 Hi, I'm Zor. Welcome to Unisor Education. Today's lecture is called Past and Future in Minkowski space-time. 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 because every lecture including this one contains textual equivalent. Basically it's like a piece of a textbook which is completely in sync with this particular lecture. Also the website contains two prerequisite courses, Mass for Teens and Physics for Teens. The knowledge presented in these two courses is definitely a must for understanding relativity. Also there are some problems to solve, exercises. There is some functionality on this website. In case, for example, you would like to study something under somebody's supervision, like your parents or teacher. So there is an old necessary functionality to accomplish this. Otherwise you can do obviously self-study. And for self-study you don't even have to sign in, by the way. There is no advertisement on the website. The website is completely free, so use it as well. Okay, so today we will talk about, we continue talking about Minkowski's view onto theory of relativity. It came in 1907, just a couple of years after Einstein has introduced in the special theory of relativity in his paper, 1905. And it presents a certain more geometrical, so to speak, view onto theory of relativity. In the previous lecture we basically started the concept of Minkowski, four-dimensional space, which contains three-dimensional our space, which we live, plus one-dimension time. So it's four-dimensional space-time. So that's why I call it space plus time. Sometimes people even drop a plus or dash or even space between space and time and have it as one word, space-time. Okay, so it's all about Minkowski's representation of theory of relativity. Now, the problem is that four-dimensional space-time, it's kind of difficult to visualize. So for educational purposes and for the purposes of easier visualization, we usually concentrate explanation on two-dimensional space, which is like movement on the plane and one-dimensional time. So that makes it three, and three-dimensional can be in some projection actually presented graphically. So that's what I will do definitely. So we'll continue talking about two-dimensional space plus one- dimensional time, but we have in mind that it's all related to three- dimensional space plus one-dimensional time. So it's four-dimensional space-time. Alright, so that's number one. Number two. Let's assume now we are dealing with this three-dimensional space-time, two-dimensions where movement's actually occurring and one-dimensional time. So we'll have x and y here and t up there. Okay, fine. Now, in the previous lecture we were talking about presentation of the movement not on the plane where it actually occurs, because their representation is just a trajectory, but how the point moves along this trajectory is not really visualized, but we will stretch it towards time. So at every moment in time we see where exactly the point is. We will have something which is called world line which contains world points. Projection of these will be the trajectory, but the world line really represents the movement. At any moment in time what are my x and y coordinates of the point? Okay. Now, if the point moves along the plane, it's always uncertain distance from the beginning. Let's say the beginning at t is equal to 0, point is at the origin of coordinates. So that's the beginning. Time is 0 and coordinates is 0. That's where the point starts. Now, at any moment in time, t, what is the distance from the origin? Well, obviously the distance is equal to square root of x square of t plus y square of t in our plane. Again, having in mind that in three-dimensional physical movement plus one-dimensional time, that will be plus z square. But forget about this for now. So this is the distance from the beginning. Obviously it's function of t. Now, special theory of relativity is based upon two principles. One principle of relativity and the second one is principle of constancy of the speed of light. And from this followed that the speed of light is the maximum possible speed. Now, since speed of light is maximum possible speed, our B of t should be less than an equal speed of light times time. Because the point cannot be further from the beginning, from the origin, than the light can cover the distance during the same time. So this will be a very important equation which basically tells whether it is or it is not possible to consider a point at coordinates x, y as belonging to a trajectory started at point zero. So let's just, nobody likes square roots. So let's just do a different d square of t is equal to x square of t plus y square of t. And this would be x square of t plus y square of t less than equal to c square t square. So this is an equation which basically dictates the possibility of x, y to be part of the trajectory of a point that at point z, at point t equal to zero starts at the origin of coordinates. If point does not satisfy this equation, it cannot be a physical trajectory of an object which starts at this point and reaches this point at time t. Just not possible. Okay, now that's very important. But now I would like to express it geometrically because the whole concept of Minkowski space time was to present it as some kind of visualization of specialist euro relativity. But what is this? Well, first of all, let me just wipe this thing out. And now we will talk about pure geometry. If you have a three-dimensional space, let's say x, y and t, what is a surface which satisfies this particular equation? Well, it's actually quite easy. At point t is equal to zero, it's supposed to be only zero because these are sum of two squares. And if it's equal to zero, it's supposed to be equal to zero, each one of them. And then as t grows, the distance which is this distance from the t axis is also growing. Now, what if this is constant? Well, what this represents when this is equal to constant? This is a circle. So at any fixed t, this is a circle of increasing radius. Obviously, t is increasing, so the radius is increasing. So it's a cone, basically. It's very easy to show that the radius, but very easy. Square root is a radius, square root is ct. This is proportional to t, so radius is proportional to t. So if we will increase it by two times the distance from the origin, the radius is increasing by two times. So that's obviously a cone. Now, that's for positive t. For negative t, the cone goes down. So this is a cone when it's equal. Now, when it's less than equal, what is this? Well, that's inside of the cone. And greater than, this is less than, and this is greater than outside of the cone. So whatever is inside of the cone represents the world points which can be reached by point which starts at point zero, at time zero. So you can have all the world lines which represent movement on the x, y, z within this particular cone. And whatever it was with the point before it reached this zero point in the past, is within the bottom cone. So that's what I meant past and future. So all the past positions of the point are in this cone or the future position, world point. But I'm talking about world point because the point is moving only within the x, y plane. But the world point which basically stretches along the timeline would belong future there and past here. So that is basically a very important thing. We have introduced this cone which is called light cone. And the upper part represents the future possible future positions, future world point positions in the space time. And the bottom cone represents the past positions of the world point in the space time provided at point, at time zero, the point is, which is present basically. So present is time zero. So we know that the point is here. That's the origin of coordinates. Then we draw the cones using this equation. And basically we are saying, okay, the point might have been here and will be here, but definitely not outside of this cone. So this part is called usually time like part of the space time. And inside the cones, outside of the cones, it's usually called space like part of the space time. So the space time is divided, so to speak, into time like, which is inside of both bottom and top cones and outside of the cones. Okay. What's next? Next is the following. What if you have any kind of a trajectory you would like to see what exactly can happen with the point? So wherever the point is at any moment of time, you can always start from pushing your origin of coordinates at that particular point. So if it moves on the plane like this, so you can always start here at any point, have the coordinates here and build the cone here, here and say, okay, if this is the point right now at present time, the future behavior would be expressed in a space time, Minkowski space time. The world points would be here and the previous, these physical points can be represented by world point within this cone, which means that if you would view only the view points, if you have this. Now, right now I'm talking about only world points. So it moves here on the xy plane, but the world points are here. Now at any point, you can build the cone and say that the future behavior should be within this cone and the past behavior between this cone. And in this point, exactly the same thing, you can build this cone here and say, okay, future should belong to here and the past definitely supposed to be here. So it actually gives you certain restriction on the trajectory. Trajectory cannot go sideways, so to speak, on this particular picture too much, too fast, because it means it goes along the space coordinates faster than the speed of light. So at any point, it should be more vertical, so to speak, than horizontal. The horizontal component is very much restricted by the speed of light. So that's why this inside of the columns is called time-like. Okay, now the last thing which I wanted to talk about is the problem. We can basically come up with this equation or in three dimension it would be plus z square, doesn't really matter. Now let's just rewrite it, ct square minus x square of t minus y square of t. And I will use the third dimension because right now we have covered all the geometry. Now this is supposed to be greater or equal to zero, right? That's the condition. Condition of the point x, y, z, c in Minkowski four-dimensional spacetime to be a possible world point of a position of the trajectory which started at zero, zero, zero, zero, zero. Okay? Now but what is this? If you remember in the previous lecture we were talking about matrix in Minkowski spacetime and what was important for us that the matrix which we introduced which is basically a distance between two points to be invariant relative to invariant Lorentz transformation which means it will be the same in any inertial frame where we will measure this distance between these two points. So the distance between two points expressed as this expression again that goes back to previous lecture. This was called d square, I don't really need the a and b. If you remember we have introduced this c square of t b minus t a square minus minus x b minus x a square minus y b minus y a square minus z. Now we have introduced this as a distance between two points a and b where x a y a and z a and t a are coordinates of the point in the spacetime of the a point and correspondingly for b. So that was the introduction of the distance. So what we're talking here right now is this is this path. It's a distance between a point at moment t and a point at moment zero right because we were talking about if t is equal to zero x of zero equals to y of zero equals to zero is equal to zero. So at moment zero we are at the point at origin of coordinates. So this represents this distance between t b and x b and x a and x b etc. All the b's are x of t y of t and z of t and point a is zero zero zero zero. So the distance between these points is exactly this. So our condition on the possibility of the point to be within the cone is the same as I would say that the distance the Linkowski distance, this invariant distance between world point and origin of coordinate if it's positive that means that it is within the cone and it is possible to be to have a representation of a trajectory of a real trajectory of a real object. If it's not positive if it's zero then it means that our object is moving with a speed of light and if it's negative it means that it's impossible to be a point the world point which represents a position of the point that started at origin at t equal to zero. So that's how we're relating the physical movement within this Minkowski space time with a distance. So the physical movement is basically such that the distance Minkowski distance, this invariant relative to Lore's transformation distance to be non-negative. So that's the condition. That's basically it. Yeah, that's it. I recommend you to read the description which accompanies notes basically which accompanies this lecture on the unizord.com. So if you go to unizord.com choose relativity for all course then next menu topic would be Minkowski view and within Minkowski view you will see this lecture about past and future and reading it with basically is presentation of the same material in a just textual form with a couple of nice pictures. So that's it. Thank you very much and good luck.