 Hi, I'm Zor. Welcome to Ingezor Education. We continue talking about things in our space and time related to Galilean transformations. So it's all priori relativity, kind of state of physics, before major Einstein's work related to special theory of relativity. So the most important part in that time was that our space is basically uniform. Our time is absolute. It doesn't really depend on which inertial reference frame you're talking about, from which inertial frame you're observing our universe. And the Galilean transformations are basically the most important kind of transformation of coordinates, and all the laws of physics are supposed to be the same in all the inertial frame systems. That's basically the Galilean invariance principle. Now the previous lecture was about whether Galilean transformations are really what we are promising them to be when we are talking about certain laws of physics. Primarily we were considering the first and the second Newton's law of mechanics. But these are not the only laws, obviously. Today's lecture will be about two other laws which also are invariant relative to Galilean transformation, which is the good thing. The bad thing will be next. So the good thing is that whatever the laws we are discussing so far are really invariant. Today we will talk about two different laws, the law of universal gravitation law and Coulomb's law, when electrostatic field is related to. Now before doing this, I would like to consider something a little bit more fundamental, which is our uniformity of space, so to speak. And one of the most important characteristics of space is the distance between two points. Now this distance is actually part of these two laws which I was just talking about, the law of gravitation and the Coulomb's law. And that's why I would like to start today from analyzing whether the distance between two points in one inertial system is exactly the same as in another inertial system if we connect these two systems with Galilean transformation. Now more kind of scientific term, it's called matrix of our space. So the matrix is basically the distance is kind of measured in, right? So metric invariance is the name of this particular lecture. Now this lecture is part of the course called relativity for all. It's presented on Unisor.com website. Now the website is totally free, there are no advertisements, so you can just go and study whatever you want. It also has certain functionality behind it. For example, you can have supervised study if you want to, then you will probably need to sign in. But again, that's completely free and no strings attached. There are exams in certain cases, not all of them. And also the site contains two very important prerequisite course for relativity. One is called mass for teens and another is physics for teens. So I presume that whatever material is in those two sides, in those two courses, is supposed to be familiar from any other source if you wish, doesn't really matter. Before you start studying relativity. Okay, so what about metrics? Metric invariance. How do we measure the distance between two different points in our space? Well, first of all, we are talking about two different reference frames, both inertial. So inertial frame, any reference frame actually, which we are considering right now, is the Cartesian coordinates and we are assuming that our space is euclidean. Okay, so that basically means that we can use the formula for distance between two points in Cartesian system. Now what is this particular difference? So if you have two points, point A and point B. By the way, it doesn't really matter whether they are stationary or moving in the reference frame, which we are talking about. But let's consider them without any, I mean I can put A of t and B of t, but t is the time. Now we have two different reference frames. One reference frame is time and coordinates x, y and z. Another is time and lowercase x, y and z. And we are assuming that this frame is moving relative to this frame uniformly. So this is called alpha frame. This is called beta frame. Now at point time equal to zero, let me start first. That we are assuming that the time is absolute. So the time is exactly the same in both systems. At moment t equal to zero, alpha of t equal to zero coincides with beta at the same moment of time. So at moment time equals to zero, both frames coincide. And then the beta frame is moving relative to alpha frame with a uniform speed along the straight line. Now so there is a vector of speed, constant vector. Now this vector is not changing. So there is a vector of speed and this vector of speed would have coordinates vx, vy and vz. So this is the vector of speed. Now if I have vector of speed, now before I was preferring actually to discuss the situation when only x coordinate is changing. So vy and vz equal to zero. Today I would like to expand it a little bit. Actually I did expand it in the previous lecture as well. Because it doesn't really matter whether it's one particular coordinate is changing or all three along some constant vector. The formula will be very, very similar. For x conversion the formula will be basically the same as before. We are familiar with this transformation of x coordinate. It was in previous lecture many times before. But exactly the same way would be a transformation of y coordinate. That would be according to, let's get it all, times t and z of t would be equal to z of t minus vz times t. So these are formula for transformation of coordinates. This is Galilean transformation. And what I would like to know is how the difference, how the distance between two points is changing with Galilean transformation. So how this particular distance is viewed from one system if we know the other system. Now let's assume that point A has coordinates x of t, y of t and z of t and point. And let's put index A here. And this one would be xb of t, yb of t and zb of t. So these are two points and these are coordinates in alpha system. So what's the distance between these two points? Well the distance, well the distance squared in the system alpha would be, and we know about this obviously from geometry from other sources, xb minus xA squared plus yb minus yA squared plus zb minus zA squared. This is the square of the distance between two points which we know from basically from the course of geometry. Now all these coordinates they are a function of time so the points might actually shift as the time goes. Might or might not doesn't really matter because the formulas are, whether you put t or we don't put t dependence on the time doesn't really matter. The result will be exactly the same. So in this case this is the distance. And again if you want we can actually put z of time t. So all of these are also times of t. I didn't put just to shorten the distance. Okay, how does it change? Well let's see. In the beta system the distance would be similar. But lower case xA minus xB rather. xB minus xA squared plus yB minus yA squared plus zB minus zA squared equals. Well we know what the coordinates in the beta system, how they are related to coordinates in the alpha system. So xA would be xA minus dx. dx is constant. I mean that's very important because this is the component of the vector which directs the whole beta system moving uniformly relative to alpha system. Uniforming means that this vector is constant. All the components are constant, not depending on the time. So whenever I have xB minus xA it would be xA of t minus vxt minus xB of t minus vx of t, right? It would be xB minus vx times t. Well t and capital and lower case are the same so I can put this lower case. Minus xA minus dx times t squared plus. Same thing for y and same thing for z. And as you see vx times t is canceling out. And what's remaining is xB minus xA which is exactly the same as here. And with the y component would be exactly the same as this and with the z component would be exactly the same as this. Which means that the whole thing is equal to d squared alpha of t. So that's what we have. This is equal to this. Which means the distance between these two points in the beta system is exactly the same as the distance in the alpha system. That's kind of a trivial because this dx, vy and vz are constant. These are components of the vector which directs how the beta system is shifting relative to alpha system. So this is a very easy piece which proves that the distance between two points whether moving or stationary doesn't really matter. Viewed from two different inertial systems, the distance is exactly the same. What does it mean? Well it means for example that if you have a circle for instance in system alpha and then the beta system is moving relative to alpha this circle will be a circle in the beta system as well because the radius will be the same. Distance from the center point to every point on the circumference. Square of a certain dimension with a certain side would be exactly the square of the same dimensions with the same side in the beta system. So the distance is preserved and that's very important for the next two laws which I would like to consider. The first law being the gravitation law. So the gravitation law tells us the force is equal to some kind of a gravitation constant proportional to both masses and inverse proportional to a square of distance between them. You have to remember that. So this is the mass of one object, this is the mass of another object this is the distance between the centers of masses. Now this is a universal constant and the force depends basically on these parameters. This is gravitational force, attraction. Now if I am going from one inertial system to another which moves uniformly relative to the first one now mass is mass is the property of the object. G is universal constant regardless of anything. It's basically experimentally derived. So the only thing which might actually in theory be changed when one system is moving relative to another is the distance between these two masses. But we have just proven the distance remains the same whether the masses are stationary or moving as the earth is moving around sun or moon moves around earth doesn't really matter. So in both inertial systems the distance is the same and if the distance is the same then the force actually is the same the formula is the same the law of gravitation is exactly the same in both inertial systems which means that the law of gravity is invariant relative to Galilean transformation and that's very important it's the good thing actually, right? Okay, so that's one thing which is kind of obvious. Now another thing is the Coulomb's law which is very very similar. So what is Coulomb's law? Again the force is equal to some kind of constant that's electric constant and it's proportional to charges electric charges and inversely proportional to square of a distance. Exactly the same situation and for exactly the same reasons charges are properties of the object of the system which you are looking from so which means that the r is only thing to consider and it's invariant when you are moving from one inertial system to another which means that the formula actually stays and K e is just the constant it's Coulomb's constant again experimentally obtained. So as we see the metric invariance of our Euclidean space relative to transformation Galilean transformation from one inertial space to another from one inertial frame to another so this metric invariance is a sufficient condition for stating that two more laws the Newton's gravity and Coulomb's law are exactly the same and again it's kind of it confirms our initial theory which people considered absolutely correct before relativity came it confirms that all the laws of physics are supposed to be the same in inertial system they're supposed to be looking the same if we have a formula the formula should be the same so it's not supposed to be like r squared plus half of r or something like this it's supposed to be r squared so that confirms now did we consider all the laws of physics just to check? No and that would be a subject of the next lecture where I hope I will demonstrate that there are certain laws which are not invariant when we are moving from one system to another and that's a huge push towards development of relativity theory by Einstein okay, that's it for today I suggest you to read the notes for this lecture and think about if you really understand everything which is on mass routines and physics routines course you can just take a look at the menu the whole thing is menu-driven you really have to be comfortable in physics on that level prior to really immersing yourself into relativity so thanks very much and good luck