 Hi, I'm Zor. Welcome to Unisor Education. Today is a little bit unusual lecture. Usually it's all abstract mathematics, etc. Today's lecture is about physics, more precisely about theory of relativity, special theory of relativity. Einstein, many, many years ago, wrote an article called Electrodynamics, where he basically derived the formulas of special theory of relativity, how coordinates are changing from one system to another, moving relative to the first one. Quite frankly, I didn't understand many of the physical aspects of what he was writing about, and I kind of didn't like the fact that he used a lot of intuitive physical thinking in derivation of his formulas. So, as a mathematician, I was thinking about maybe it can be done truly mathematically. Whatever the formulas of special relativity, of transformation of coordinates he could come up with, can it be really just derived very simply? I'm sure many other people did something like this before me, so I would not pretend to be like an inventor, discoverer of anything new. But for me, for myself personally, that was something which was new, and I didn't read about this before, I just did it myself, and I was quite satisfied with the results. Now, why am I basically including this in this particular course? Well, because it's about systems of linear equations. It's a practical implementation of the systems of linear equations to a purely physical problem. So, my task was, from certain physical aspects, derive system of linear equations, and then basically solve this system to derive the formula of transformation of coordinates. And that's exactly what I'm going to do right now. I will try to be as mathematical as possible. However, there are certain physical aspects, but quite elementary actually, which you would agree with me that even a mathematician can actually accept without any kind of a problem. All right, so, let me go, and I would like to start from certain principles, certain axioms, if you wish, on which the whole system of equations is built. So, the first thing is, what are we talking about? We are talking about the following. Consider we have two systems of coordinates, one and another. So, these are upper cases, X and T, and these are lower cases. Now, X represents a space dimension, and T represents time dimension. So, as we know, our physical dimension is basically 3D, we have X, Y and Z, and we can add the time parameter, like a force dimension. Now, I would like to concentrate on the movements within one particular coordinate, X, it's much easier. To extend it to three-dimensional case is basically truly a technical aspect, which I don't want to deal with right now. So, let's consider we have a one-dimensional space and a dimension, one extra dimension, which is the time. So, this is one system of coordinates. Another system of coordinates is moving relative to the first one with a constant speed of 3, as measured in this particular coordinate system. So, if you are an observer sitting here, this particular coordinate is moving with the constant speed V. Okay. So, that's something which we can consider as given. Now, next thing is, let's consider there is certain process which we observe in this particular system, and this process is changing basically. It's object, if you wish. It's changing its coordinate, which is X, and obviously the time also is changing. So, it produces certain trajectory in this particular system. Now, coordinates which we mark capital X and capital G in this particular coordinate system are describing position at time of this particular object in this system. Now, how about an observer which is sitting in this particular system, which is moving, and observes exactly the same process? Well, obviously whatever coordinate X and T where in this coordinate, in this coordinate system will be different in this, because this system is moving. So, the question is how based on just this information that this system is moving relatively to this system with a constant speed V, how the coordinates will look in this system if you know coordinates in this system? Classical Newtonian mechanics actually says very simple. Time is exactly the same, and as for space coordinate, so if this thing has coordinate X, then this system, the coordinates of the same point in this system would be minus VT, right? Because this system is moving, so at time T it will move by this particular distance. So, if you take this particular point, at time T coordinate X in this particular system, in this it will be X minus VT because the whole coordinate will move relative to this one by the amount of VT. So, this is a simple Newtonian approach. Well, great, but now we had a very interesting observation, which was physical observation, the experiment of Michelson, which actually says that measured in both systems, moving and standing still, speed of light is exactly the same. Now, it's impossible in the Newtonian sense, because in this particular case speed, which is actually the ratio between the increment of the distance divided by the increment of time, obviously the speed would be different in this particular case, and the difference would be V. However, if you measure the speed of light in both systems, according to the experiment of Michelson, speed moves with exactly the same speed, light moves with exactly the same speed in both systems. And somehow, it means that these equations, well, either they do not reflect the way how light is actually moving, they are not universal, or we have to, you know, change equations to make them universal. And that's exactly what Einstein did. He came up with a different transformation formula, not this one, but different one, which if used, it will actually show that really speed of light moves with exactly the same, is exactly the same in both systems, standing still and moving. And that's what I would like actually to derive using the systems of linear equation. So, I don't have this, but what I do have is that there is certain object which is light in this case, but it doesn't really matter from the mathematical standpoint. There is certain object, speed of which is exactly the same as measured in both systems. This is another, like an axiom, which I would like to add. So, one axiom is that the system, lowercase xt is moving with a constant speed relative to this one. And the second axiom which I would like to add is the speed of some object is constant, it's c, and yes, physical essence of this object is light, but it's irrelevant. So, there is an object which has a constant speed as measured in this and in this coordinate system. Alright, so that's given. Now let's talk about mathematics. First of all, we obviously assume that at moment g, which is equal to zero, both coordinates are at the same place at the same time. So, in the beginning of the time, we have both coordinates at the same place, which means that if g equals to zero, then x equals to zero, which is the coordinate system. At this moment, lowercase g and x both are equal to zero. That's what it means that both systems coincide at moment g. So, this is also a given thing. It just makes our life easier. What else I'm looking for? Well, I'm looking for transformation of coordinates, not the ones which I have just written, which are Newtonian transformation, which we know are not true for life, but some other transformation. So, basically, from purely mathematical standpoint, I would like to write a transformation of this kind, a linear transformation from capital X, capital T coordinates to lowercase x and t coordinates. So, from constantly positioned, from a standing still coordinate system, transformation into the coordinates of the moving system. And that's what, basically, we are looking for. Now, in the elementary case of Newtonian mechanics, as I was saying before, t is equal to capital T, and lower x is equal to capital X minus dT. But in, in, in spinon, in spinon mechanics, in the theory of relativity, where the light has constant speed c, we are looking for something else. And now, our problem, actually, our task is to find these coefficients. Well, first of all, because of this condition, if T is equal, if capital T is equal to zero and capital X is equal to zero, then the lower capital, lower t and x are equal to zero. So, the systems coincide at the very beginning. What follows is that these are equal to zero. So, we don't have these three constants. That makes our life easier. So, we are looking for this particular system. Now, let's think about what kind of conditions on P, Q, R and S we can impose to basically find these values and that what actually will constitute the transformation of co-ordinate. So, we are dealing with a problem of finding one, two, three, four, four unknown variables. And for these four unknown variables, I would like to put together a system of linear equations. The result of solving which would give me these coefficients. And that's what actually is needed. And that's where the mathematics of it will take. All right. So, we know what we want to find. We don't need these pictures anymore. And if we will, I will draw it again. So, let's think about what kind of equations we need. Now, there is some piece of knowledge which you might not actually be familiar with. It's very simple one. And it's very simple to formulate. It's a little bit more involved to basically explain and maybe in some further lectures I will dedicate certain amount of attention to this particular principle. The principle is as follows. You see, we are transforming co-originates from one system to another. But we don't want something like, let's say, the distance between two different points to be different if I measure it in this system rather than in that system. Now, as a consequence of this, we will probably don't have, we would say, angles to measure differently in one system versus another. Because these linear transformations of co-ordinates can be stretching, can be changing the position, et cetera. So, I don't mind to change the co-ordinates in some way except I don't want the distance between the points to be measured differently. So, if I will measure the distance between two points, let's say, capital X1 and the capital T1 and capital X2 and capital T2, and then I measure based on this transformation the corresponding co-ordinates in the moving system, lower case, X and T. I would like to have the same results. How can I achieve this? Well, there is a very simple rule which is called orthogonality of the transformation. So, transformation should be not deforming and the criteria if you have an equation, system of equations like this to transform co-ordinates from one to another is very, very simple. It's P times S minus Q times R is equal to 1. I don't want to get into why I know about this. There is a whole theory of this. It's known, it's established, well-established in mathematics and I realize that you might not know it. However, just take it as a phase, basically. This criteria is the one and only criteria which is needed for this transformation to be non-deforming. And this actually constitutes my first equation. I have to say this is not a linear equation, but nevertheless the others will be a little bit better, but we are looking for linear transformation of the co-ordinates. Now, we've got this particular equation. That's one. We have four different variables in four equations. All right, let's continue. Okay. Now, let's think about this. Since one co-ordinate system is moving relatively to another co-ordinate system at constant speed V, then let's think about what happens at moment t. So this is our first equation, second equation. Let's talk about moment t and x is our co-ordinate and I'm looking only about point which has a co-ordinate x is equal to 0. So what happens with this particular point, what exactly are co-ordinates of this particular point in the moving system? Well, if moving system is moving at the constant speed V, it means that this point which is the beginning of co-ordinates at moment t would have co-ordinates like this. This is one. This is another. So these are x and t. These are x and t. So what is co-ordinate of this system? Well, at any moment t, at any moment t, its co-ordinate along the x is 0 because it's the beginning of the co-ordinate at any moment t. Now, what is this particular co-ordinate relatively, this particular point has co-ordinate in this particular system? Well, since this is moving with a constant speed V, at the time t, it will be on the distance V t from the beginning, speed multiplied by time. So what I would like to say that this particular point has co-ordinates t 0 in this co-ordinate and capital T V t in this co-ordinate. So what does it mean? It means that our transformation and let me write down again our transformation here. So we were talking about x is equal to px plus qt and t is equal to rx plus st. So if I substitute instead of t and x, I will substitute these which are the time and co-ordinate of the 0 point of the moving system. In the moving system, I will get t and 0. So let's just substitute it and what we will have. Instead of x, I will put V t. Instead of t, it will be t. Instead of this x, it will be 0 and instead of tt. So my system would look like this. z is equal to p V t plus qt and t is equal to r V t plus st. This is something which I don't really need right now but I do need this. So I just factored out t. But now let's think about this way. This is supposed to be true for any time moment t. How can it be done only by having this equals to 0? Since t can be innocent. And this constitutes my second equation. So p V plus q is equal to 0. This is my second equation. p V plus q is equal to 0. This is my second equation. And again, this equation I basically derived based on the fact that the moving system of co-ordinate has a beginning with lower case x co-ordinate is 0. And the same point in the non-moving system has x co-ordinate as V t. Because V is a constant speed at which it's moving. And from this I have derived this particular equation. Now, next. Now let's remember this speed of light being the same in both systems. It's very interesting consideration. What is the speed of light in the moving system? If I have an object which is moving in the space relatively to the moving system, its co-ordinate will be t and c t. Always. So at any moment t, if this particular object, a ray of light, started from the beginning of the co-ordinate, then at time t it will be on this distance from the beginning of the co-ordinate. Now, what will be its co-ordinates in the system which is not moving? Well, that will be its co-ordinates. Because it's exactly the same thing, exactly the same speed, the time would be capital T. And the distance it will travel be c times t. So what's interesting is that this particular and this particular co-ordinates, one in the moving system, another in the system which then still are supposed to be related by the same original equation. So that's basically what I wanted to say. So if x is equal to capital X t, lowercase x is equal to this. So now let's go back to our original system of equations which is x is equal to p x plus q t and t is equal to r x plus s t. Instead of x we will substitute c t with a capital T and we have to get c t with a lower t because this is lowercase x. So we will have c t equals t instead of x c t, capital C t plus q t and t is equal to r c t plus s t. That's what we get. Now I need the relationship between p, q, r and s, right? So what I will do, I will substitute this t instead of this. And what I will get? I will get c times this thing which is r c square t plus s c t equals p c t plus q t. That's what I will get. And obviously t can be factored out and reduced so I will have r c square plus c s is equal to p c plus q plus q. That's right? r c square, yes. So this is my third equation which connects p and q and r and s. We need four, right? We have four independent variables. Now the fourth equation is exactly similar to the third one but instead of light going from the beginning of the coordinates towards this direction, send the ray of light to the opposite direction. Now no matter which direction I will send the ray of light, its speed is exactly the same and its c no matter what. But now the equation would look just very, very slightly different because at the moment capital C, the position relative to this system would be minus c t. The position relative to this system would be t minus c t. So this coordinates are transformed into these coordinates. And we will use exactly the same approach as before. Lower case x which is minus c t is equal to p capital X which is minus c t plus q t. And t is still r minus c t plus s. Now substitute this t into this. What do I get? I get minus c times this. So it would be minus c times this. It would be minus r c square t. Then minus s c t equals minus p c t plus q t. That's what I will get. Now again reducing by t, I will get r c square minus s c equals minus p c plus q. This is my fourth and the last needed equation. Now c is a constant. So I have everything here which is needed to find p, q, r and s because everything else c and v are constants. So all I need is to solve this system of equations which by the way is relatively easy and I will show you how. These are linear, these are linear and these are linear. So three linear equations except one has the power of two basically. So the way how I will solve it is for instance I can express this q as a function of d. q is equal to minus p v and substitute it into this and this. Well before doing that I can actually eliminate something. So let me add and subtract these two together. If I will add my s c would disappear and my p c and minus p c would disappear. So I will have r c square equals q. Well actually two r c square equals two q but I will reduce by two. Now if I will subtract from this I will subtract this. Now my r c square and q would disappear but what is left would be s c and p c. So s c equals to p c which basically means s is equal to p. Now I can use this, substitute it into this and I will get minus p v and what do I have now? I have q, r and s all expressed in terms of p and I can substitute it into the first equation q, r and s and I will get basically a second degree quadratic equation for p. Okay let me try. I hope I will not get lost in these calculations. So instead of s we can put p so it's p square minus q, r, q and r. So q is minus p v, r is minus p v divided by c square and that's equal to one. So this is a quadratic equation which gives me the value of p. Alright let's try to solve it. So we have p square minus minus and minus so it will be still minus. p square v square divided by c square equals to 1 am I right? Looks fine. From which p square equals to 1 over 1 minus v square c square. So I factored out p square. I have 1 minus v square over c square and then I divided 1 over. So p is actually a square root of this. Now if I know p I can basically find all the rest of these. So I don't need this anymore I hope and I'm basically ready to write down the transformation. x is equal to p capital x. So instead of p I will put this. So it's p divided by plus q, plus q t. Now instead of q I will put minus p v, oh this is x obviously and this is t minus p v divided by same, no I'm sorry. Instead of p I have to, instead of q I have to put this. So instead of q I have to put this. Now so it's v over this. Right so I can put minus here and I think this is the formula. Or I can factor out, I don't need this p. This is 1. That's what it is. So p is 1 over this x plus q which is v over this with a minus sign and t. That's the right way. So what I will do here I will do it this way. It's a common denominator so I will put it here. x minus v, x minus v t on the top and I will have this. That's my transformation of x. Now transformation of time. By the way speaking about transformation of x, if you remember Newtonian mechanics had only x minus v t. Now the Einstein's relativity theory adds this particular multiplier and that's what actually makes the whole thing different. This is, this quotient, this factor multiplier is a hallmark of a special theory of relativity. That's what makes speed of light the constant. Now about transformation of time. Well in Newtonian mechanics time is absolute. It's not changing from one system to another. In this case it does change. So it's r x plus s t. So instead of r and s we will substitute its values. So r e is minus p v over c square p is this. So it would be minus v over c square and this multiplier. That's r which is v over c square minus and over this. Right. Plus s t plus s which is p which is the same thing. So it's t. And here we can say that it's t minus v over c square. So it's t minus v over c square times this same famous factor. So this is the answer to our problem. This is how time and space coordinates are transformed in theory of relativity. Again in the Newtonian mechanics we have only t. Lowercase t and capital T are the same because the same absolute. In Einstein's relativity theory this is a shortening of time. So the system is moving and the time is shortening. If you read some popular things about relativity theory you would notice something like paradox of twins etc. And that's actually what explains it. This is the reason, mathematical reason. So this lecture is basically an exercise in how pure mathematics can solve physical problem using a relatively robust and rigorous approach. I basically assumed certain things about coordinate system moving relative to another coordinate system with a constant speed. And what's very important, the result of experiment made by Michael Sun that the speed of light is exactly the same whether it's measured in a constantly positioned system versus moving relative to this coordinate system. Now out of these two, well you can call it axioms if you wish or prepositions, everything else is derived purely mathematical, purely logical without any kind of intuition, physical consideration, thinking, etc. And these are the equations which exactly correspond to what Einstein derived in his famous article about electrodynamics more than a hundred years ago, which was the beginning of the theory of relativity. Okay, so just an exercise in systems of equations. As you saw my system of equations for P, Q, R and S was almost linear, I mean three linear equations in one quadratic. And that gives me a linear transformation of coordinate from one system into another relatively according to the relativity theory. There are notes to this particular lecture. They are at unison.com as usually where all these prerequisites and calculations are listed. I do recommend you to go through these notes again. It would be just a nice exercise and again it's something which is a rare in this particular course, a rare connection to some real life. It's not just abstract math, it's an application of abstract math to a physical problem. That's it. Thank you very much.