 Hi, I'm Zor. Welcome to Inuser Education. Well, let's continue talking about zero relativity. And today we will start a new chapter, which is called Minkowski view. Before I presented the Galilean view on zero relativity, well, on physics, basically, on the world and the universe, then Einstein view, and this is Minkowski view. Now, Minkowski view basically is just a continuation of Einstein. Einstein's view was drastically different from Galilean view. Minkowski is basically the same as Einstein, just viewed from slightly different angle, more geometrical, even if you prefer. Now, this lecture is part of the course called Relativity for All, presented at Unisor.com. This is a completely free website, no advertisement, even signing is not necessary. There are two prerequisite courses, of course, is mass proteins and physics proteins, which are prerequisites, as I said, for this course. Relativity is more advanced. So you have to know basically everything, whatever is in mass and physics. And that includes in mass, for example, obviously, calculus, vector, algebra, et cetera. And physics is, well, it's very, physics course contains some very important topics about electromagnetic field, Maxwell equations, et cetera. Okay, now, the chapters before this, like especially the Einstein's view chapter, contains lots of information, which we consider is to be known already. These are part of the same course, which are obviously as well prerequisite to this lecture. Okay, so now let's talk about Minkowski's space time. Now, let me start from something very simple. If you have a plane and you have a point moving in a circle, can you tell that this particular picture represents the motion? No, it represents only the trajectory. Because along this trajectory, we can move either uniformly with the same angular speed or with a different speed or even in a different direction, and the trajectory would be the same, a circle. How can I represent a motion much more completely in a graphical form? Well, here's how. Let's say I have x and y axis on the plane. I will introduce a new time axis, and for every moment in time, I will put point in a three-dimensional now space, retaining x and y as they were here. So let's say at time zero I was here. So this is the point. Let's say point times equal to one, I was here. So I have to move from this point to this. And then basically if you will continue, if it's a regular movement, circular movement, you will see the helix as full representation of this movement. If I'm moving in this direction one circle, which is this, and then I start moving to another direction, my helix would be in a different shape. But in any case, whatever my movement is, the curve, in this case, helix or anything like that, would be different and it will completely represent my motion. So this is something which has been known before Minkowski obviously that this adding the third dimension after x and y, adding the time, would be beneficial to graphically represent the motion. Now, here's a little problem. Our world is three-dimensional, like x, y and z. If I will add the time, it will be four-dimensional and they cannot visualize it. So that's why people in their examples, in their lectures, in their textbooks, et cetera, they were drawing only two-dimensional movement plus third-dimensional time because it can be represented and viewed. But in formulas, they were using all three-dimensional x, y and z, and the time t. So that's what we will do as well. So for visualization, we will use just the plane movement, but as far as the formula would be concerned, of the distance, I will use all three-dimensional. And that's kind of understood why. Let's just do a couple of examples of this thing. So let's say my point is at rest. Here's my x, y. And my point is somewhere here. x0, y0. How, in a three-dimensional space plus time, the graph of the motion in this case, well, motion quotes obviously because it's at rest, how it will look. Well, at any given time, the point is exactly at the same place. So x and y coordinates will retain and time will grow. And the straight line parallel to my t-axis, which projects onto the point. So the x of t would always be x0, y of t would be always y0, and t would be basically t. I can probably use a different letter here. I will use, for example, tau, tau, tiot, tau, is equal to tau. So that basically kind of gives me coordinates based on some parameter, tau, whatever the parameter is. It can be from minus infinity to plus infinity. Minus would be corresponding to past, and plus would correspond to future, and tau is equal to zero is current moment. Okay, that's nature. All right, now, what if situation is slightly different? Let's say, starting from this point, x0, y0, I go along some straight line in a uniform movement. So constant speed, constant direction, constant velocity. That's how it was before. Here I am at point t equal to zero. How this particular be represented in the space plus time. Well, let's start with x and y coordinates. Since I'm moving along the straight line with a constant speed, this would be something like this. And this would be something like this. So I'm adding a speed component, speed along the x, speed along the y, and that's how my movement is done. At t equals to zero, I'm here. At negative t I will be here. At positive t I will be here. Now, how will it look with addition of the third dimension? Well, very simple. It will just grow. This projection, obviously, onto my real trajectory. So this is a graph in a three-dimensional space plus time. But obviously it's a straight line because this is the equations which define straight line in 3D. And obviously the projection will be always into x and y. So next, what if I'm moving... Let's do it simpler. I'm moving along x-axis, not only, but with acceleration a from zero. So at tau is equal to zero, I am at point zero. So that would be y below would be zero because I'm moving along x-axis. And x would be equal to a t squared divided by 2. So that's the... from Newtonian mechanics, well, actually, kinematics, how my distance along the x-axis would be if I'm moving with acceleration. All right, so what is this? I should say tau. I'm sorry. So how this would look on the graph? Well, x is a parabola. t is the same as tau, so x would be a parabola. So in the plane, x-t plane, I will have a parabola. And that's the representation. Now, if I have only this parabola with whatever the parameters the parabola has, I can have these equations, obviously, which means I'm moving with constant acceleration. So I'm deriving just trajectory by itself doesn't really tell me much about how I move with acceleration, without acceleration, etc., back and forth. But with this parabola, I know exactly how I move. I'm moving with constant acceleration. So this is an example. Okay, I just wanted to be comfortable with, in this case, three-dimensional, but again, as far as the formulas will be concerned, four-dimensional space plus time used by Minkowski to interpret theory of relativity by Einstein. So Einstein's view are kind of transformed into more visual representation. That's basically where Minkowski started his work. Now, being as it may, let's talk about something else. So we know that representation of the movement in the space time, space plus time, sometimes people even omit space or minus or plus in between or dash, they put space time as one word. So we basically kind of comfortable with what it represents. Now, what's the most important property of any space we live in, three-dimensional, our X, Y, Z space or four-dimensional space plus time, what is very, very important characteristic? Two points must have a distance between them. Well, in Galilean approach, we're not considering the time. Time is absolute. We have only our three-dimensional world and we have a Cartesian coordinates, X, Y and Z. Then the distance, square root distance, between two points, would be equal to difference of X coordinates plus difference means between Y coordinates square plus Z square. Well, A minus B or B minus, it doesn't matter because it's a square. So this is the square root distance in Cartesian coordinates in Galilean mechanics and what's important, this distance is retained if I'm moving from one inertial system to another with Galilean transformation of coordinates when the time actually is absolute. Now, we did actually approve it when we were explaining the Galilean viewpoint. Now, here comes Einstein and he says, hey, Galilean transformation of coordinates from one inertial system to another is not exactly what it is. We need Lorentz transformation. Lorentz transformation, which involves time and it involves the time dilation and involves certain modifications of the lengths whenever the object, whenever reference frame is moving. So we need Lorentz transformation actually of coordinates and that's how in reality it is. And as an approximation, only the Galilean distance between two points is valid. So how can I define the distance between two points? Not like this obviously. So it will be invariant relative to Lorentz transformation because I would like the difference between two points to be the same if measured in one inertial system or measured in another inertial system which is moving relative to the first one. We know that length is sometimes getting shorter, longer, etc. I mean, that's along our real representation of the lengths. But we need something which is called a distance which we can kind of use as an invariant from one system to another. So it's not our distance in meters basically but it's some quadratic form in this particular case which retains its value when moved from one inertial system to another. And in a previous lecture about metrics in the Einstein view the last lecture is about metrics we have come up with this formula c t b minus t a square square t square minus x b minus x a square minus y b minus minus square minus z. We have come up with this formula, this expression to be invariant relative to Lorentz transformation. So if we represent our universe as a four-dimensional which is three-dimensional space plus one-dimensional time and use Lorentz transformation of coordinates from one to another and Lorentz transformation of all four coordinates including time then this particular expression was proven to be invariant relative to Lorentz transformation which means this distance, so to speak distance is retained the value from one inertial system to another. Now obviously if this is invariant then any function of this also is invariant. If I multiply it by five it will also be invariant. So sometimes people using this sometimes people using its negative which is plus, plus, plus, plus here and minus here. It doesn't really matter in certain cases it's one, certain cases another. What matters is that there is a concept of distance in four-dimensional Minkowski space. Now again for visualization we are using only three-dimensional two-space x, y plus time t but in any case now what's also very important by the way I have to put squares here c also is squared, sorry. So what's also important is that it's more beneficial if let's consider again two-dimensional space plus one-dimensional time. So on the time it's better to measure it not in time units like in seconds but in c times g which is basically meters or centimeters or whatever where c is the speed of light. Speed times time would be again unit of lengths why is it the same basically? Just to make the same measurement units on every axis, meters, meters and meters, right? And since c is constant in any inertial reference frame it doesn't really change the character of the curve or whatever we are drawing in this space it just stretches it a little bit during in one particular direction or another. So that's basically all I wanted to talk about this is how the Minkowski view on the theory of relativity begins so it begins with creating a four-dimensional space time with three-dimensional space in one time it also involves measuring time not as just seconds but time is measured in terms of lengths like light second if you wish where c is the speed of light, multiply just a measurement unit doesn't really matter to unify all the units to the same basically standard and this expression as being invariant lengths invariant definition of the distance between points A and B and points A and B are considered to be four-dimensional we have x, y, z and t coordinates and obviously we have a Lorentz transformation from one coordinate system to another which is moving uniformly relative to the first one so that would be the definition of the distance and it's invariant that's important okay, that's it I suggest you to read the notes for this lecture you go to unison.com you choose the relativity for all and that would be the Minkowski views menu first lecture in there okay, thanks very much and good luck