 Hi, I'm Zor. Welcome to Indizor Education. Today we will continue talking about Lorentz transformation and its effect on time, which we did discuss before when talking about time measuring lecture just before the Lorentz transformation. But now, equipped with Lorentz transformation, we will try to derive basically the same formula but more mathematically or theoretically whatever you want to call it. Now this lecture is part of the course called Relativity for All, presented on Unizor.com. On the same website you can find two prerequisite courses, math for teens and physics for teens. Now, obviously both mathematical and physical knowledge are mandatory to understand theory of relativity, so either you take those courses or maybe you have already in possession of the proper knowledge of math and physics. But in any case, these are prerequisites which are available on the website. Also every lecture on this website, well including this one of course, have a very detailed textual notes which basically like a textbook. So you have at the same time the video presentation and the part of the textbook if you wish related to this particular lecture. Whatever, maybe I might miss something during the lecture then I'll put it in the notes. But basically they're supposed to be about the same thing obviously. Now also the website contains some educational functionality. For example, you can take exams as many times as you want. There are some problem-solving exercises and also there is a provision for supervisor study. So if you are basically given some kind of an assignment by teacher or supervisor or parent, if it's a self-study for example, then it can be arranged through this function knowledge of the website. And the website is completely free. There are no advertisements, no strings attached. Sign-on is optional and there is no information except maybe the name and the password if you want to, but you can be completely anonymous and take the whole course without any kind of signing in if you don't want to. Signing in is actually necessary for this functional provisions like if you have a supervisor, then you need basically to arrange the relationship between you and supervisor and that's why both have to sign in. But just for self-study without any kind of restrictions you can just go and do whatever is necessary. Okay, back to time. Now let's consider we have two inertial systems. One will be called alpha. I will put only one space coordinate X. Y and Z coordinates will not be important because they will be completely the same in both this system and another system which I call beta. I will use lower cases for beta. And beta is moving with speed V along the X axis of alpha with retaining the parallelism of all the axes. And at time T is equal to 0, alpha coincides with beta completely. So at the moment of time 0 they are exactly overlap each other. But in any case before and after the time 0 beta is moving towards the along the X axis of alpha. At the same time you can say that its alpha is moving along the X axis of beta with the speed minus V. So alpha also moves with the speed minus V relative to beta. So it's all relative, it's all symmetrical so to speak. Now in the lecture which was called time measuring we considered a light clock in the beta system which basically send the light from one particular point up to the mirror and let's say the lengths of these distances are and the mirror is reflecting back and we accept this beam of light at the same point. Now this is all happening in the beta system. So the point is not moving anywhere, the mirror is not moving anywhere, they are stationary in the beta system and the time of this experiment from sending the beam of light up to the mirror and back is a certain duration which we have measured basically as 2R divided by C. That was our timing of the event, 2R because the speed of light is C and the distance is R to the mirror and R back. Everything is stationary so nothing is moving, everything is exactly the same. And that's what we called a timing for the beta observer in the beta system because it's a beta system, it's a lowercase t and it's a beta observer, that's why the beta is here. At the same time we were trying to evaluate time of alpha observer of the same beta event or beta process, beta experiment and we have found this formula where gamma is 1 over square root of 1 minus V squared divided by C squared. V is the speed. Now if you don't remember how we obtained this formula you can obviously go back to this lecture, it's called time measuring and you will see that that's basically the formula which we obtained analyzing how this beam of light is going to the mirror and back from perspective of alpha observer because since beta is moving so with alpha is seeing actually that beam of light goes this and then this since beta is moving. Basically calculating whatever the distances we came up with this formula. Okay, fine. Being as it may, now we are equipped with a little bit more, I would say, mathematical and theoretical tool called Lorentz transformation. So we have come up with the transformation of coordinates between two systems like this, alpha and beta and this transformation looks, that was a previous lecture actually, this transformation looks like this. If we are talking about using alpha coordinates as the base and you would like to find out what's the beta coordinates look like that would be the formula like this. x minus vt divided by square root of 1 minus v square c square and t, so lowercase x and lowercase t are prerogative of beta system uppercase x and uppercase t are the alpha system that would be t minus vx divided by c square divided by the same square root. So this is Lorentz transformation which we have derived in the previous lecture how to obtain the coordinates of something in the beta system if we know the coordinates and time by the way in the alpha system. Time obviously is important because as the time goes by beta system is moving and that's why the same point in alpha for example would have different coordinates in beta as the time goes on, right? Now obviously this e is completely inversible because of this if beta is moving with the speed v relative to alpha alpha is moving with the speed v relative to beta which means we can all reverse this just changing v to minus v and we will have x is equal to lowercase plus in this case v and lowercase t divided by the same square root and capital T is equal to also v changed to minus v that would be 2 t plus sorry lowercase t v lowercase x square divided by the same square root. Okay, so these are mutually symmetrical transformation from space and time coordinates of one system into space and time coordinates of another system. So that's done, this has been covered already and now my purpose is to use this very powerful apparatus to basically evaluate how the time is changing the duration of time is changing from one system into another basically not repeating the same logical well I would say experimental kind of logic which we were using in the lecture time measuring when we came up with this relationship I'm talking about this one. So the duration at alpha system of the event which is happening in beta system would be by gamma where gamma is one over this square root would be greater. By the way this is less than one so one over square root would be greater than one so the gamma is increasing factor so capital T beta would be greater than lowercase the proper time, the time where event actually is happening is always the smallest and in any other frame that would be greater. Okay, so now let's again use this system and again we will consider that there is some experiment actually if you wish doesn't really matter right now we're not talking about light clock or anything like this any kind of a process any kind of an experiment is conducted in the beta system so it's local to beta it's basically everything is at rest in the beta system but the entire beta system is moving relative to alpha so let's consider that at some point at some point let's call it x0 in the beta system it's a fixed point at rest not moving we conduct certain experiment it takes time from T begin to T end so my space coordinate is the same during this experiment I'm just standing at point x0 in conducting experiment which takes some time now my question is if I observe this particular experiment from the alpha system how long will it take? Alright, well we do have basically two different two different co-ordinates x0 and T begin one space time point and another is x0 T end so x0 is x T begin or T end is the T or case T and all I have to do is substitute into these formulas to get correspondingly x0 and T begin I would rather put it x begin because I'm not really sure whether that would be the same and x and T end so from these two I have to convert into these two somehow okay so let's just do the first one and I will use these formulas so x begin equals x0 plus v T begin divided by well if you don't mind let me just multiply by gamma so it would be where gamma is 1 over this square root same thing as this and T begin that's the time experiment begins I will use this formula which is gamma times T begin plus v x0 divided by c square okay good now let's do conversion of the end of the event in one system into another so we will have x end equals to exactly the same formula x0 we still have plus v T end and T end is equal to gamma or T end plus v x0 divided by c square okay so I have an end I have a begin I have an end I have a begin space coordinates and in time coordinates so what follows is my duration delta T that's the duration of the whole experiment in the alpha system from T end to T begin well obviously this thing will cancel out and I will have gamma T end minus T begin which is equal to gamma delta T where delta T is duration of the event in the beta system and delta T capital is the duration in the alpha system which corresponds completely to this formula so we have derived the same formula this is a good confirmation I mean though if you have derived to the same formula from two different theories that's always good right so we have done that now what's interesting is let's compare the space coordinates you see space coordinates are different now and what's the difference delta x is equal to we will subtract from this this gamma times x0 will cancel out but this one will not it will be gamma times V times T end minus T begin which is delta T so as you see the experiment which is at rest basically at point lowercase x0 in the beta system obviously it's just there we are moving with we are within the beta system and observe this experiment it stands still the time goes by but the location is not but if you view it from the alpha frame that the beginning and the ending of the experiment will be in two different space locations well why? because during the time delta T while experiment is moving will move by V by the speed V so that would be the distance and then there is still this factor gamma which means that there is this is a relativistic effect actually which you know kind of maybe not exactly considered as obvious and it's not obvious so the presence of this denominator is definitely not obvious the achievement of the relativistic theory which basically added this particular factor to make the whole thing much more precise if the speed V is very very low relative to speed of light C then we don't feel this so the gamma would be very very close to 1 and that's why experiments of 19th century didn't really show any kind of problem but more precise experiments proved that this is the right way to do it and so the relativistic approach was many times confirmed experimentally not only through the theory which we kind of built ourselves so that's an interesting point obviously and again as I was saying we were just using the Lorentz transformation to confirm that the time actually is perceived differently by the person which is in the local beta system where experiment is conducted actually and stands still and by the person who is in another inertial time frame system which well you can say moves relative to this one or this more relative to that one doesn't really matter but in any case there is definitely a speed V plus or minus whatever you want to consider it so if there is such a speed that the time is perceived differently and the shortest perceived time is the proper one by the observer which is local to the experiment which we are talking about the timing of which we are measuring well that's it I would suggest you to read the notes for this lecture so you go to unison.com choose the course called Relativity for All and in this course there is like a part called Einstein View and this lecture is in that particular part of the course so thank you very much and good luck