 Hi, I'm Zor. Welcome to Indizor Education. Today we will talk about space, not about space exploration. Now, space is the place where our physical processes actually take place. So, if time, which is the subject of the previous lecture, answered the question when something happened, the space answers the question where it happened. Now, we obviously understand intuitively what space around us actually is, and that's not the purpose of this lecture. The purpose of this lecture is basically approach the space from a mathematical and physical standpoint. Now, time was a particularly physical concept. Space is actually borrowed from mathematics. Now, in mathematics, we have many different spaces and it's kind of equivalent to the word set with certain properties. In physics, we are talking about only one particular, I mean classical physics, we are talking about one particular set with particular properties. It's a three-dimensional continuum. It's a regular, three-dimensional space as known from mathematics. Now, that basically implies that in mathematics, we have usually the coordinate system. There are many different coordinate systems. This is borrowed completely from mathematics into physics. So, the classical physics considers the space around us as basically a three-dimensional, mathematically known concept and it's implied that to know where exactly something happens in space, we need a regular Cartesian or some other coordinate system in this three-dimensional space and I have to actually tell you that most likely we will use the Cartesian system, which in turn implies that we have to have a special point as origin of the Cartesian system. We have to have three axes, x-axis, y-axis and z-axis, mutually perpendicular to each other, which basically define our Cartesian system and we also need a unit of measurement of the distance along each axis and this is something which we probably should talk about more in terms of physics rather than mathematics because in mass we can allow anything to be the unit of the lengths on each axis. In physics, there are certain standards. Now, the most important standard as the unit of measurement is a meter. Now, meter is, well, something about that size. More precisely, it was defined certain time ago as one ten millions of the distance from the north pole to equator, so this is our planet Earth, this is equator, this is north pole, so the distance along the meridian, it's a quarter basically of the total meridian, so one ten millions of this distance was considered to be a meter. In other words, one forty millions of the equator, which roughly the same thing, especially if you consider that Earth is a regular sphere, which is not. Now, obviously, contemporary physicists cannot be satisfied with this type of standard unit. Some time ago, they made, I think it was an alloy of platinum and iridium, this metal rod actually, they made and put two marks on it, which was considered to be the standard for the meter and it's probably still in Paris somewhere and the duplicates of this were spread around the world in different countries. And again, that wasn't really precise enough, well, because you have to maintain certain temperature, humidity, whatever it is. Now, a contemporary standard for one meter is based on the speed of light in the vacuum and it was measured and it was actually considered to be one over two nine nine seven nine two four five eight of the distance covered by the light in vacuum in one second. So, in one second, there is certain distance which light covers in vacuum. So, one over whatever, can't even read it, of this distance is considered to be the definition of a meter. And this is very precise and obviously, it is related to one second. Question is, what is one second? Is it defined? Yes, in the previous lecture about the time, we were talking about how one second is defined. It's something related to change of the energy state in the atom of season. Again, there is a certain number of these changes and certain number of these changes is, by definition, is one second. So, we have defined unit of measurements for the time, that was a previous lecture, and now we have defined the international unit of measurement for the lengths. Now, obviously, there are many other measurements. In the United States, we use miles, somewhere else, we might use somewhere else, and inches, by the way. And obviously, there are certain coefficients which are transforming one into another. But again, international standard is a meter and it can be divided into thousands of a meter, which is called millimeters, millions of the meters, micrometers, nanometers, whatever picometers, I don't know. Anyway, so, what's important is we have defined our system of coordinates, let's consider it's Cartesian coordinates, which means origin, axis, and the unit of measurement of the lengths per on each axis. That allows us to define a position of any point in space. Now, I would like to refer you to the term of continuity. This is a very important characteristic of our space. Again, it's borrowed from mathematics. In mathematics, there is a term called continuum, and that's the set, which basically, for instance, the set of all points on the line, or all points in space. So, we borrowed that term into the physics, and we consider our three-dimensional space to be continuous in the sense that, let's say, if you take two points and the line in between, then every point on this line also belongs to the same space. So, it's completely tightly packed with points. There are no holes or anything like that. Every point in the space is a valid point. Okay, and every point is defined by three coordinates. Now, assume it's Cartesian coordinates. Actually, we will probably do some other coordinates, like, for instance, cylindrical coordinates, whenever we will study rotation. But it doesn't really matter. Whatever it is it is. Right now, we will consider Cartesian coordinates. Now, what's important is that if you have these Cartesian coordinates, every point can be characterized by three coordinates. Now, if this point, and we're talking about motion right now, right, so if this point is moving as the time goes by, the position becomes actually a function of time. So, these three functions, which are coordinate functions, define at any moment time a certain position, certain point A in the space. Now, that's fine. However, I would like to slightly change the view point. You see, if you have a three-dimensional space, and you have a point, which something like this. So, if this is x, this is y, and this is z, then this piece would be x-coordinate, this piece would be y-coordinates, and something like this, and this piece would be z-coordinate, right? Now, at the same time, I can consider these three numbers, x, y, and z, as coordinates of a vector from the origin of the coordinates to this point A. Now, well, basically three numbers, they can obviously define the point, but they can also define the vector, which goes into this point from the origin of the coordinate system, right? Now, why I would prefer to use the vector in this case? Here is why. Let's consider that your point has moved from one position, let's say at moment T1, this is A1, it moves to A2, x of T2, y of T2, and z of T2. So, it moves, this is A1, and moves to, let's say, A2. Now, how can I specify this segment from A1 to A2? Well, I can specify it by its length, obviously, and that requires the Pythagorean theorem. At the same time, if I am really talking about vectors, that's much easier, because if this is not just a point A1, but this is a vector A1, and this is a vector A2, then I can always say that A2 minus A1, and this is the vector algebra, is equal to a vector which is called a displacement. And the coordinates of this vector are very easy to calculate. It's x2 of T minus x, I mean x2 T2 minus x of T1. That's my x coordinate, right? That's how the difference between vectors is calculated. Now, y of T2 minus y of T1 and z of T2 minus z of T1. So, this is my displacement vector. Vector algebra is much more convenient when you're dealing with points in this three-dimensional space. Just because, for instance, the displacement from a point to a point in vector form is actually calculated very easily, and you know what it is. Because if you have just two points, then how do you qualify the displacement? Yes, one thing is the length, but you still need the direction where exactly you are displacing the point A1 to the A2. So, somehow you have to incorporate, in your explanation of how A1 moved to A2, you have to incorporate not only the distance, but also a direction. And that's why it's very important to have this vector representation. It does not make our life any more difficult. It's actually the other way around. It makes it much simpler, because in vector format, the motion is very easily expressed. Motion from a point to a point is expressed in vector format much easier. Okay, I would like also, above and beyond continuity of the space, which we know, I would like also to talk about other two very important properties. Now, the first important property is very much similar to the analogous property of the time. Now, remember when I was talking about time, I said that, well, if you have an experiment and it produced certain results, and then you repeat exactly the same experiment, next day or next hour or whatever, you must have exactly the same results. That means that the time is homogeneous. There are no preferable point of time relative to another point of time. All moments of time are exactly the same rights. It's uniform. The time goes uniformly. Exactly the same uniformity, but now in terms of location, we are applying to our physical space. So the space is uniform as far as location is considered. If you have an experiment in one particular location, and then you make exactly the same experiment, which means exactly all the conditions must be exactly the same, and you make this experiment in another location, results must be the same. So this uniformity relative to location is called homogeneousness. So our space is homogeneous. That's very important. Location to location, there are absolutely no difference. This point has exactly the same space properties as another point. Again, as long as our experiment, which we repeat here and there, is exactly repeated under exactly the same conditions. So homogeneous. The second property, which I would like to point your attention to is, let's say you have an experiment in one particular location, and then you turn it around. Again, our intuition tells us there should be no difference. I mean, obviously, if you are considering something like making an experiment in a magnetic field of the Earth, and you're turning and your experiment involves somehow magnetism, then there might be certain differences. But why? That's because not every condition is exactly the same. When I'm talking about every condition, means every condition should be the same. And if you turn your apparatus, which is making your experiment and make your experiment again, all the conditions must be the same. In case of my example about magnetic field, obviously the condition is not the same, because the position of the apparatus relative to the magnetic field will be changed. Now, if nothing's changed, then the turning around should not really result in any differences in the experimentation. Intuitively, we understand that that might be the case, right? And there is a special name for this. Now, we are talking about isotropic on the space. Isotropic plate. Our space is isotropic. So, let me just summarize. We have continuity, obviously, but that's not the physical part of the space. That's actually coming from mathematics. Now, homogenous, homogenous, something like this. I hope I spelled it correctly. Homogenous, that means that if you take your apparatus and put it in another location, must be the same result. And the third is isotropic, which means it's exactly the same whether you're turning or not. So, this is kind of a symmetry regular to angular movement. And this is the symmetry relative to transformation of the position, transposition of whatever it is. And continuity, again, as I was saying, that's coming from the mathematics. So, these very important properties, and if you remember, we were talking about undefined concept, which we don't really want to point. This is the time or this is the space. Space is also kind of undefined. But it has properties which we can really use. Now, in this particular case, we're using the property of being three-dimensional, which means we can specify a certain coordinate system with three-dimensions. And it also has these properties of being homogenous and isotropic. Symmetrical, so to speak. Symmetrical relative to position and symmetrical relative to angular movement. Well, that's basically all I wanted to share with you as far as the space is concerned. I think it should be very important to understand that space is still kind of an abstract concept, which obviously we consider that this real space, which we live in, corresponds to our abstract concept. So, whenever we're studying this abstract concept of space, we hopefully inevitably get the properties of our own space. But again, don't forget that this classical physics is just a model of the real world. It's our model, which is in our mind. We have imagined that it corresponds to this model pretty well. Well, apparently in the 20th century, there were newer developments, which were more precisely corresponding to our real world, and that's theory of relativity and quantum mechanics and whatever grew out of those things. Well, they are more precisely formulated models. Right now, we are talking about the models, which originated in 17th century by Newton, Leibniz and other guys, and they correspond to our real world pretty well. Now, what else is very important as far as these symmetries are important? Well, there was a very interesting theorem proven that from these symmetries we can derive certain conservation laws, like conservation of energy, conservation of momentum, etc. We will talk about these conservation laws later on. But for now, I would like you really to admire the thought, which was behind this, that from the... Now, this is the space. Now, the time we were talking about also has a continuity and the time is homogeneous. So, from the homogeneous of time, we can derive the law of conservation of energy. Well, I don't know what you think about this, but quite frankly, when I learned about the possibility of this, and I'm not in a position to present you the proof or something like this, but just admire this as a fact, from something like uniformity or homogeneity of the time, we can derive the law of conservation of the energy. That's a pretty remarkable thing. Now, as far as these symmetries of our space seem relative to change of location, again, also it can be proven that the result of this is the law of conservation of momentum of motion. And from isotropic property of our space can be derived the law of conservation of angular momentum. Again, we didn't really talk about this. I just want you to feel how great laws, which basically guide all our physical world, how they can be derived from symmetries, from the symmetry of the space or the symmetry of the time. Great achievements. Okay, basically, that's all I wanted to talk about as far as the space is concerned. Thank you very much and good luck.