 Hi, I'm Zor. Welcome to Inezor Education. Today will be the first lecture where we will see just the first look at theory of relativity, one particular very narrow aspect of theory of relativity. This lecture is about time, how we measure time and how the time is measured in different inertial systems. Now, this lecture is part of the course called Relativity for All. It's presented in Unizor.com, together with two other courses, which are, I would say, prerequisites, mass for teens and physics for teens. Now, the whole website is totally free. There are no advertisements, so it's open for everybody. You don't even have to sign in. However, there are some additional functionalities which you might be interested in. For example, supervisory study. So where there is a parent or a supervisor kind of supervise your flow of educational material which you have to study. Then you do have to sign in and your supervisor as well. Also, every lecture has very detailed notes on this website. So, in any way, I just recommend you to watch this lecture from the website and read the notes for every lecture. That will be much more beneficial than if you just found the lecture by itself somewhere on YouTube or wherever. Okay, so let's go back to time. First of all, there are a couple of very, very fundamental things which we have to agree, and physicists based on their theoretical and experimental data obviously agree. These are principle of relativity and constant speed of light in all inertial frames in, let's say, in vacuum. It's in the same kind of category in other media as well, but that's much more difficult. So, that's why we're talking about space, vacuum, empty space, and all inertial frames have the speed of light exactly the same. That's basically a result of, again, experimental data like Michelson-Molley experiment which we were talking about before, like Maxwell's equation and some other considerations. Okay, so these are two fundamental things. Speed of light constant in all the inertial frames in vacuum and we will deal only with inertial frames in vacuum and principle of relativity. Let me just, again, repeat about principle of relativity. It basically means that there's supposed to be no way you can distinguish one inertial system from another. All the laws are supposed to be exactly the same. So, if you are immersed in one particular inertial system, there is no experiment which you can make, which you can conduct, which would reveal that this particular inertial system is moving with, you know, certain speed relative to, let's say, another inertial system. And there is no, like, one particular inertial system which is hooked to our space. And it's basically like at rest in the space. Space does not have this type of thing. So every inertial system is exactly the same as another and you cannot distinguish one from another. All the processes are happening in exactly the same way. So if you, let's say, take certain apparatus, build it in one inertial system, and it behaves a certain way, it works the same, a certain way, then you take this apparatus and you move with this apparatus to another inertial system. It will behave exactly the same thing. There will be no difference in the behavior of this apparatus. If you grow a corn, let's say, in one inertial system and then you move to another inertial system, which basically has the same kind of different, well, the same conditions. It's a different system, but the same conditions, the same soil, the same corn seed, etc., whatever it is. It will grow in exactly the same way with exactly the same speed, regardless of whether one system is moving relative to another or not, in the direction of moving, because as long as they're all inertial, no difference whatsoever. You will not be able to recognize. Okay, being as it may, there is a very important time property which we have inherited from the time of Galileo, etc. I was talking about this time property when I was discussing the Galilean transformation. Namely, time is absolute and universal. So if you have two different inertial systems, then the time is exactly the same and there is absolutely no difference. If you look from one inertial system onto events in another inertial system, you will see exactly the same thing as if the person who is immersed in that other inertial system sees these same events. And that is something which I'm going to deny today. So the absolute character of time is something which theory of relativity really have put an end to. Time is not absolute. It's not like universal, whatever. So if you see certain event from inside the one inertial system and you see the same event from another inertial system which moves relative to the first one, the timing would be different. Okay, so this is something which, to tell you the truth, I was amazed myself when I learned about this. I was probably in school at the time, long, long time ago. I was reading the book. I think it was a book written by Einstein and maybe a rumor or something. I don't remember really. And they were basically explaining what I'm going to explain to you right now. And it seems to be logically correct. Everything was fine. But as a result, we have come up with this particular difference in time variation from different systems, from different inertial systems. So I will basically do the same thing. It's really simple. It's just plain algebra, if you wish, and a little bit of logic. What I believe is very important is that some people, and Einstein definitely was one of them, they were thinking much deeper than the rest of us about very simple things. Time seems to be a simple thing. And when you think about this a little deeper and you're thinking about certain thought experiments which you can conduct, and you see that it's not that simple. I mean, that's the revelation, basically. So that was a revelation for me some time ago, and I will attempt to basically convey it to you. Okay, so first of all, what we need is, we need some apparatus to measure the time. And this apparatus should be relatively simple and universal, applicable to any inertial frame. And here is what I suggested. Let's consider you have some kind of a rigid rod made of some metal or whatever. It doesn't really matter. And on one end, you have a source of light. On another end, you have a mirror. Now, what happens is we send a ray of light towards the mirror. Well, we send it everywhere, but one of the rays, exactly one way actually reaches the mirror, reflects back, and we have some kind of a sensor device here. And the length of this rigid rod is fixed, and we can install this particular device anywhere. And the time between sending the ray of light to the mirror and getting it back is our unit of time which we can measure something with some kind of other events. Now, this is an event by itself. Sending the ray up and getting it down, that's the event, and it has certain time, obviously. And using this unit of time, we can measure some other events. But that's besides the point. So we're talking about this particular case. We took this particular device and we have this particular kind of device to measure time and to observe it from different standpoint. You can actually imagine, like, we have some kind of a room or whatever. Car doesn't matter. And we installed this device here. So this is the source of light. This is the mirror, and light goes up and then down, and that's the event which we are talking about. Now, let's consider now two different inertial systems. One inertial system is this car. Well, let's say standing still doesn't really matter, but at some particular state. So this is alpha. And the alpha reference frame has certain coordinates. X, Y, Z, and time T. Now, at some time, we have decided to have exactly the same car, but the car will be moving with some speed B. Okay, now this same car represents a different inertial system. So the first inertial system is called A alpha when the car stands still. And another system would be beta, and I will use lower case coordinates, X, Y, Z, and time T to describe events in that particular system. In this case, it's moving car. Okay? Now, considering the principle of relativity, if we have an alpha observer, which means observer which is in the system alpha, it sees this event and it has certain time while the ray of light goes up and down. Now, when the car is moving and you take another observer which is local to this moving car, let's call it beta observer, it will basically see exactly the same thing because we know that the principle of relativity states that all the events are supposed to be exactly the same, which means that the timing of ray going up and down in the moving car for the beta observer who is inside that moving car should be exactly the same as for the alpha observer who was observing that particular event when the car was standing still. Okay? Now, this speed V is the speed of beta system relative to alpha system. And let's just talk about this thing the following way. So, let's say this is x, y and z and this is lower case x, z and y. This is upper case. So, the whole system is moving only along the x-axis. It just makes my life easier and you will understand it better. So, let's just forget about y. Now, let's say this distance is r. So, our source of light is at the origin of this coordinate system in here and here. Now, this one is moving with the speed V. This one is standing still. And let's analyze what exactly the alpha observer sees inside this car if he can see. I mean, he can analyze it just logically. So, alpha observer analyzes how this particular event is occurring in the beta reference frame. So, again, we know that alpha observer sees within the alpha system event as such and the beta observer within beta system sees exactly the same thing. But I'm talking about cross reference. So, alpha observer who stands on the ground, let's say, when the car was on the ground, this particular observer sees events in the moving car. And how can this particular alpha observer evaluate this event and how long does it take? Okay. So, that's the most important part. So, I'm talking about cross evaluation from an observer in one system looking at events in another system. From the Galilean standpoint, from the standpoint of old physics, this particular guy should see exactly the same thing as the moving, as the beta observer sees inside the car. But, again, if you think about this more thoroughly, you will see that this is not the case. And here is one. Now, what happens? The ray of light starts here, which means at origin, let's say, at some moment we started our experiment, the origin is at point x0, okay? Origin of the beta system relative to the alpha coordinate. So, this is x0. At this point, we have sent the light everywhere. Now, the light ray, which goes vertically up, would not actually hit the mirror, because during the time it goes up, the mirror will move to the right, because the car moves to the right, which means it's not the vertical ray which will reach the mirror. It will be the one which is going at some angle, because by the time we reach the top of the car, the mirror will move here. So, it's not this ray, it's this ray will reach, and then it will reflect, and again it will be here, x1. So, by the time ray of light would be reflected and sensed by the sensor which is here, the trajectory of the light would be like this, not up and down, but at the angle, and then it will reflect at the same angle, and it will be accepted at further point. So, by this time the ray goes up and down, the whole system, the whole car in this case, will move from x0 to x1, so that would be a new location of origin. Now, the light goes with the same speed everywhere, regardless of the speed of inertial systems relative to another inertial system. So, it takes time, and by this time obviously it's a longer trajectory. It's longer than this one, which means there should be a difference in time. So, the event which took certain time going up and down for the beta observer, the alpha observer sees it differently. It sees that the light goes along the longer trajectory. The same event, which happens with one interval of time, during one interval of time in one system, in this system in alpha reference frame, takes longer. So, local event for the beta observer is shorter by time going up and down vertically than the event observed, the same event, but observed by the person who is outside of this particular system. Well, let's just evaluate how exactly they are related. Now, I'm using these letters to basically signify the time in the alpha system and lower case in the beta system, which is moving, and I will use indices. So, t-alpha would be the time for this event for observer in the alpha system of the event which is in the alpha system. So, capital T means that I'm using the observer in this alpha system and the event itself occurs in this alpha system. So, what is my time? My time is whenever I go up and down, so r and r divided by the speed. So, it's 2r by speed of light. Now, what if I'm in a moving car? I'm using the lower case because it's a beta observer and I am using the event which happens local in the same. That's exactly the same thing. We are talking about principle of relativity. So, these two timings, this is the time of this event going to the mirror and back in the alpha system for the alpha observer and this is event which happens in the beta system for beta observer. It's supposed to be exactly the same. What I'm trying to calculate right now is t-beta. Now, what is t-beta? That's how the alpha observer uses alpha coordinates in alpha time, how it sees event which happens in the beta frame. That's what I would like to understand. So, from the perspective of alpha observer, event is going this way and then this way, right? So, let's say the time of this event is t-beta. Now, what happens during this time? Well, during this time I know that this is the time the whole system moved to the right with the speed v, which means that my x0, x1 segment equals to speed of moving the car v times my t-beta, right? Now, let's divide it by 2. Let's call it pq. Now, obviously, x0, q, x1 is a socialist triangle. The angle of reflection is the same. So, that's kind of easy. This is r. Now, this is half of this, right? So, the length x0, q, x0, q by theorem, by Pythagorean theorem, square is equal to r square plus half of that. r square plus 1 half of this segment, x0, x1, which is this one. 1 half of vc beta square. Okay? Okay. Now, the light goes all the way this and this. So, it covers basically two lengths of this. So, if I will have double this length, that would be the trajectory covered by the light during the same time capital T beta. So, c times capital T beta is the length of this thing. So, it's double square root of r square plus 1 half vt beta square. So, this is kind of equation, if you wish. So, what do we do with this? We don't do this anymore. So, we will just square it. So, c square t beta square equals 4r square plus 1 half vt beta square. And this 4r square equals ct beta square minus this 1 half square would be 4 and this 4 will be canceling each other. And v square t beta equals c square minus v square t beta. Simple algebra, right? Okay, fine. From this, now, this is how our event happens locally. This is local for alpha, this is local for beta, but they are the same because of principle of relativity. So, from here, we can see that 2r is equal to c times t beta from which 4r square is equal to c times lower t beta. So, what is lower case t beta? That's how the event is seen locally by the beta observer. What is capital T beta? That's how exactly the same event in the beta reference frame is evaluated by the person which is in the alpha reference frame. So, these things are equal to each other, which means ct beta square of this square, obviously. Equals ct beta square square minus, well, I can put this here, minus v square t beta square from which follows that t beta square is equal to t beta square c square divided by c square minus v square. But let's just divide it by c square both things. And we will see that it will be 1 minus v square divided by c square. Let's just get rid of the squares and put square root here. And 1 over square root of 1 minus v square over c square is traditionally called gamma. So, t beta is equal to gamma t beta. So, what does it mean? By the way, v is obviously much smaller than speed of light. So, this is smaller than 1, so this is greater than 1, which means my capital T beta is always greater than lowercase t beta, which means that locally event is viewed as a shorter event than if you look at the same event from outside from another inertial framework. It's called time dilation, basically. There is a term for this time dilation. Now, this is very important, which means what? The time is not absolute. How I, from the alpha system, see events in the beta system happening is not the same time, they take longer. As I see them, they take much longer, much or not much, depending on the speed v, obviously. The greater the speed v, the closer this is to 1, which means the closer denominator is to 0, which means the greater would be gamma. So, as v is increasing, gamma is increasing as well, which means that events from the outsider, events in some particular framework, from outside of that framework, would seem to be taking longer than they seem to be from the local guy, whoever is in the beta system itself. That's what's very important, and that's one of the first things which I personally learned about zero relativity, that the time is not absolute, and people are saying that the time goes slower or whatever. I mean, this is kind of a very general kind of description about the time. It's much more precise to say the following, that if there is a certain event, event is supposed to be taking certain time for a local to this event person, and it takes for an observer from the outside, this event takes longer. Basically, that's what it is. Well, so time dilation is very important kind of property, and that's one of the first things which you can be introduced in studying theory of relativity. Not only the time actually is, how should I say, subjective. Well, it's subjective because it depends on who exactly is kind of sensing the flow of time. If it's a local guy and the inertial frame is moving, it's one speed, and if it's somebody from outside look at the events which are happening in this moving car or whatever, he sees it differently. Now, but from the symmetrical standpoint, if beta is moving with speed v relative to alpha, it means alpha is moving with speed minus v relative to beta. You can consider beta to be basically stationary, and alpha moving with the speed minus v. And considering we have v square here, it doesn't really matter whether it's plus v or minus v, which direction is moving, it's unidirectional. It all depends only on the absolute value of the speed. So from the observer at beta, if he looks at events which are happening in alpha frame, the result will be the same. From alpha event it would be, for alpha observer event would be shorter than it would be evaluated by the person outside in the beta frame, which means in our equation that would be very similar, but I should change beta to alpha and alpha to beta. So if we are talking about alpha event, this is how it's viewed by an inner alpha observer, and this is how long it takes for an observer from the beta system, the same direction. So alpha observer sees events in beta system as taking longer than local guy, but the beta observer sees events in alpha system also taking longer than for alpha observer, the local guy. In some way, I mean, it's strange because there is no absolute, everything is relative. It's not like event really takes longer in one thing than another. No, event is exactly the same. It's just seen longer from the outside than it takes according to the senses of the local guy. In some way it might be similar to, let's say you have two people looking at each other from a distance. Each one seems to be smaller than he really is because of perspective, right? So if you have A and B, A looks at B, and B kind of seems to be smaller than the real size. Same thing if B looks at A, also smaller, but none of them is smaller than another, it's just the viewpoint. What's important is the viewpoint. If you look from the distance, it looks smaller. That's it. If you look here from another inertial system, the event takes longer. That's it. Okay, I do suggest you to read notes for this lecture. As I was saying, it's on Unisor.com, the course called Relativity for All, and if you will open that menu option, you will see the Einstein view onto Relativity versus Galilean view, which was kind of predates. Okay, that's it for today. I thank you and I wish you good luck.