 Hi, I'm Zor, welcome to User Education. So we continue talking about relativity, special theory of relativity, and right now we are only approaching this particular topic. So we are still at the end of 19th century, and the previous lecture was about reference frames basically. So how do we measure where exactly the object is, and how it moves within this system of coordinates. So we have chosen Cartesian space coordinates, and we have introduced inertial reference frames, which mean that any particular object which has no outside influence from the forces or all forces are canceling each other. So this particular object is moving in inertial reference frame along a straight line with uniform speed. So that was kind of definition of inertial frames. And what other very, very important point was in the previous lecture, that all inertial frames basically are the same in the properties which you can observe. All the physics laws are exactly the same. And as an example, if you are in a moving, very smoothly moving train car along a straight line with constant speed, you will not be able to detect that you are moving relative to, let's say, other objects outside of the train. So all the physics are the same. It's called principle of relativity. And what's important, this principle of relativity is basically taken into the next step into theory of relativity. So the principle of relativity is basically accepted as a true principle. And in this particular case, all the inertial systems are exactly the same. Okay. Now we will talk about two different inertial systems which are moving against each other. And the movement is supposed to be the longest straight line with constant speed. And we will choose a point, we will choose the way how they are moving these systems so that their axes are always parallel to each other. So we will assume that at time, sometime zero, both systems are coinciding. So one system will be called capital X, Y and Z, but they will drop Y and Z and the time will be measured as T. And another system, which I will put here, I will use lowercase letters. So this is another inertial system which is moving along the x-axis. So x and x are coinciding. At point time zero, these points coincide. So the systems are at the same position, basically, at time zero. And then we assume that this system, we will call this one alpha and this one beta. We are assuming that the beta system is moving along the x-axis with constant speed V. So now our point is that somehow knowing where exactly the point is in one system, this is point P, let's say. We know its coordinates and we would like to find out its coordinates after certain time T in another reference frame, in reference frame beta. So knowing alpha coordinates, we would like to find beta coordinates. Now, why this is important, we will actually see later. So for now, let's just assume that this is our task. Let's concentrate on this task. But it does have a purpose, I promise. So how can we determine the coordinates? Now, the other two axes, Y and Z, they are basically different. They're three-dimensional space plus the time. But I wanted the time to be represented. That's why I dropped other two-dimensions, Y and Z. And they're assuming that since all the axes are parallel to each other and there is no movement along Y or Z, movement is only along the x-axis. So these two systems are completely coinciding in the beginning and then moves along the x. So Y and Z remain exactly the same. So I can basically tell that one more very important thing is, we are assuming that the time is absolute. It's universal. There is no difference in time progressing in two systems. Now, that is something which will be changed as we will go into the relativity part. But before the relativity was actually introduced, time was considered absolute, which means if we are standing, let's say, on the station and we are moving in the train, which is moving very smoothly, it's inertial, two inertial frames. Now, the time will be exactly the same in both cases. So if the watch of one person is the same here and here, when it was the beginning of time, the watch is going with exactly the same speed here and here, so the time is exactly the same. So always at all the time, t is equal to t, capital T and lowercase t, this is time in alpha, this is time in beta, they are the same. What about Y and Z? I have already told, they are not changing, they are the same because moving only along the x-axis, so the only x coordinate is changing. How? Well, it's actually very simple. Let's assume that the p point p is stationary in reference frame alpha. Now, it has certain coordinates. Again, Y and Z not really important, what's important is this coordinate xp. Well, which means that this distance is equal to xp, right? Now, if this system beta is moving with constant speed v, it means that after certain time t, this distance will be equal to v times t, right? This is speed, this is time, so this distance, the origin of coordinates of the beta system moved by this particular distance from. So, what is the coordinates of the p in this system now? Well, since we moved to the right, coordinate of p would be xp of t would be equal to original xp minus v times t, right? So, we are interested in this distance. Well, it's negative, right? So, we are subtracting, not from this, subtracting this, but from this subtracting that. If point p is a little bit further, it would be a little bit better understood if you wish. Let's say p is somewhere here. Now, this distance is xp and this distance is v times t. So, this distance, which is coordinate, x coordinate of point p in this system would be, again, xp minus v times t. So, this is the transformation of coordinates. It used to be xp, which is stationary, it doesn't change with the time, but this one is changing with the time. Okay. Now, what other coordinates are changing? Well, I just put it this way. These are not changing and time is the same. So, this is transformation of coordinates from one inertial system, from alpha to beta. Okay. Now, we just assumed that the point p is stationary in alpha. What if it's not? What if it's moving with the time? Does it change anything? Well, if position p is actually a function of time, basically all we have to do is put here, sorry, capital T. It doesn't really matter that it's moving. It's still exactly the same thing. Original position and the x is changing by vt. So, if p have moved to somewhere else, at point capital T, which is exactly the same as lower kt, because we assume that the time is exactly the same, it doesn't matter that it moved. Wherever it moved, since origin is moving by vt, v times t, then we have exactly the same situation. So, now, one more little thing. Do we really depend on a concrete point p? Well, it can be any point. We didn't really have anything specific about this point. So, in general, I can say that x of time t is equal to capital X of time t minus v times t and these guys remain the same and t is equal to t. So, this is transformation of coordinates. Now, what are the coordinates? Well, coordinates in the alpha are t, x, y, and z. Coordinates in beta are lowercase, t lowercase, x, y, and z. So, from these four, we have transformed into these four. This is called a Galilean transformation. So, this is the transformation of coordinates, which was kind of obvious, very much understood by everybody. It seems to be very natural. It corresponds to our intuition, if you wish. There is absolutely no contradiction in this. So, this is called a Galilean transformation. Now, what about this Galilean transformation? Now, what's interesting is it's actually reversible because, again, think about this. Inertial frames are exactly the same in sense of physical laws. So, if inertial frame beta is moving with the speed v relative to inertial frame alpha, it means that inertial frame alpha is moving with the speed minus v relative to frame beta, right? And there is absolutely no difference, which means that all these laws are completely reversible. Instead of v, I will use minus v. And what will I have in this case? I will have that capital X of t is equal to lowercase of t plus v times lowercase t. And capital Y is equal lowercase y. Capital Z is equal to lowercase z. And capital T is equal to lowercase t. Exactly the same except minus and plus. Because beta moves with the speed v relative to alpha, but alpha moves with the speed minus v relative to beta. So, these two transformations are completely reversible. They are inverse to each other. Now, this kind of adversity, if you wish, can be very easily expressed in vector and matrix form. So, let me just do this for completeness. So, let's first have transformation from alpha to beta. So, we need a vector t, x, y, and z to be product of certain matrix of transformation times, as matrix times vector, the alpha coordinates. Now, these are all functions of time, of course. So, what's the matrix here? Well, this matrix is very simple, 0, 0, 0, 0, 0, minus v, 0, minus v1. 0, 0, 0, 0, 0, 0, 0, 0. Okay, let's see if it works. t is supposed to be product of this row by this column, right? 1 times t is t, 0 times x, y and z would be 0. So, t is equal to t. Next, x of t minus v times t plus 1 times x of t. So, that's exactly what we need. x of t minus vt and 0 the rest. y times this. So, this times this. Lowercase y would be 0 times t, 0 times x and only y will be here and z will be 0. And same thing with 0. So, this is exactly correct product of matrix of transformation by the vector of coordinates in one reference frame. And that gives me coordinates in another reference frame. Now, considering v would be an opposite movement. So, in theory my reverse transformation matrix should look very similar. So, that should be t, x, y and z of t vector. It should be equal to, okay, what's the matrix? Matrix should be exactly the same except I should have plus v instead of minus v, right? Let's check it out. Times t, x of t, y of t and z of t. Okay. Now, 1 times lowercase t and 0 the rest. So, product of this row by this column that gives me my first element. Now, product of this times this, v times t, 1 times x and 0 gives me x. And the same thing the rest of this. So, these two matrices are supposed to be inverse to each other. Well, which means what? Which means if we will multiply them as matrix by matrix, we should have a unit matrix with only 1s on the main diagonal. As an exercise, please do it and you will see that exactly this times this will give you the unit matrix. All 0 except the main diagonal which we will have once. And v will be canceling each other, okay? And also, it's very interesting, the determinant of this matrix is 1, which actually means the matrix is orthogonal. It preserves the matrix. Okay. These are all mathematical kind of considerations because, you know, it's actually a matrix of transformation. That's what basically it is. From one system of coordinate to another, it's a linear operator if you wish. But in any case, physicists don't really like to do this type of things. They don't really think much about these inverse matrices, etc. But it doesn't really matter. The Galilean transformation can be expressed as this matrix, as two inverse matrix from alpha to beta or for beta to alpha. And it's really very, very simple. Only the x coordinate is changing because we are moving only within the x direction. And it's changing very, very simply. The coordinate is equal to capital lowercase x, in this case plus v times times. Or if it's lowercase from uppercase, it's the minus v. Okay. So that's basically it. I wanted to introduce just a concept of what is Galilean transformation. This concept was completely dominant until Einstein actually started working on this. Not that people did not understand how different coordinate system can be differently transformed into each other. Definitely, there was a very, very important transformation, which is not this one. It's called Lorenz transformation. The famous physicist Lorenz actually invented it for one purpose only. But that would be the subject of the next lecture. Not about Lorenz, but about Galilean transformation, which actually was not exactly what people expected it to be in certain cases. And that's when Lorenz transformation came to replace it. And that would be already used in theory of relativities. So this is it. That's for introduction into Galilean transformation. Please read the notes for this lecture. So you have to go to Unisor.com, the course called Relativity. And it's within the first part of this course where I'm just talking about principles of relativities. And it's called Galilean transformation lecture. Thank you very much and good luck.