What is Affine Connectivity?

Affine connectivity. This term says little, even to a human., university educated. but, I hope to explain with fairly simple examples, what is it and why is it of decisive importance for physics.

Hope, you have already looked at the articles about relativity, tensors, pseudo-Euclidean and metric, and have some idea of ​​how, how at each point in space-time, sets of units of measurement can be selected in different ways. I will continue with the assumption, that this has already been done and all points of a certain region of space-time are equipped with such sets of scales and there is an image of this region in the form of a numerical continuum, described by the coordinate system. For simplicity, let the space be two-dimensional and the coordinates of the point are denoted as xⁱ, i=1,2. The scale itself, units, in every point are represented by contravariant vectors eⁱ₁ , with components 1 and 0, and eⁱ₂ , with components 0 and 1. Notice, the components of these vectors in a given coordinate system are everywhere the same, at every point. In addition to the scale vectors, it is possible at each point to determine a set of vectors of infinitesimal displacement from it – dxⁱ. There are infinitely many such vectors, one for each path, leading from the point. But traditionally they are spoken of in the singular. – infinitesimal displacement vector, because usually one vector is always meant from the whole set, one way.

For ease of discussion, let us consider the classic approximation. Such, that the point in the image represents a large enough piece of the world, in which several items are placed at once, and observers with their different sets of scales too. In some “point” (and in the image it will be exactly the point) of such space we spent continuous in time (let my first coordinate, x¹, it will be time) measuring various properties of some selected object. Let these properties be represented by a scalar s and vector Pⁱ as a function of time at a given point. That is, as values, known on the timeline, passing through this point in space. And the values Pⁱ are the same for every moment in time. And some other observer did the same, but by its scale, with results s and P^i’, famous, obviously, on the same line in space-time. Besides, scales from another set were also mutually measured. And it turned out, that from the point of view of the second observer both vectors, depicting our scope, have components not only not equal to the above values, but generally change in his time. Also, like its scale in our measurements. Here's a question for you – what measurement results can be compared with what in such a situation, and be sure, that the comparison results will be the same for both observers (in both coordinate systems)? After all, only such results can be discussed by everyone and accepted by everyone without objection.. The answer is. Scalars can always be compared, for any observers – after all, they do not depend on the choice of units of measurement. And the vectors depend. And if at one point there are units, which they were measured alone, and already in the next one they have changed, then how to compare them? You can say, that there is no need to do nonsense, but it was necessary to force the second observer to take good (namely, coinciding with our) watch and do not fool us. But he will say, that we are such slobs, our watches are bad, not his. Who is right? You can agree to use the same clock. Will solve this problem? The third will come, with my watch and all over again. We will persuade him too… And the fourth, fifth, etc. etc.? Well, check all your clocks at the same time. What then? You know, that they have such a property to change in the course of their existence, compared to any others. Moreover, the rulers also behave the same way.… How to solve this problem without squabbles and contracts, how to avoid doubts once and for all? I would like to hope, that there are such ideal objects in nature, which never change anywhere, and they are at every point in space-time, so they can be used as the best, we need “good ones” watches and rulers. Then point out to everyone and everyone, that it is enough to use only these scales for measurements – and the question is closed. And other possible measurement procedures – bad, you can forget about them. The concept of Euclidean (and pseudo-Euclidean) space, it is on this belief that it is based. By the way, the first article is devoted to this issue. A. Einstein on special relativity (“To the electrodynamics of moving media”).

Only here is the problem. How to really make sure, what “good ones” the clock is everywhere and everywhere, and besides, they show the same time? And the rulers too. We know for sure, it is not so. So we can't get away from recognizing the equality of all measurement procedures without exception.. And you need to learn how to work with measurement results, which they give. What does it mean? This means, what every observer must admit, that its sets of scales can change when moving from point to point in space-time. Respectively, and when comparing the measurement results of one object at different points, you need to take into account, that not only the object itself could change, but the scale too. Affine (linear) connectivity is that structure, which explicitly describes the potential change in scale and allows you to work without problems in such a situation.

The scale is depicted by a contravariant vector eⁱ_n. The subscript n at the bottom here denotes the scale number, and not the components of the vector. May we admit, that when shifting to an adjacent, infinitely close point, the scale will differ from its value at the current point in the first, linear approximation in displacement dx^j from a point by some value deⁱ_n. What does linear approximation mean? By the way, It is with this approximation that the definition “affine”. It means that, that it is possible to write n relations for each i-th component

deⁱ_n= {Eⁱ_j}_ndx^j .

In this formula, the index j implies the summation. In our two-dimensional case

deⁱ_n= {Eⁱ₁}_ndx¹ +{Eⁱ₂}_ndx².

Further, we will follow this convention always – if the indices above and below in the formula are repeated, then this is the sum of all the values ​​of the index.

Symbols {Eⁱ_j}_n the coefficients in the above expansion.

On the other hand, scales at the shifted point are also vectors of the same kind. And their values ​​can be considered as the result of the action of some transformation on the values ​​of the scales at the starting point:

eⁱ_n+deⁱ_n=eⁱ_n +{Uⁱ_k}_n e^k_n.

This ratio should be understood as follows — changes in scale should be considered proportional to the scale itself. Ie. need to track relative changes in scale. After all, the scales themselves are used to measure changes., existing at this point, no others. Therefore, the independent quantities are relative changes, not absolute. Respectively, the coefficients introduced above can also be written as a convolution with the reference vectors (that is, with a set of scales).

{Eⁱ_j}_n= – Gⁱ_jk e^k_n

Why did I put a minus here, I'll tell you later. Now in excuse – I enter the notation myself, well, it's convenient for me so, why not? Let's substitute this ratio instead of symbols {Eⁱ_j}_n . It will turn out

deⁱ_n= – Gⁱ_jk dx^j e^k_n.

note, on the left is not a vector! And odds Gⁱ_jk not tensor! It is very important. And now I will rewrite it all again in a different way.

Fromⁱ_n= deⁱ_n + Gⁱ_jk dx^j e^k_n = 0.

This equality is valid in any coordinate system, for any choice of scales. And notice, now the vector is already on the left! Equal to zero by definition. Why? AND in my coordinate system, any scale of mine, which I carry with me, for me, by definition, always coincides with myself! It is for others that he can change, and for me – not. And what is it I wrote here so complicated? Nothing special, just formalized the scalability assumption, wrote down this possible change attributed to the scales themselves and, as a first approximation, as proportional to the displacement from my point. Instead of the changes themselves, I introduced the coefficients Gⁱ_jk, which in my coordinate system will be a function of the point and with their help it is possible to link the measurement results at neighboring points (in other coordinate systems, of course, also, but there these coefficients will be other functions). This is why mathematicians called this structure connectivity, and Gⁱ_jk coefficients of affine (linear in displacement) connectivity. You can also meet with that, what are they called symbols Christoffel. But this name is usually used only in the particular case of Riemannian spaces.

Well, finally the subject of this article is before you. Let's discuss now, what else can be said about connectivity, besides what has already been said. And why is it such an important structure for the space of time.

The physical meaning of connectivity is clear enough from the very method of determining the connectivity coefficients. These are the rates of relative changes of objects., selected in this procedure of measurements as units when going from point to point in the described space. This is not a tensor, but a more general geometric object. How its components are transformed when moving to other coordinate systems is extremely important for mathematics. (and for physics too), but here we do not need to know this. It is only important to understand, what in another coordinate system, the connectivity coefficients will also represent the rates of relative changes in objects, but others, namely those, which are the units in the new measurement procedure. Connectivity has another, a more familiar meaning for physicists. Physically, connectivity is nothing, as a complex of potentials of a single physical field. But now I'm not going to dwell on it.. I will dwell in more detail on that, what does the presence of connectivity give for mathematics (and, Consequently, for physics too).

Spaces, in which at each point an affine connection is defined (its coefficients as functions of a point are given in a certain coordinate system, and therefore in all the others too), are called spaces of affine connection. Riemanov and, respectively, Euclidean spaces are special cases of spaces with affine connection. The presence of connectivity makes it possible in a covariant way (that is, with consistent results for any valid coordinates) not only perform algebraic operations with tensors (with measurement results of selected objects) but also differentiate, and even integrate them. The results of these operations are again tensors, which are mathematical images of the measurement results of selected objects. This means that the results of operations can also be matched to certain measured objects..

The affine connection is used to define operation of covariant or absolute differentiation of tensor quantities on such spaces. The symbol is used to indicate this operation. D, unlike the symbol of the usual differential d. But there is one more operation, closely related, which is called the parallel transfer of vectors and other tensors along the curve. Most often, affine connection is introduced by mathematicians precisely with the help of the concept of parallel transfer. Naturally, nothing additional appears, the emphasis of the presentation just shifts a little. I paid attention to the fact, that changes in scale components are measured by the scales themselves. But the same can be expressed as a deviation of the scale in the adjacent (infinitely close) point from the scale transferred there parallel to itself from this point. The words “parallel to myself” are equivalent to the ratio Fromⁱ_n= 0. Means it, which by definition, in the given coordinate system, all basis vectors (units, necessary and sufficient to describe the space) translate parallel along any coordinate line, outgoing from this point. I.e, when offset from point, basis vectors, transferred there in parallel, coincide with existing at the new point. This definition can also be interpreted and vice versa. – parallel transfer is such a transfer, at which the transferred vector coincides with the existing at the point, where is it transferred. The components of the transferred vector are assigned the values ​​of the components of the vector, existing at a neighboring point. The definition guarantees such a property in any coordinate system only for basis vectors. Still need to clarify, that the coordinate line is, outgoing from point, to which one of the basis vectors is tangent, that is, the line is exactly in the direction of this vector. And here are the other vectors, existing at this point, are by no means obliged to translate parallel along such a line in any coordinate system. But! There is a coordinate system, in which the given vector is carried in parallel along some line (that is, its absolute differential D along the whole line is zero)! This is that coordinate system, in which the given vector (if it is contravariant, of course) is one of the vectors of the basis, one of the units. And the corresponding line is the coordinate. As should be clear from the above, the result of parallel translation of any vector depends on the path of this translation.

I want to emphasize one most important property of connectivity. We recorded its coefficients as the results of measurements of changes in our scales with the same scales.. It sounds good, but how to make such measurements? Indeed, from the point of view of the existence of the scales themselves, they inevitably remain identical to themselves at all moments of their existence.! By the way, this is precisely what is written by the relation Fromⁱ_n= 0. We have by no means turned a blind eye to this problem.. What does it mean? But what. Yes, really, there is no direct way, by measuring the set of scales itself, selected to create the current image of the world in this area, set the type of connectivity in this particular coordinate system. But this is not very important either.. It is enough to recognize and take into account, that the selected scales, Maybe, change from point to point. Yes, in this sense, connectivity will contain a certain amount of uncertainty. But everything, with regard to the relationship between the measured values, will be quite definite. As you understand, the remark is about the connectedness description of relationships in the real world. And in the world of pure mathematics, who is just studying, what opportunities does this tool give her, there is no such problem at all. We count, that the connectivity is given, and basta! And you know, what's funny? It turns out, if the connectivity coefficients in space are known as functions of coordinates, then everything is known about such a space!

Yes, of course, for, to describe it “all” the richest apparatus was developed, about which I will say only a few words here. Although connectivity is not a tensor, but generates (connected components can be formed using algebraic operations and / or differentiation) several very important tensors. These include the torsion tensor and curvature tensor, along with their packages. Let me remind you, what does this mean, that some connectivity properties can be obtained as a result of measurements of objects. Further, the classification of spaces according to their properties begins. – under such conditions, it turns out that, with others – this. Respectively, spaces are named – equiaffine, Римановы, affine, Euclidean… Euclidean you know best. How are they characteristic in terms of connectivity?? But what. At first, these are such spaces, in which there are special, sets of scales identical at all points. If you choose one of these sets of scales for building a coordinate system in space, then such a system will cover the entire space and the coefficients of affine connection in it will be equal to zero everywhere! Secondly, these scales also allow you to form a metric, also the same at all points in space! I.e, in such (and only in this!) space and the opportunity to have “good ones” units.

 

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