning at t equals t equals to zero, we will send a ray of light along the x-axis. Or if you wish we will shoot one photon, which is, you know, the smallest piece of electromagnetic oscillations. Now it will move with the same speed in both systems. Right? So what is basically the equation which describes position of this photon? In one system it would be this equation where c is the speed of light. In another system it will be the same c, but different time, different space coordinate and different time. So one system is moving relative to another, but equation should be the same. That's the principle of relativity. And the constant c of the speed of light is the same in both reference frames. Alright, fine. So let's try to again substitute it into these equations and see what happens. x is equal to p times c times t plus q t. Okay? So what we are going to do is c is equal to r times c times t plus s times t. Okay? We got that. Now since this, so I will multiply this by c and equate these two. So I will p c t plus q t is equal to r c square t plus s c t. Now t we can cancel because that's for every t. And what we have is the equation p c plus q equals r c square plus s c. Okay? We've got the second equation. We are looking for, basically for four different equations for four different variables which we will be able to solve eventually to get the coefficients, to get the real transformation from one coordinate system into another. Okay? Now, let's send the same photon or the same ray of light into this direction. So the speed will be minus c. Okay? What is the equation in this particular case? Well in one system it's minus c t and in another system it's minus c lowercase t. Exactly the same as before. And we will do exactly the same as before. We will substitute this into this. We will have x is equal to p, x is minus c t plus q t. T is equal to minus r c t plus s t. Then we multiply this by c square c and equate and we will have and we will cancel the t in the same way. Minus p c plus q is equal to minus r c square plus s c. Okay? Am I right? Zero equals to minus r c minus p c plus q. Why do I have it reverse? Oh, I know why. Because I should put minus c t. So I should multiply it by minus. That's my mistake. Now I can equate them and I will have plus here and minus here. Okay? Now that's better. Okay? So we have three equations with three unknown. Okay? I'm not going to solve them now because I have only three but what I can do I can express everything at least three q, r and s in terms of p. That would be better. Okay? Now v is already established. I mean q is already established as a function of p. Okay. Now then what? What if I will do with these two? I will add them together and I will have two q equals to r c square which means q is equal to r c square and since q is already expressed in terms of p I will have minus p v equals r c square. Right? Now if I will subtract them I will have q will cancel, r c square will cancel and I will have that two p c equals two s c from which I have obviously s and p are the same. So I have this one. I have q is equal to minus p v. I have this one which is s is equal to p. What else remains? r. r means minus p v divided by c square. r is minus p v divided by c square. So everything is expressed in terms of p. Now this particular system of two equations is kind of natural for mathematician to express it in the vector and matrix form because if this is vector x t, sorry, oh, lower case, x t out is round. It's equal to matrix p q r s multiplied by vector x capital and t capital. Right? This is how it's written in form. p times x plus q times t is x. r times x plus s times t is t. That's how matrix is multiplied by vector. Now instead of q r and s I can really substitute these guys. Right? So let me just substitute it. So that would be p s would be also p and q would be minus p v and r would be minus p v divided by c square. That's my matrix. Okay, great. So this is the matrix of transformation of this vector into this vector. Now let's just think about it. If beta is moving relative to alpha according to these transformations, if it moves towards the increasing of capital X coordinates, at the same time principle of relativity again alpha is moving relative to beta with a speed minus v. Right? Relative to beta. So just from general consideration I can say that if these are, if this is transformation of one into another, then from purely physical considerations and principle of relativity I can say that x t is equal to inverse transformation, which is transformation with a speed minus v relatively to lowercase x and t. And what is how this matrix looks like? If it's minus v instead of v it would be p, p v, p v divided by c square and p. So from purely physical consideration and principle of relativity, if beta is moving relative to alpha with a speed v, then alpha is moving relative to beta with a speed minus v in beta coordinates. Right? So the same type of equation but instead of v we put minus v, which means we change these two signs, we get an inverse transformation. Now what follows from here, what follows is that these two matrices are inverse to each other. So their product p minus p v divided by c square p times this matrix which is p, p v, p v divided by c square and p equals to unit matrix 1, 0, 0, 1. This is my unit matrix from which I can derive the equation from which I can get the p. That's my last equation. Okay? So what is this equation? Alright, we have four different equations, right? Okay, p minus v times this, what will be? p square minus p square v square, right? p times p, no, first, no, p square v, no, p square v divided by c square. p times p minus p v square, right? Minus p v and p v divided by c square. Okay? So first by first. Now first by second row, p times p v would be p square v minus p v times p minus p square v. Okay? Now the second row by first column, minus p square minus p square v divided by c minus p square v divided by c square. Alright? And second by second would be p square v square divided by c square minus p square v square divided by, wait, once more, p v times p square v divided by c square with a minus and plus here, okay? And plus here. p times p p square v divided by c square. That's how it is. And now this times this, second row by second column would be, okay, minus p square v square p square v square divided by c square plus p square. Am I right or am I right? No, something is wrong. p times p again this times this, here minus, minus, minus p v times p v divided by c square. So that's minus. Okay? Now this is zero and this is the same and it's supposed to be equal to one and one. So what do I have as a result? I have p square minus p square v square divided by c square is equal to one. From which p square is equal to one divided by one minus v square divided by c square and p is equal to square root of this. And that's what I wanted to get. This is my last variable which I basically expressed everything through this. Now usually physicists use the symbol gamma for this particular coefficient and as a result we get q which is equal to minus p v which is equal to minus gamma v. Okay? r is equal to minus p v divided by c square which is minus gamma v divided by c square and s is equal to p which is equal to gamma and p is equal to gamma. So we have all my p is equal to gamma. Okay? So gamma is one over square root of one minus v square divided by c square. d is the speed of beta relative to alpha c is the speed of light. All these coefficients are like this. So we can have now the x is equal to minus gamma x q minus gamma v t and t is equal to r which is gamma v divided by c square x plus s t plus gamma t. Why do I put minus here? This is plus p. p is gamma. All right? x is equal to gamma x minus v t and t is equal to t. Oh, this is minus. r is minus. I'm sorry. r is minus here. So just a little bit differently x is equal to x minus v t divided by square root of one minus v square divided by c square. That's gamma. And t is equal to, again, gamma would be, it would be t minus v square c, not plain v, not v square plain v divided by c square x divided by square root of one minus v square c square. So these are transformations of alpha coordinates, capital X and capital T, into beta coordinates, lowercase x and lowercase t. And again, y and z coordinates are exactly the same. Lowercase y is equal to capital case y and lowercase z is equal to capital z. So this is so-called Lorentz transformation. And that's what I wanted to talk about. As you see, it's not exactly such a simple thing. If c is speed of light, and it's very large number, v, if it's not really very large, what if v is just the speed which we are used to move around, then we can say that the ratio v over c is basically almost zero. Now, if it's zero, then you will have x is equal to x minus vt, and t is equal to, if this is zero, just t lowercase. These are Galilean transformation, if you remember. When beta is moving with speed v relative to alpha, the new coordinate x would be just, you subtract this, and the time would be the same. So Lorentz transformation is basically, is changed to Galilean transformation, if the speed v is very, very small. So that's why we didn't really have any doubts when in, let's say, up to 19th century, we used these Galilean transformation because our speeds, our experiments were with a very low speed v and relatively too very high c, especially c square. Obviously, these are negligible factors. Well, that's it. Our purpose was basically to derive these Lorentz transformation. And as I said, if our speeds are not very high, the Lorentz transformation are approximated by Galilean quite well. That's it for today. I suggest you to read the notes for this lecture. Again, it's in unison.com. You choose the relativity for all course, and Einstein's view, the menu, and among them you will find Lorentz transformation. That's it. Thank you very much and good luck.