 Hi, I'm Zor. Welcome to InDesert Education. We will talk about metric today. Well, first of all, what is metric? Metric is basically a distance between two points in space. Now, in our space, which we consider for a very long time to be three-dimensional, and we usually use Cartesian coordinates x, y, and z. In our world, which we consider three-dimensional, we use Euclidean metric as a distance between two different points. Well, let's say you have two points, point A, with coordinates x, a, y, a, and z, a, and you have point B, which has correspondingly. Then the square of a distance between these two points is, again, this Euclidean distance between two points, expressed as in coordinate form. It's different between x square plus difference between y's plus difference between z. Now, obviously, this formula is a result of basically applying Pythagorean theorem. I consider this fact to be known to everybody, and there is no point actually to talk about properties of distance in Euclidean world. Now, going to physics, up until the beginning of 20th century, in physics, we were considering time to be absolute, and the distance between two points was measured exactly like this. Now, we were using also the Galilean transformation from one inertial frame to another. Remember this, if you have alpha system with x, y, and z, and time t, and beta system with lower case, where t is equal to t is absolute, time, always the same. And in a simple case, when the beta system is moving along the x-axis with speed v relative to alpha system, the transformation was very easy. x was x minus v times t, y was equal to y, and z was equal to z, and t is equal to t as well. And if you will apply this transformation to this formula, so if you will express instead of capital X, Y, and Z, you will use lower case and replace lower case x with its Galilean transformation from one system from another. You will see that the distance in alpha, which is like this, would be equal to distance in beta, which is basically very similar, but instead of upper case, I'm using lower case. So, this is invariance of Euclidean metric relative to Galilean transformation. Okay, everybody was happy. Well, until, until we had problems. Now, these problems led to development of theory of relativity, the special theory of relativity. And in that system, the Galilean transformation was considered not exactly precise. So, there is certain factor and these factors like constancy of the speed of light in any inertial reference frame, which led to replacing, basically, the Galilean transformation with much more precise Lorentz transformation. And in Lorentz transformation, we have time participating in a much greater degree, actually, in this. So, now we're talking about Lorentz transformation. So, instead of this, time is no longer absolute. And the Lorentz transformation is equal to gamma times x minus b times t, where gamma is 1 over square root of 1 minus v square divided by c square. So, if v is very small relative to speed of light, c, then this gamma is very close to 1 and that's why it actually, a good approximation with Euclidean with Galilean transformation, sorry, without the gamma. So, gamma actually is this particular coefficient which is changing from Galilean to Lorentz transformation. Now, let's just assume that we are still moving the beta system relative to alpha along the x-axis. So, y and z are staying as they are, but time has a different transformation. Time is not equal exactly to t. Again, there is a gamma and there is a t and there is another very important factor, x. Again, if v is very small, then gamma is very close to 1. This is very small, which means it goes to 0 and we have t is equal to t. That's how the Galilean system assumes that the time is absolute. In theory of relativity and Lorentz transformation, time is also changing from one system into another. Remember, time dilation and stuff like this we were talking about in the previous lectures. So, this makes the whole thing much more complex and what's most importantly, this is no longer equal. So, our Euclidean metric, which is this in beta system and this in alpha system, they are not equal to each other. So, if we are using Euclidean metric as a distance between two different points A and B and we take into consideration only space coordinate x, y and z, we will see that there is no equality. I mean, if you will substitute instead of lower case x, y and z, z's, you will have the t's, we will be screwing the whole picture. There is no equality here. But look, I mean, it's nice to have something which is invariant if you are going from one system into another. So, this is called, by the way, it's called quadratic form. Now, this quadratic form would used to be invariant relative to Galilean system of transformation, but since it's no longer invariant in Lorentz, we can't really use it as a distance between two points. Because, you know what, if there is a distance between two points in space, we would like to have this distance invariant if we are moving one inertial system, in any inertial system which is moving relative to each other. Because we are actually talking about that the inertial systems are supposed to have everything the same. So, all we can say right now that Euclidean metric is no longer one of the properties which is remaining invariant, which means it's not the proper distance. We cannot measure the distance between two points regardless of the time when all these measurements are taken. So, time needs to be somehow involved in the formula if we would like to invent something new. And that's exactly what we want to do. We want to invent a new metric, the metric which is more precise than this one, which takes into account time, and what's the most important is invariant relative to Lorentz transformation, which means the distance between two points, the new distance, let's call it relativistic distance. So, the relativistic distance between two points where not only space but also the time is involved must be invariant. And we are going to invent that formula. Okay? Right. Let's just think about how we do it. So, let's take a look at this formula. We would like to modernize it somehow. We have to take some time into account and we are hoping that we will find the new formula for distance, relativistic distance between two points. Well, put yourself into shoes of the first person who just didn't know this new definition of the distance and he wanted to invent it. Quite frankly, I just don't know who did. Maybe Lorentz did it, maybe not. I'm not really sure myself. Definitely not in Einstein because he already had this formula. So, he in his article in 1905 just mentioned, okay, consider this formula. I mean, it's obviously, you can check it's very easy that this is invariant relative to Lorentz transformation. Well, we are trying to get into the head of the person who invented this thing. So, he has to improve this formula somehow. Well, remember what we did when we were trying to invent basically the Lorentz transformation. I didn't give it to you as a formula or something like this. We were trying to devise it using certain general consideration. The postulates like constant speed of light and that the laws of physics are supposed to be exactly the same in all inertial systems. Right? And we had some system of equation, very simple linear equations which we solved and we came up with formula which seems to be correct. I will try to do the same thing. I will offer some very simple modernization of this, which depends on certain parameters which I will try to find out and if we will find it, that means we are going to unlock something and that's exactly what we need to do. So, how can we improve this particular formula to add something to it, to modernize it so it will be dependent on time as well and the Lorentz transformation is preserved. I mean, it's invariant relative to Lorentz transformation. Well, let me just give you the simplest solution to this. And again, solutions, if you want a solution, you basically start with something simple. Check if it works. And, well, if it works, great. The simple case is always better if it works. If it doesn't work, we will try to think something else. Okay, my first attempt was d square. I use capital G right now, which is basically Lorentz invariant distance between two points. Lorentz invariant metric. So, I will try basically just to modernize this formula. Isn't that simple? I mean, if I lived long time ago when the formula was not really invented yet, I would start from this. Now, what's my next step? Well, very simple. Substitute x, y and z in a simple case of one-dimensional along the x-axis. So, y is always basically and z is equal to each other. So, I would just take x and t and just check and I would fail. Because if you will just define this metric like this and you will compare d alpha. This is d alpha, actually, because I'm using capital and corresponding d beta of a, b, where I'm using Lore case. And this is t b minus t a square. Right? I would compare them expressing x and t using these formulas. Well, I will not have an equality. Okay, fine. Let's just do something more clever. Now, what's more clever than this one? Well, more clever than this one, I'll just use some kind of constant coefficient k times this. Now, that's the simplest linear improvement. I'll just multiply by unknown to me coefficient. And then I will try to equate these two things and consider this as an equation for k. And I will find k. k is equal to something. Now, these are basically linear equations relative to k. So, the most important problem is not to get lost in all these squares, etc., etc., because they're all kind of constant, right? This is the coordinates of a point and b point and the time when we took the measurements, etc. Now, so it's a simple algebraic exercise. The only thing is it's simple in principle, but it's long in calculations. Well, let's consider we did this. How then the k would be expressed as a solution to this equation d alpha is equal to d beta? Well, it will be expressed in the terms of xb, xa, I mean, the capital X and capital B and capital Y, B and capital A and capital Z and capital T. So, all these are parameters. And by the way, there will be another parameter, v and c, right? Because they're all participating in the transformation. So, all lower cases will also have v and c here, right? So, we will have a simple linear equation where k is expressed in terms of the capital coordinates, capital time, speed of one system going relative to another system and speed of light. That's not a good solution. Because if it's dependent, distance cannot be dependent on the definition of distance. I mean, different coefficients. We cannot use different coefficients to measure distance between these points relative to distance between their points. So, the coefficients should be the same, not dependent on exactly the points where we measure. So, in this particular part of space, this coefficient will be one and in this part of space it will be two. It's not good. Our coefficient k must be a real constant, okay? So, it must depend only on the constant. What's the constant? There's only one constant which is c, speed of light, in all the inertial system. Everything else depends on our experiment, the speed of beta system moving relative to alpha, the point exactly where we measure this distance, etc. So, that's an interesting observation and it's kind of put some doubts into somebody's mind who is trying to approach the system by basically resolving this as an equation for k. However, somebody was courageous enough to go through these calculations and, lo and behold, k is equal to minus t square as a result of these algebraic calculations which we did. Independent of location, independent on speed, the b. Only dependent on the speed of light. Now, I did it. I don't want to do it here because it's kind of lengthy and messy, but I did it in notes for this lecture. So, if you have any interest in basically having this equation and resolving it actually for k and you will see that everything cancels out except this one, I'm encouraging you to really take a look at these calculations. It's like, you know, a page or something like this because it's just too lengthy. That's the only problem. There is no difficulty. You don't have to invent something crazy. No, it's just a linear equation relative to k, but there are a lot of all these coefficients, squares, etc., etc. So, whenever you equate this to this, you will have a large formula basically which is linear relative to k. So, everything is simple but lengthy. But that's basically what I wanted to tell you that we have a formula for minus c square times this. This is a formula for metric minus c square in a relativistic term. So, if you put this particular definition of metric in our relativistic physics, in our relativistic space which includes the time, you will have invariant. So, this would be equal to this where beta is reference frame moving relative to alpha. Well, this same formula is written in slightly, I would say, more compact way, if you wish, that the d square s. Now, this is now not just the same letter d. This is differential actually. So, if you have two points infinitely close to each other, then it's equal to dx square plus dy square plus dz square minus c square dt square. Again, dt dx dy differential basically, so it's very, very small intervals. Why do we need it in differential form? Well, because the lengths we might not really measure along the straight line. We can measure using differential. We can actually have a curve in our space and divide it into infinitely small pieces and the lengths of each infinitely small pieces, square of the lengths, would be expressed as this particular thing in any inertial reference frame. Now, whether you will pay attention to this differential form or not is really not important. We will talk about this maybe later. But what's important is that to Euclidean metric we add another component with a minus c square coefficient and now this is invariant relative to Lorentz transformation. So, this is a relativistic matrix or matrix in the theory of relativity. And what's important, this is the first actually like glimpse that our space might not be three-dimensional. Maybe it's four-dimensional where the time is a fourth dimension. But this is another story and we will talk about this another time. Well, basically, let's say that I do suggest you to read the notes and yeah, do go through these calculations in the notes. I had some problems. By the way, when I was doing the calculations I probably like made three mistakes. So, it's only my or two mistakes, but my third attempt I think was really successful because I was just, you know, forgetting some square or some code or whatever. It's lengthy calculations. Well, whatever it is. Life, you know, life is massive but whenever you're coming up with some very interesting result this is kind of an interesting, relatively simple formula from very long and messy calculations to get this formula is really a good find. I mean, for every scientist who ever first did this and somebody did. I just don't know who did it. It was really a very satisfactory result that this formula represents invariant relative to Lorentz transformation definition of a relativistic matrix or relativity matrix. Okay. Okay, that's it for today. Read the book. I mean, the notes for this lecture. You can find it on Unisor.com. It's a relativity for all course and in this course you have to choose the part called Einstein view and this lecture is one of those in the menu for Einstein view. Thank you very much and good luck.