Why ?

You can refuse to answer this question – the world works like that and that's it. Yes, the world works like that. But the world doesn't know, what “pseudo-Euclidean” space. Yes, and we are up to Г.Минковского did not know. To the majority, even now this word does not mean much.. So what's behind the statement “Space-Time pseudo-Euclidean in the vicinity of the point“?

I will try to show, that there is one very trivial fact behind this statement, generally speaking, non-property (“by law”) the world as such, but the quite understandable property of the existing limitations in the ability to describe this world.

Probably, indication of that, what we – any experimenter, is the subject alive, whether the object is inanimate – it doesn't matter – we are parts of the world, but in no way equivalent to it as a whole, is a fairly common place. But how much follows from this general passage! In particular, also then, what we call the pseudo-Euclidean space-time.

To better understand, what pseudo-Euclidean , let's figure it out first, what Euclidean.

We get the concept of Euclidean space at school. For simplicity, we will talk about two-dimensional space, unimaginable, nor for immediate implementation. Probably, any person as an example of an area in Euclidean space will suggest looking at a piece of paper. And it will be right, but not completely! Actually, a sheet of paper is an example nested hierarchies of some, more general than Euclidean, spaces – diversity, spaces of affine connection, Riemannian space, affine (linear) space and only then Euclidean. Yes and that, if only we already mean, that the position of a point on the sheet is described using coordinates determined in some way.

What is the catch? The thing is, what many points become space only by specifying some well-defined way (ways!) assign numbers to these points – “address” – which allow them (points) distinguish from each other.

How do we do it? Nothing could be easier! We take right triangle, select on the sheet point, we call it the origin (countdown) and draw two perpendicular lines through it – coordinate system axes. We postpone equal intervals on each of the axes from the origin, let's say, in one centimeter and done! we made Euclidean space on this sheet of paper. From each point, perpendiculars can be dropped on both axes and two coordinates can be assigned to the point – number of units on each of the axes, separating the projection of the point from the origin. You can also say, that in our construction two mutually perpendicular axes pass through each point. At the same time, as proved Pythagoras, we have a well-defined Euclidean distance from our point to the origin, and with it the Euclidean distance between any two points on a sheet of paper. I want to emphasize – it is all of the above that together makes a sheet of paper an example of a region in Euclidean space.

And if instead compulsory right angle between the axes we allow any angles (but always the same coordinate systems in a given implementation)? Letter, ruled with a slope. What's easier? Who is older, can still remember such calligraphy books in elementary school. Can? Yes of course you can. Such a leaf remains an example of a Euclidean space? Not. it will be already an example of affinity (linear) space. More general.

Times more general, then we have lost something. Then, what is in Euclidean space and what is not yet in affine. What is it? Pythagorean theorem and Euclidean metric, an allegory of the presence of which is the Pythagorean theorem. We've lost the Euclidean distance between points. And here is a well-defined linear distance between any two points, ie. we still have a linear metric. Only the distance is not calculated using the Pythagorean theorem.

And now let the angles between the axes change when going from point to point. What will happen? Our sheet has ceased to be an example of affine space too.! But here he is, hasn't gone anywhere! What an example is he now? Easy to guess, what an example of some even more general space – Риманова. And the distance between the points is still or not already? There is also, metric still exists. But this is no longer the same linear distance., to calculate which it was enough to know the coordinates of only two any points. Now the distance must be calculated by integrating along the path (ie. accumulate little by little, moving along some line, leading from one point to another). The distance turns out to be the same, how many ways! But! Among all the distances, there is one – the greatest (or the least). Way, which gives such a distance is called a geodesic.

But let's leave this exciting path for now.. She will lead us away from our goal – pseudo-Euclidean. Easy to understand, Prestaurate pseudo- means, that the Euclidean is, as it were. And we lost her here long ago, still on the first step to freedom. Means, we went a little the wrong way, when did the angles between the axes (but an illustration of, that any agreement is extremely important for the final result we have received!)

so, the angles between the axes remain straight! I emphasize – this agreement, no more! But what is also important at the same time – we have the practical ability to adhere to this agreement. We have right-angled triangles. Solid, good ones, perfectly unchanged right-angled triangles. The truth is? True unchanging? okay, let's leave that too “for later”.

So what else can we easily and immediately change in our construction of coordinates for Euclidean space? Like what – units. Centimeters, inches, elbows, planted. Yes, and meters and kilometers – also other units, not centimeters.

Who told us to lay the same units on both axes? We will always plot centimeters along one axis., by another inches. How, we will finally coordinate Europe with England and America. We have the right? Why not? We have! That's just…. Yes, we are definitely something lost. So what? Well, of course, distance again… Moreover, now it's overhauled. Not only Euclidean, but in general, metric distance. Indeed, does it make much sense to mix inches with centimeters in some formula? Well, let's add 5 inches with 3 centimeters. And what do we get? Yes, not good. But the lack of distance in a given space does not close the ability to describe points on a sheet of paper and thus. But this will again lead us away from pseudo-Euclideanism.. Means, we must keep the distance. Which means, what units on all axes must be the same!

Okay, the units on both axes are the same. And then what will we release? Well, eg, let the axes will be curves, not direct. Oh, again we will lose distance… And if we allow units (together, for both axes simultaneously) change from point to point, as the corners were allowed, and what led to Riemannian space? Not, again the distance will disappear. So what else can you free?! There is nothing else left, tried everything!

Not, something we missed. AND it is really connected with the choice of units of measurement along different axes, only harder, what have we done so far.

Note, how good we are, convenient to manipulate a sheet of paper. We apply our triangle and so, and so. We rotate it as we want, carry over. Why is it possible? Because, what triangle exists outside sheet of paper. Is not part of that space, to describe which is used. Does this leave any imprint on the result? Imposes, and what!

Then, that the units are outside of my sheet of paper, allowed me to avoid many of the caveats in the previous reasoning, which should inevitably appear, if i originally meant, that the units of measure are the internal objects on this sheet. As a matter of fact, I typed into this sheet, what I wanted – which units, how they change from point to point, not caring, whether they really exist there or not. I implicitly imposed on that area of ​​space, which he modeled with a sheet of paper, a certain structure, which I did not even mention. This structure is called an object affine connection, makes sense of the rate of relative unit change, implementing this coordinate system (in it) when moving from point to point. AND space becomesI, eg, Euclidean not simply because, that we do not allow non-Cartesian coordinates. And therefore, what's in it there are objects, which can be used as units, generating Cartesian coordinates and in which (in Cartesian coordinates) this structure, affine connection, everywhere, at each point zero. What does it mean, zero affine connection? Yes, very simple – all these units are the same everywhere. Ie. with fully self-consistent, internal description of the geometry of some space, the main thing – are there such coordinates, as we need, is it possible to implement them with internal objects. And for us – in our case, when we put units of measurement on a sheet of paper from the outside, as we want – everything is allowed.

In particular, we use a triangle – ie. both scales at once together, with a given angle between them at a given point and an implied equality of units on both axes. Further, our triangle can be transferred without changing these ratios to any point on a sheet of paper and rotated as desired, including, So, that one axis can be aligned with the other (like two instances of a triangle in one place at once) and their units can be compared directly. Ability to transfer the entire benchmark (triangle) unchanged means, that the connection is zero and the space is Euclidean. And the ability to turn makes it possible to confirm, what was meant – selection of identical units, guarantees it. But if such opportunities (turn) you have not? What will happen? This is where we, finally, and grope a path to understanding that, where does the prefix pseudo come from.

Imagine, that you live inside this sheet of paper, you are part of it, line in it. AND, of course, you consider yourself straight. (At least, straighter than everyone else. And what? You have the right, until proven otherwise.) Your existence realizes your time (don't feel the connection – lifetime – the most familiar phrase, is not it?). Your existence – it's straight on (in) sheet of paper. There are other direct. And curves too. You even communicate with them somehow.. At least, sometimes intersect or exchange something (send a point, which, buried in another line, comes back to you). In this way, you know, that your world is two-dimensional, at least. You – one dimension, there is something else – means there are more than one measurements. This is how you build the image of your world as a two-dimensional space.. What? Your unit, your time scale always with you and, by itself, you consider it the same in all moments of your existence. it realizable you scale. Here the idea of ​​Euclideanism has already appeared. Did not notice? But what about – your scale is unchanged, the same, by definition. (By your definition, but what do you care, if others have their own definitions? As long as you try for yourself, we will agree with others later.) But there are two dimensions! In the frame, you need to have two identical (and unchanging) scale. This is where you should envy me. I am sitting over a sheet of paper with my triangle, I'm not blowing my mustache. What do you do? Where to get the second scale? There is no, because he is on your line and that's it! Answer – and come up with. Let it be. And not some kind of overwhelming, and exactly like this, as you need – ie. orthogonal (perpendicular) to your time scale, and, of course, constant everywhere. Master – barin. What he wants, then he comes up with. Your dear, realizable scale is constant. And the invented should not be worse. Your world has become (two-dimensional) Euclidean. Wherever you are, you have two great scales to describe it. One temporary and one, let's say, spatial… what? Ah, you don't go everywhere? okay, we will moderate the claims – all this is so beautiful only in your surroundings, ie. peace (its description by two-dimensional space-time) Euclidean locally, in the vicinity of every point of your (lines) existence.

Euclid?! Let, I can make sure with my triangle, that my units for both axes are equal, turning triangle. And so you can? Not? And why? Ah, you only have one realizable unit, time scale. And how are you there inside a sheet of paper do not spin, she is the only one and will remain. Well, no way it is impossible to combine the realizable scale with the imaginary. That must always be orthogonal to the time scale.. After all, we imagined him as such. And a point. Well, your scales are not the same! AND it must be admitted explicitly. In your mathematical image of space-time, the time scale cannot turn into the space scale in any case.. And in Euclidean space can. How can this be depicted mathematically? This is where it appears pseudo-Euclidean. She also depicts the inequality of scales in the frame. Their fundamental difference from each other.

so, we have two fundamentally different scales. Hence, it is desirable to depict the corresponding coordinates with different numbers.. And what choice do we have? Right, real and imaginary numbers – this is exactly what the names mean, what do we need. Imaginary = Imaginary. Let the time coordinate be represented by a real number (measured by the realizable scale), and spatial – imaginary (measured by an imaginary scale). Space-time has the properties of Euclidean in the sense, that between any two points can be defined in an invariant way (with respect to the whole group of our Cartesian coordinates) distance, calculated according to the Pythagorean theorem: r²=t²+x²

Here only x is an imaginary number, but we can't see it in any way. Let's make the entry explicit – let the spatial coordinate explicitly contain the imaginary unit : ix . Then the distance, computed literally as Euclidean, turns out to be actually different: r²=t²-x² since the square of the imaginary unit gives minus one. So we got something like Euclidean space., but no, other – pseudo-Euclidean.

Although the use of imaginary numbers suggests itself, but it is not necessary, if we focus our attention, how is it very often done, on keeping the invariant form to calculate the squared distance using the minus sign instead of plus (not to be confused with purely spatial distance, it is usually called an interval) when transforming coordinates. But then it becomes not obvious the difference between the spatial and temporal coordinates as measured by fundamentally different scales.. Well, probably, can I still remind, that historically in physics the imaginary coordinate is usually assumed to be time. We are very used to drawing spatial coordinates on paper and assume them to be valid.. What to call, for result, generally, not that important, if only the interval is calculated correctly. but, inverted terminology never contributes to an easy understanding of the essence of the matter.

Okay, we found out, what lines, existing in a sheet of paper and wanting to describe it from the inside, will be forced to locally use the pseudo-Euclidean space as an image of their immediate neighborhood. Well, our physical world? Yes, it will be more difficult, of course. We had to come up with as many as three additional spatial units. For the rest, we are no better than lines on paper and our capabilities are no more. That is why we also describe our world locally with pseudo-Euclidean space..

Say – all this is not true! Here, look I have good triangles, to measure spatial gaps! Realized by objects from our world. Do you want – wooden, do you want – metal, do you want – plastic. Yes? Didn't you forget, what to find out this very distance, you need to look at the two ends of the label, representing the unit? AND a period of time will pass between these events, how are you not tricky. And the real, not an imaginary unit should give you a spatial coordinate (anywhere far from the origin) instantly, to any given, single moment in time. So the word “look” more important in your statement than others. And its presence refutes the very statement. You cannot instantly assign spatial coordinates to anything in this best of worlds..

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