What it is? Is spacetime homogeneous and isotropic? Why I decided to pay special attention to this issue?

First answer to the last question. The thing is, what's in books and articles, concerning the properties of space and time, as popular science, and scientific, written by very worthy scientists, you can often find statements like:

• Spacetime is homogeneous and isotropic (good still, if it mentions, on what scale). More often, true, claim, that space and homogeneous, and isotropic, and time is only uniform.
• It is the homogeneity and isotropy of space and the homogeneity of time that are the cause of the laws of conservation of energy-momentum and angular momentum.

Both of these statements are wrong. (the second partially), and their widespread occurrence leads to serious problems in understanding physics. One of these problems is the identification of the homogeneity and isotropy of space-time with the quite natural requirement that the results of physical experiments be independent of the choice of the coordinate system in space-time, in particular from the choice of the origin and direction of the axes. Moreover, such independence is too often interpreted overly straightforward.. This is why I want to discuss these concepts..

First, let's give definitions of these words themselves., to avoid any misunderstanding in their understanding.

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Uniformity. In mathematics, in the language most free of contradictions, such regions of space are considered homogeneous (space in the broadest mathematical sense), whose points have all properties exactly the same. Examples of such spaces, homogeneous as a whole, are one-dimensional infinite or closed lines, have no self-intersections and no points allocated on them, as well as Euclidean spaces of two or more dimensions. To more complex spaces like this, still easily accessible to our imaginations, can be attributed to the surfaces of spheres of arbitrary radius (example of enclosed space, each point of which has the same positive curvature) and surfaces, formed by the rotation of one branch of the hyperbola around the x or y axis (example of infinite space, each point of which also has the same curvature, but already negative). Of course, there are other examples of homogeneous spaces, and very. But I think that already given is quite enough, to understand the meaning of the word homogeneity in relation to the concept of space. Pretty clear, what space, generally inhomogeneous, may contain some quite homogeneous areas. The simplest example is a line segment. All points of a line segment, except for its ends, have the same properties.

And in everyday life, not so accurate, we assume something homogeneous then and only then, when its arbitrary components look exactly the same to us. Large enough capacity (but not too big), into which pure water or some other clean liquid is poured, gives us a visible example of, what we call homogeneous substance, three-dimensional. As well as a fairly smooth paper or table surface give us the same example of a homogeneous substance., having two dimensions. But in everyday life we ​​are already accustomed to understand, that this homogeneity may be the result of not true (ie. all without exception, when choosing any, including arbitrarily small parts of the substance) properties of the substances themselves, and that approximation, in which we consider them. I mean, that taking a good magnifier, we can see even on the smoothest paper certain roughness in some places, or even fibers, of which this paper is composed. Sometimes it happens that way, that by running your palm over the seemingly smooth (ie. homogeneous) we risk getting a splinter on the table. And with the help of a microscope, foreign inclusions can also be detected in clean water., eg, bacteria. Moreover, we already know also, that all substances we know, no matter how homogeneous they seem to us at the everyday level, ultimately consist of molecules and / or atoms, divided emptiness. Ie. with due pickiness, they turn out to be completely heterogeneous. Let's note this most important point: in the real world, some parts of it can in certain approximations described as homogeneous, being at the same time highly heterogeneous in other approximations. These examples illustrate well the fact, that the critical property of the approximation, on which depends whether the description will talk about the homogeneity or heterogeneity of the substance in question, is the choice of sizes of those parts of the world, which in the description are associated with points (in the case of a strict mathematical language) or which seem to us to merge (in everyday language). Ie. from the point of view of experience, the question of homogeneity or heterogeneity of one or another part of the world is rigidly connected with the choice of the scale, units, less than which everything is supposed to have no dimensions. Besides, associating homogeneity with the adopted approximation, ie. actually slightly changing the very concept of homogeneity, we can go ahead and talk about “partial homogeneity” as homogeneity in only one property of a point, or by an incomplete set of its properties. To this set of properties, which is the same, persists from point to point. But should be clear, that such extensions should be clearly stipulated. “Partial uniformity” does not mean full homogeneity of a given area of ​​space.

Isotropy. This is also the same. But not all properties of all points of the region of space. One set of properties is highlighted, inherent in any point — are considered directions from this point, ie. connections this particular point with all neighboring. Clear enough, what if we talk about homogeneity can be applied to discrete spaces (sets of unconnected points), then isotropy implies the presence of connections between points (elements) space. Which means we are talking about continuous spaces, continua. Isotropy at a given point implies, that connections with all neighboring points without exception (different directions from a given point) exactly the same. Let's mark the words “isotropy at a given point”. Their presence means, that the concept of isotropy, generally speaking, applies to individual points of a region of space. When they talk about the isotropy of all space or some of its region, then imply the fulfillment of this condition for all points of space or its region. And this also requires homogeneity of space or region., at least partial, at least for this property. In the same time, if space is homogeneous, then, if it is continuous, it is automatically and isotropic, since in the definition of homogeneity we are talking about the coincidence of all the properties of points without exception.

Should add, that the concept of isotropy at a point also allows us to speak of limited isotropy, excluding some areas. for instance, directions at point, located on a spherical surface in three-dimensional Euclidean space, all are the same in the sense of three-dimensional space, if the belonging of a given point to a sphere is not essential. And only two-dimensional space remains isotropic, if at the same time we strictly follow precisely the belonging of the point under consideration to the distinguished sphere.

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Now it will be easy for us to understand, is spacetime homogeneous and isotropic. If we are talking about space-time as a whole, a single object, ie. about the image The universe, then the answer is clearer than clear — of course not. The universe contains all and all kinds of objects, parts of it, which are different from each other, and space-time as an image, the description of such a universe is not uniform (and therefore not isotropic) by its very meaning. There is no one in a homogeneous universe (and nothing) would ask the very question. All points in it (parts of it) are exactly the same and therefore nothing actually exists. However, in a largely inhomogeneous Universe, some subregions are not excluded, possibly homogeneous in the absolute sense, or in limited — in a certain approximation. This is even more true in relation to isotropy. The presence of such areas, parts of the world and priori cannot be denied. Our conclusion will not change if we select only space in space-time. At first, for the entire universe at once this simply cannot be done. But even for its individual parts, local, local instantaneous sections, which with a certain stretch can be considered a space, separated from time (local space), this space as a whole cannot be homogeneous. All for the same reason — it will contain sections of objects that differ from each other, existing in this area of ​​space-time.

If we are talking about some hypothetical mathematical space, in which we put this image of the universe — and mathematics, how does language allow for such a way of describing the world — then such, containing space, it is quite possible to choose homogeneous and isotropic. Known, that any complexly organized space of a finite number of dimensions can be considered as a subspace in some Euclidean space, but already in a much larger number of dimensions. It is possible. But is it necessary?

For several reasons, quite substantial (experience is most often mentioned as one of these reasons, which considers the behavior of the water surface in a bucket, hanging on a rope, and experiencing torsional vibrations), in my time Newton assigned space and time to the role of receptacles. Arena, on which all physical phenomena occur, but these arenas themselves are not subject to their influence. Note, that there is a certain inequality between these two containers. Space is meant to exist in time as a whole.. Therefore, one can also speak of their totality as a single container. It was quite appropriate to assume that these absolute space and time are homogeneous. (and isotropic). And one could not believe. But the description of the world in such a container would become much more difficult., than that, which was developed by Newton and other scientists.

But already at that time, when such a description of the world or physics was formed, which we often call Newtonian physics, some scientists (and Newton himself including!) clearly saw a certain fragility and inconsistency of the concept of the containing space. What did it cost, eg, the need to take into account the so-called forces of inertia when describing many physical processes. These forces, one side, made the description very difficult, on the other hand, they created difficulties in understanding the very foundations of the description — are they real or fictitious (eliminated by the correct choice of reference system)? Which particular frame of reference is the most “correct”? Complicating the description of the world, pushed out of the front gates of theory, came back through the back gate and laughs at the butler.

The situation became even worse after the creation Einstein General Theory of Relativity. One of the real, “correct” forces, and just that, whose description is so closely associated with the name of Newton, gravity, turned out to be almost exactly inertial, destroyed by choice of frame of reference. Almost everywhere, except for those places, where are the gravitating, massive bodies. Well, in such places and Newton's theory failed, attributing there infinite meaning to the force of gravity. Not enough, and properties of space-time points, still containing these massive bodies, and not only massive (in the sense of the presence of precisely the rest mass), turned out to depend on the location of these bodies. And the energy-momentum tensor, describing the presence of something truly physical in the image of the world, created by Einstein, is absolutely alien for space-time.. Situation, let's face it, poorly satisfactory for a good theory. In this sense, Newton's theory is more consistent.. Space and time are the arena for physics, their properties are clearly postulated and do not depend on physical phenomena. Point. All the rest — physics subject. Physicists, talking about various subjects, forces, processes, etc. Discovered something new? Well and good. This has little effect on the overall picture of the world.. Let's add new strength to the set of already known, a new state of matter or something like that. Now the situation is half-hearted. Containing space-time, one side, changes its properties depending on physical bodies or processes, existing or occurring in one or another of its areas. This happens as a result of communication curvature tensor space-time with the energy-momentum tensor of matter (this connection is written by the Hilbert-Einstein equations). On the other hand, space-time itself also partially points to physical bodies, how should they move, ie. exist, what areas in this space-time to occupy. This is postulated with the statement, that the lines of existence of point massive bodies (provided they have no electromagnetic properties) are geodesic lines of space-time. But for charged massive particles this is no longer the case.. The unsatisfactory nature of this picture of the world is obvious. Here or all, or nothing (as in Newton's picture of the world). This is why Einstein spent most of his life searching for a formulation of a unified field theory., making the most extensive range of electromagnetic phenomena dependent on the geometry of space-time, and not only defining this geometry. Let's not go further in this direction, this would take us too far from the issues of homogeneity and isotropy under consideration. In the image of Einstein's world, the properties of space-time, including homogeneity and isotropy, are entirely determined by the properties of the energy-momentum tensor, physical, external to geometry. If this tensor is homogeneous (or isotropic), then we can count on the repetition of the corresponding properties in geometric structures (although not necessary). And if not, then guaranteed, that space-time itself is homogeneous (or isotropic) will not. The latter is strictly true, because. at least one geometric structure, one of the convolutions of the curvature tensor, tensor Richie will not be the same for all points of space-time. Quite obvious, that due to the existence of relatively compact massive objects — planets, stars, galaxies — it is not necessary to speak of the homogeneity of the energy-momentum tensor. Nevertheless, this does not prevent a huge number of scientists from assuming the distribution of matter in the Universe to be uniform.. Notice, not within a star system or galaxy — it's too absurd. Namely in the universe. They say galaxies distributed more or less evenly in the volume of the Universe and when describing the Universe using the equations of general relativity, all this matter can be assumed to be uniformly smeared over the points of space-time. Let us recall my comments on the possibility of believing in physical substance., which is highly heterogeneous on one size scale, quite homogeneous on a different scale. This is probably the very case? And everything in such reasoning is agreed? There is an approximation in which the universe (as a kind of spatial section of the universe) homogeneous? But no. The trouble is, what all such reasoning should be applied not to the universe (cm. article The universe as a whole. Big explosion) as a whole, and to the universe as spatial sections of the total space-time at certain points in time. If sections of this kind are well defined for sufficiently small regions, parts of space-time (for which it is easy to determine the time common for the region itself), then already for large areas it is very difficult. And for the Universe it is completely impossible. Smallness here is determined by the ratio between the spatial dimensions of the considered area and the duration, which in the selected approximation is set equal to zero. After all, it is necessary to include only those points of space-time in the section, which correspond to the same moments in time. Averaging is a physical procedure and all arguments, which Einstein used to formulate the Special Theory of Relativity, fully applicable to this procedure. It is possible to average only over areas with spatial dimensions small compared to the duration of this averaging procedure. In the sense, what the transit time of the signal between the farthest points of the averaging area should actually be zero on the selected time scale. So to speak, if for us a second is a small period of time, then we have the right to average the substance (or the energy-momentum tensor) in areas small compared to from centimeters. Here from this, obviously, the speed of light and areas with sizes much smaller 300000 kilometers can be averaged. Quite clear, that our usual ideas about relatively homogeneous substances, accessible to our direct experience, very good, obey this condition with a large margin. But on an astronomical scale, from galaxies onwards, make spatial sections, anyway, apply Einstein-Hilbert equations (differential equations, written for an infinitely small neighborhood of the point!) you need to be very, very careful. And they simply cannot be applied to the Universe itself.. And woe to that, who doesn't understand this…

My point is, what, since four-dimensional space-time is enough to describe all the relationships in our world, then it is not required to introduce any containing space. Keyword here containing. May be, sometimes its introduction will be useful to someone, will help to understand something easier, why not? But all, what with its help can be understood about the structure of the world, must necessarily be expressed in terms of objects and relationships between them, points and properties of these points, strictly owned by the world, as a space of four dimensions. After all the enclosing space is nothing more than fiction in terms of experience. And anyway, when we talk about space-time as a space of four dimensions, then we are not talking about this hypothetical containing space. We clearly know, what's the point of this, four-dimensional spaces are not the same. So this space is not uniform. (and not isotropic). Who doubts it, let him try to eat a stone instead of bread, or leave the room not through the door, and through the wall… All this on a scale of measurements, close to our size. When changing to smaller sizes, even those substances, what seemed to us homogeneous, becoming more and more heterogeneous. If we turn our gaze to astronomical dimensions, then and there the same. Star systems are highly heterogeneous — massive objects, stars and planets, very small and separated by huge regions of almost empty space. Galaxies too. Yes, only “empty” space, separating the stars, can be approximately assumed to be homogeneous. But the energy-momentum in it is smeared very little compared to the stars.. And it is no coincidence that the closest to homogeneous areas are precisely the areas with the poorest matter., massive objects. Emptiness is maximally homogeneous for us. And there, where is there something, there is no uniformity in nature. Only sometimes, as an approximation, yes and that, usually, first approach, you can use such an idea of ​​some areas of space-time. Yes, when talking about the homogeneity of the universe, then they talk about areas not even galaxies, and many large ones, than galaxies. Such large cells, in which the number of galaxies is assumed to be approximately the same in each, emit in the entire visible universe about a thousand. Moreover, that for all this thousand cells, it is simply not possible to determine the total time, consider such space as a satisfactory cross section of space-time, in my opinion it is completely unacceptable. And to describe such a universe (just a thousand points, how can they form a continuum, even approximately?) using differential equations is completely funny. But they do it and do not think. The equations were written by Hilbert and Einstein, then you can apply…

Isotropy is somewhat more complicated. Since massive matter in the Universe has a clear tendency to group on different scales of measurement into fairly compact objects, and the empty space surrounding these objects on these scales can be just considered approximately homogeneous (planet or star in space, atom in gas), then in the point-like approximation of these compact objects on such scales we find a good approximation to isotropy three-dimensional directions exactly at points, associated with massive objects. This isotropy is disturbed to some extent when included in the field of view (to the zoom scale) neighboring such compact massive objects. And this happens at all similar levels of approximation.. What's in the gases (liquids, solids), what is in star systems.

Now let us turn to the question of the relationship between homogeneity and isotropy and conservation laws. I will say right away, there is a certain connection, but not so global, in no way requires for the existence of conservation laws of global homogeneity and isotropy of space-time. What “preservation” something? This word means the sameness of this very something in different points of space-time. Very close to the concept of space-time homogeneity. Only here is the concept “preservation” does not require the same at all points space-time nor of all quantities, characterizing the point, not even one of the complete set of quantities. Just the opposite, preserving something is usually clearly associated with something well distinguished from the rest of the world. Conservation of energy-momentum is usually associated with uniformity., and with isotropy conservation of angular momentum. Moreover, the preservation of these values ​​takes place in time, in the process of existence of the object of the world, which is characterized by them.

What is the existence of an object? Implied, that at all times during the existence of an object certain characteristics, by which this object and highlighted from the outside world, remain identical to themselves, ie. the same, the remaining. It's not obligatory all object characteristics. Acceptable, that some part of its characteristics may change. But there are main, unchanging, which define the object as such. If the object is depicted as a point in space, then its existence in space-time is necessarily depicted by the line. And the only geometric characteristic of such a point object, related to its existence (line) the tangent vector turns out. More precisely, two conjugate vectors — tangent and scalar parameter gradient, whose change describes the actual existence of the object, his own time. Both of these vectors can be associated with the energy-momentum vector. Covariant native gradient (scalar) time directly. Then, that it is the energy-momentum vector is detected immediately, as soon as the proper time in its scalar form is identified with the number of events, accumulated on a given segment of the object's existence, which in physics is usually called action. And the tangent vector becomes proportional to the energy-momentum vector when the classical metrics. We will not go further into these details., here for us only that is important, that both of these vectors must be the same at all points of the line of existence of a point object. Yes, this means a certain homogeneity of the points of this line. Lines of existence, not all space-time. In fact, this statement is true only in the classical approximation, when every the point of existence of an object is an event. In the quantum approximation, when only a chain of discrete events can be specified in the history of an object, only these events themselves are the same. In the quantum approximation, only a discrete set of events is homogeneous (points) on the path of the object. But and in the classic, and in the quantum approximation this homogeneity “partial”, by no means entailing the homogeneity of the entire space-time or at least some of its region. The line of existence is not an area in the exact sense, because. has the dimension (=1) less, than the dimension of space-time (=4).

Quite obvious, what can we say about the isotropy of the line of existence of the object?. One direction, time, determining and determined by the order of events in their sequence on the trajectory of the object is the selected. But we can talk about isotropy, the sameness of all directions in a small three-dimensional spatial region of a spatial section, surrounding every point of existence of an object. This is directly related to, that the object in our approximation is represented by an isolated point in space. And for a point in space, around which at least in a small area there is nothing, all directions are equal. They will become unequal, if other objects are taken into consideration, close enough to the given. In this case, it is also not necessary to talk about the general isotropy of points in space.. For the above reason, and also because, that for points from a small spatial area around the object, the direction to the object itself is clearly different from all other directions. If we restrict ourselves to a small spatial area, containing no other objects, then each point of existence of an object (for every moment of its existence) will have the property of isotropy in three spatial dimensions. In the language of physics, this property means the conservation of the angular momentum of a point isolated object.

Let's see now, are the concepts of homogeneity and isotropy related to coordinate transformations. On coordinate transformations there are two views — passive and active transformations. Under passive transformations (but in fact, only such transformations I usually call coordinate transformations) a very simple thing is understood. Let there be a region, in which many different observers assigned unique coordinates to each point, each in his own way. Then, for each point, you can find the conversion factors from some one coordinates to all the rest. These coefficients represent (n x n) matrix, with nonzero determinant. Here n denotes the number of dimensions of space, in our case 4. The coordinates of the point themselves {x} are a collection of n numbers. One of the simplest coordinate transformations is to change the origin, position of a point in space, to which all coordinate values ​​are assigned, equal to zero. This transformation is described by an incomplete matrix, and the column of quantities, to which the origin is shifted along each of the coordinates. Besides coordinates, at any point can be determined (if we are talking about physics, then using measurements; in mathematics they are simply attributed to the point) different sets of numbers. The number of numbers in each specific set must be the same in different coordinates, but the values, in general, change when moving from one coordinates to another, transformed. But, Nonetheless, in each coordinate system, these sets of numbers have well-defined values (generally different for different observers). Depending on the transformation law, these sets, geometric objects, have names, such, like scalars, vectors and so on. Quite obvious, that all point properties, the values ​​of the sets of numbers in it (except the coordinates themselves) do not depend in any way on the choice of the origin of coordinates (and from any choice of coordinates too). Ie. properties of points do not seem to depend on specific values ​​of coordinates. It's very easy to mix two completely different things here.. Point properties depend from specific values ​​of coordinates given that, that a specific binding of coordinates to these points is selected. Changing coordinates in this case is equivalent to moving to another point. When coordinates change during transformations, this is a completely different case., than that, when coordinates change from point to point. It's so easy to forget… Especially considering that, that in many mathematics courses attention is focused on active coordinate transformations, when the coordinate system itself does not change, but it is the point under consideration that changes (in case of broadcasts) or the chosen direction (in case of turns). Active transformations are just linked with the concepts of homogeneity and isotropy and with the corresponding conservation laws. But you cannot mix them with passive ones completely., although there is a certain connection between them. It is framed in a geometric concept connectivity. This connection is, that the entire group of possible transformations of passive coordinates is repeated in the connectivity. Those changes in measurement procedures, which are possible at every single point, are also possible when shifting from point to point. But this does not mean at all, that a shift from point to point is equivalent to a transition to a different way of assigning coordinates to points. Thus, to clarify the question of the presence or absence of homogeneity is possible only with the help of active transformations, ie. displacements from point to adjacent, but not through passive transformations, transitions to other ways of assigning coordinates to a given single point.

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