What is a metric? What is it for? Is it a physical field?

Metrics in our time is firmly connected with the theory of gravity, thanks for the work Hilbert and Einstein together with Grossman. However, in mathematics, it was introduced long before that. If I am not mistaken, among the first to use it in one way or another explicitly, were Rieman and Gauss. First, we will try to understand its role in geometry and only then we will see, how the metric became the main structure of general relativity, General Theory of Relativity.

Today, there is a fairly detailed and clear definition of metric spaces of a fairly general form:

Metric space (“metric”) such a space is called in mathematics, where for any two of its ordered points (that is, one of them is named first, and the other – second) a real number is defined, that it is zero, then and only then, when the points coincide, and the inequality “triangle” – for any three points (x,and,from) this is the number for any pair (x,and) is equal to or less than the sum of these numbers for the other two pairs, (x,from) and (and,from). The definition also implies, that the number is non-negative and does not change (metric symmetric) when changing the order of points in a pair.

As usual, as soon as something is defined, so this definition is extended and the name is extended to other, similar spaces. So it is here. for instance, pseudo-euclidean spaces strictly formally will not be metric according to the definition given above, because. they “metric” number, interval, can be zero for two different points, and also its square can be a negative real number. However, practically from the very beginning, they are included in the family of metric spaces, simply removing the corresponding requirement in the definition, expanding the definition.

Besides, the metric can also be determined not for all points in space, but only for the infinitely close (locally). Such spaces are called Riemannian and in everyday life are also called metric.. Moreover, it was Riemannian spaces that made the metric so famous and attracting attention as mathematicians, and physicists, and familiar even to many people, little related to these sciences.

In the end, here we will discuss the metric in relation to Riemannian spaces, ie. locally. And even locally unknown.

Formal mathematical definition and its extensions – this is the result of understanding and clarifying the concept of a metric. We'll see, from what this concept grew, what properties of the real world it was originally associated with.

All geometry originated from those concepts, which were originally formalized Euclid. So is the metric. In Euclidean geometry (for simplicity and clarity, we will talk about two-dimensional geometry, which means about the geometry of the plane) there is a concept of the distance between two points. Very often even now the metric is called the distance.. Because for the Euclidean plane, distance is a metric, and the metric – distance. And that is how it was understood at the very beginning. Though, how will i try to show, this applies to the modern concept of metrics only in a very limited, with many reservations and conditions, sense.

Distance on the Euclidean plane (on a sheet of paper) seems extremely simple and obvious thing. Really, using a ruler, you can draw a straight line between any two points and measure its length. The resulting number will be the distance. Taking the third point, you can draw a triangle and make sure, what distance is it (for any two points on the plane) exactly satisfies the above definition. Actually, definition and was sketched one to one with the properties of the Euclidean distance on the plane. And a word “metrics” originally associated with measurement (with meter), “metrization” plane.

And why did it take to measure distances, carry out this very metrization of the plane? Well, why do they measure distances in real life?, probably, has its own idea. And in geometry they really think about it, when the coordinates were entered for, to describe each point of the plane separately and uniquely from others. A coordinate system on a plane will obviously be more complicated than just the distance between two points.. Here is the beginning of the countdown, and coordinate axes, and distances (how to do without them?) from the origin to the projections of the point on the axis. Why do we need a coordinate system seems to be clear – it is a solid grid of lines perpendicular to each other (if coordinates are Cartesian), completely filling the plane and thus solving the problem of addressing any point on it.

It turns out, metrics – distance and coordinates – distance. Is there a difference? Coordinates entered. Why then the metric? There is a difference, and very significant. The choice of coordinate systems implies a certain degree of freedom.. In Cartesian systems, we use straight lines as axes. But we can also use curves? We can. And all sorts of twisty ones too. We can measure distance along such lines? Of course. Distance measurement, length along the line is not related, what line is it. The curved track also has a length and you can place milestones on it. But the metric in Euclidean space is not an arbitrary distance. This is the length of the straight, connecting two points. Straight. And what is it? Which line is straight, what is the curve? In the school course, straight – it is an axiom. We see them and grasp the idea.. But in general geometry, straight lines (in itself this is the name, label, no more!) can be defined as some special lines among all possible, connecting two points. Namely, as the shortest, shortest. (And in some cases, for some mathematical spaces, on the contrary, longest, having the greatest length.) It would seem, we caught the difference between the metric and an arbitrary distance between two points. It wasn’t there. We went down the wrong path. Yes, everything is correct, straight – shortest arcs in Euclidean space. But the metric – it's not just the length of the shortest. Not. This is her secondary property.. In Euclidean space, the metric is not only the distance between two points. Metrics – this, first of all, image of the Pythagorean theorem. Theorems, which allows you to calculate the distance between two points when you know their coordinates, the other two distances. Moreover, it is calculated very specifically., as the square root of the sum of the squares of the coordinate distances. Euclidean metric is a non-linear form of coordinate distances, and quadratic! Only the specific properties of the Euclidean plane make the connection of the metric with the shortest paths, connecting dots, so simple. Distances are always linear functions of path offset. The metric is the quadratic function of these displacements. And here lies the fundamental difference between the metric and the intuitively understood distance, as a linear function of displacement from point. Moreover, for us, in general, distance is directly associated with the displacement itself.

Why then, why on earth is a quadratic offset function so important? And does it really have the right to be called distance in the full sense of the word?? Or is it a rather specific property of only Euclidean space (Well, or some family of close to Euclidean spaces) ?

Let's take a small step aside and talk in more detail about the properties of units. Let's ask a question, what should be the rulers, so that you can draw a coordinate grid on a sheet of paper? Solid, rigid and unchanging, you say. And why “rulers”? One is enough! Right, if it can be rotated as desired in the plane of the paper and carried along it. Noticed “if a”? Yes, we have the opportunity to use such a ruler in relation to the plane. The ruler itself, plane by itself, but the plane allows “attach” to yourself our line. And as applied to a spherical surface? How not to apply – everything sticks out of the surface. I just want to bend it, give up hardness and rigidity. Let's leave this line of thought for now.. What else do we want from the line? Hardness and toughness actually mean something different, much more important for us when measuring – a guarantee of the unchanged chosen line. We want to measure with the same scale. Why is it needed? What do you mean why?! To be able to compare measurement results everywhere in the plane. No matter how we turn the ruler, no matter how they move her – some of its properties, length, must be guaranteed to be immutable. Length – this is the distance between two points (in a straight line) on the ruler. Very similar to a metric. But the metric is introduced (or exists) in plane, for points of the plane, where does the ruler? And at the same time, what the metric is just the image of the unchanging length of the abstract ruler brought to its logical conclusion, separated from the outermost ruler and assigned to each point of the plane.

Although our rulers are always external objects for the distances they measure on the plane, but we also think of them as internal, scales belonging to the plane. Consequently, we are talking about a common property, as an external ruler, and internal. And the property is one of the two main ones - the quantity, then, which makes the scale a unit of measurement (second property of scale – this direction). For Euclidean space, this property appears to be independent of the direction of the ruler and its position (from points of space). There are two ways to express this independence.. The first way, passive view of things, speaks of the invariance of the quantity, its identity for an arbitrary choice of admissible coordinates. Second way, active look, speaks of invariance under displacement and rotation, as a result of an explicit transition from point to point. These ways are not equivalent to each other.. The first is simply a formalization of the statement, that the magnitude, existing in the given place (point) the same regardless of point of view. The second also claims, that the values ​​of the quantity at different points are the same. Clear, that this is a much stronger statement.

Let us dwell for now on the invariance of the scale value for an arbitrary choice of coordinates. Op-pa! Like this? To assign coordinates to points, you already need to have scales. Ie. this very line. Other coordinates – what's this? Other lines? In fact, this is how it is! But! Then, that in the Euclidean plane we can rotate our ruler at a point as we want, creates the appearance, that the coordinates can be changed, without changing the ruler. This is an illusion, but such a pleasant illusion! How we got used to her! Talking all the time – rotated coordinate system. And this illusion is based on some postulated property of scale in the Euclidean plane – invariance of its “length” at an arbitrary turn at a point, ie. on arbitrary change of the second scale property, directions. And this property takes place at any point in the Euclidean plane.. The scale is everywhere “the length”, independent of the local choice of the directions of the coordinate axes. This is a postulate for Euclidean space. And how do we determine this length? In coordinate system, in which the selected scale is a unit of measurement along one of the axes, we define it very simply – this is the very unit. And in the coordinate system (rectangular), in which the selected scale does not coincide with any of the axes? Using the Pythagorean theorem. Theorems-theorems, yes there is a little deception here. Actually, this theorem should replace some of the axioms, formulated by Euclid. It is equivalent to them. And with further generalization of geometry (for arbitrary surfaces, eg) rely precisely on the method of calculating the length of the scale. As a matter of fact, convert this method into the category of axioms.

Let's repeat something now, what lies at the heart of geometry, which allows you to assign coordinates to the points of the plane.

It is a unit of measurement, scale. Scale exists anywhere. Has a value – “the length” and direction. Length is invariant (does not change) when changing direction at a point. In rectangular coordinates in Euclidean space, the square of the scale length, directed from a point arbitrarily, is equal to the sum of the squares of its projections on the axis. Such a geometric quantity is also called a vector. So the scale is a vector. AND “length” vector is also called the norm. Okay. But where is the metric here?? AND metrics with this approach, there is a way to assign a norm to any vector at each point, a method for calculating this norm for an arbitrary position of this vector relative to the vectors, constituting the base, benchmark (those, which determine the directions of the coordinate axes from a given point and have a unit norm by definition, ie. units of measure). It is very important that, that such a method is defined for each point in space (plane in this case). In this way, it is a property of this space and its internal vectors, and not objects external to the space.

Let, but already at the very beginning we gave the definition of metric spaces. Why a new definition? And is it consistent with the old? But why. Here we have indicated exactly how, this very real number is determined. Namely, the distance between the points is “length”, vector norm, connecting these points (in Euclidean space). Then, that the vector has some norm, independent of the point of view (frame selection) is the definition of the vector. The most important condition, which also makes the space metric, is the requirement, so that vectors with a given norm exist at every point in space in all directions. And this definition is quite consistent with the one given at the very beginning.. Is there any other way to define the metric on some space? Basically, can. And even in many ways. Only these will be completely different classes of spaces., not including Euclidean space even as a special case.

Why is Euclidean space special for us?? Well, how is it what? At first sight, it is precisely these properties that the very space possesses, in which we live. Yes, on closer inspection, not quite like that. But there is a difference between “not quite like that” and “not at all like that”?! Although the set of words seems to be the same. So our space-time, if not Euclidean, then under certain conditions it can be very close to it. Consequently, we must choose from that family of spaces, in which Euclidean space is. This is what we do. But anyway, what is so special about euclidean space, which is expressed in certain properties of its metric? There are quite a few properties, most of them have already been mentioned above. I will try to formulate this singularity rather compactly.. Euclidean space is, that it has the ability to select scales (that is, enter the coordinates) So, that it turns out to be completely filled with a rectangular grid of coordinates. Perhaps this is when the metric at each point in space is the same. Essentially, it means, that the scales necessary for this exist at every point in space and they are all identical to one single. One ruler is enough for the entire space, which can be transferred to anywhere (in an active sense) without changing its value, and its directions.

Above I put the question, why is the metric a quadratic offset function. He still remains unanswered. We will definitely come to this. Now mark for yourself for the future – the metric in the required family of spaces is a quantity invariant under coordinate transformations. We talked so far about Cartesian coordinates, but I will immediately emphasize here – this is true for any coordinate transformations, which are admissible at a given point in a given space. The quantity, invariant (unchanging) when transforming coordinates has another special name in geometry – scalar . Look, how many names for the same – constant, invariant, scalar… Maybe there is some more, doesn’t come to mind right away. This speaks of the importance of the concept itself. So here, a metric is a scalar in a certain sense. Of course, there are other scalars in geometry.

Why in “a certain sense”? because, what, the concept of a metric includes two points and not one! And the vector is connected (defined) with only one point. So I misled you? Not, just said a little more, what to say. And I must say, that the metric is the norm of a non-arbitrary vector, but only vectors of infinitesimal displacement from a given point in an arbitrary direction. When this norm does not depend on the direction of displacement from the point, then its scalar value can be considered as a property of only this one point. Wherein, it still remains the rule for calculating the norm for any other vector. Like this.

Something doesn't fit… The norms are different for different vectors! And the metric is a scalar, the value is the same. Contradiction!

There is no contradiction. I said it clearly – calculation rule. For all vectors. And the specific value itself, which is also called the metric, calculated according to this rule for only one vector, displacement. Our language is accustomed to liberties, defaults, reductions… So we used to call a metric both a scalar and a rule for its calculation. Indeed, it's almost the same. Nearly, but not really. It is still important to see the difference between the rule and the result., with its help received. What's more important – rule or result? Surprisingly, in this case, the rule… Therefore, much more often in geometry and physics, when they talk about the metric, mean exactly the rule. Only very stubborn mathematicians prefer to speak strictly about the result.. And there are reasons for this, but about them elsewhere.

I also want to note, that in the more usual way of presentation, when the concept of vector spaces is taken as a basis, the metric is introduced as a scalar pairwise product of all vectors of the basis, benchmark. In this case, the dot product of the vectors must be determined in advance. And on the way, which I followed here, it is the presence of the metric tensor in space that makes it possible to introduce, define dot product of vectors. Here the metric is primary, its presence allows you to introduce the dot product, as a kind of invariant, linking two different vectors. If a scalar is calculated using the metric for the same vector, then it's just his norm. If this scalar is calculated for two different vectors, then this is their dot product. If this is also the norm of an infinitely small vector, then it is quite permissible to call it simply a metric at a given point.

And what can we say about the metric as a rule? Here we have to use the formulas. Let the coordinates along the axis with the number i be denoted by us as xⁱ. And the displacement from a given point to an adjacent dxⁱ. Draw your attention – coordinates are not vector! And the displacement is just a vector! In such notation, the metric “distance” between this point and the adjacent, according to the Pythagorean theorem will be calculated using the formula

ds² = g_I dxⁱ dx^k

On the left is the square of the metric “distance” between points, “coordinate” (that is, for each individual coordinate line) the distance between which is given by the displacement vector dxⁱ. On the right, the sum over the coinciding indices of all pairwise products of the components of the displacement vector with the corresponding coefficients. And their table, coefficient matrix g_I, which sets the rule for calculating the metric norm, is called the metric tensor. And it is this tensor in most cases that is called the metric. The term “tensor” extremely important here. And he means, that in a different coordinate system the formula, written above will be the same, only the table will contain others (in general) odds, which are calculated in a strictly specified way through these and the coordinate transformation coefficients. Euclidean space is characterized by, that in Cartesian coordinates the form of this tensor is extremely simple and the same in any Cartesian coordinates. Matrix g_I contains only ones on the diagonal (at i = k), and the rest of the numbers are zeros. If non-Cartesian coordinates are used in Euclidean space, then the matrix in them will already look not so simple.

so, we wrote down the rule, defining metric “distance” between two points in Euclidean space. This rule is written for two arbitrarily close points. In Euclidean space, ie. in such, in which the metric tensor can be diagonal with units on the diagonal in some coordinate system at each point, there is no fundamental difference between finite and infinitesimal displacement vectors. But we are more interested in the case of Riemannian spaces (such as the surface of a ball, eg), where this difference is significant. So that, we admit, that the metric tensor in the general case is not diagonal and changes when passing from point to point in space. But the result of its application, ds², remains in this case at each point independent of the choice of the direction of displacement and from the point itself. This is a very strict condition. (less rigid, than the Euclidean condition) and it is when it is fulfilled that the space is called Riemannian.

You probably noticed, that very often I put words in quotation marks “length” and distance”. This is why I do it. In the case of a plane and three-dimensional Euclidean space, metric “distance” and “length” appear to be exactly the same as normal distances, measuring rulers. Moreover, these concepts were introduced to formalize the work with measurement results. Why then “seem to coincide”? Fun, but this is exactly the case, when mathematicians come together with dirty (they do not need) water splashed out of the bathtub and the baby. Not, they left something, but that, what is left has ceased to be a child (distance). It is easy to see this even on the example of the Euclidean plane..

Let me remind you – metric “distance” does not depend on the choice of Cartesian (and not only) coordinates, let's say, on a sheet of paper. Let in some coordinates, this distance between two points on the coordinate axis is 10. Can I specify other coordinates, at which the distance between the same points will be equal to 1? No problem. Just set aside the new unit as a unit along the same axes., equal 10 previous. Has the Euclidean space changed from this?? What's the matter? And the point is, that when we measure something, it is not enough for us to know the number. We still need to know, what units were used to get this number. Mathematics in the form familiar to everyone today is not interested in this. She only deals with numbers. The choice of units of measure was made before the application of mathematics and should not be changed anymore! But our distances, lengths without specifying scales do not tell us anything! But mathematics doesn't care. When it comes to metric “distance”, its formal application is indifferent to the choice of scale. At least meters, even fathoms. Only numbers are important. That's why I put the quotes. Do you know what side effect this approach has in the mathematics of Riemannian spaces?? But what. It makes no sense to consider the change in scale from point to point. Only changing its direction. And this despite the fact, that changing scales with the help of coordinate transformations in such geometry is quite a common thing. Is it possible to include in geometry a sequential consideration of the properties of scales in their entirety? Can. Only for this you have to remove a lot of conventions and get used to calling things your own, correct names. One of the first steps is to realize that, that no metric in essence is and cannot be a distance. She is, certainly, has some physical meaning, moreover, very important. But another.

In physics, attention was drawn to the role of the metric with the emergence of theories of relativity. – first special, then general, in which the metric became the central structure of the theory. Special Theory of Relativity was formed on the basis of this fact, that three-dimensional distance is not a scalar from the point of view of a set of inertial, moving relative to each other uniformly and rectilinearly physical reference systems. Scalar, the invariant turned out to be another quantity, which was called the interval. Interval between events. And to calculate its value, you need to take into account the time interval between these events.. Moreover, it turned out, as the rule for calculating the metric (and the interval immediately began to be considered as a metric in the combined space-time, event space) different from the usual Euclidean in three-dimensional space. Seem to be, but a little different. Corresponding metric space of four dimensions, introduced Herman Minkowski, began to call pseudo-Euclidean. It was Minkowski's work that attracted the attention of physicists, including Einstein, to the importance of the concept of a metric as a physical quantity, not just mathematical.

The General Theory of Relativity also included physical reference frames accelerated relative to each other. AND, in this way, was able to give a description of gravitational phenomena at a new level in relation to Newton's theory. And she was able to achieve this by giving the meaning of the physical field to the metric – and value and rule, metric tensor. At the same time, she uses the mathematical construction of the Riemannian space as an image of space-time. We will not go too far into the details of this theory.. Among other things, this theory states, what the world (space-time), which has massive bodies, that is, bodies are attracted to each other , has a metric different from the Euclidean metric so pleasant to us. All the statements below are equivalent:

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  Physical statement. Point bodies, having mass, attracted to each other.
  
  
• 
  
  In space-time, which has massive bodies, you cannot introduce a rigid rectangular grid everywhere. There are no such measuring devices, which allow you to do this. Always as small as you like “cells” the resulting mesh will be curved quadrangles.
  
  
• 
  
  You can select a scale with the same magnitude (the norm) for all space-time. Any such scale can be moved from its point to any other point and compared with the existing one there.. BUT! Even if the offset is infinitely small, the directions of the scales being compared in the general case will not coincide. The stronger, the closer the scale is to the body, possessing a mass and the greater this same mass. Only where there are no masses (true, here's a question for you – what about the scale?) directions will coincide.
  
  
• 
  
  In the field of space-time, containing massive bodies there is no such coordinate system, in which the metric tensor at each point is represented by the matrix, zero everywhere, except diagonal, on which the units are located.
  
  
• 
  
  The difference between the metric and the Euclidean is a manifestation of the presence of a gravitational field (gravitational fields). Moreover, the field of the metric tensor is the gravitational field.
  
  
• 
  
  Space-time, containing massive bodies has a non-zero point at each point curvature.
  
  

Many more similar statements could be cited., but now I would like to draw your attention to the latter. Curvature. It is something, what we haven't discussed yet. How does it relate to the metric? By and large – no! Curvature is a more general concept than a metric. In what sense?

Family of Riemannian spaces, including Euclidean spaces, itself belongs to a more general family spaces with affine connection. These spaces, generally speaking, do not imply the existence of such a quantity, as a metric, for each of its pairs of points. But their necessary property is the existence of two other structures, related to each other – affine connection and curvature. And only under certain conditions for curvature (or connectivity), in such spaces there is a metric. Then these spaces are called Riemannian. Any Riemannian space has connection and curvature. But not vice versa.

But one cannot also say, that the metric is secondary to connectivity or curvature. Not. Existence of a metric – it is a statement of certain properties of connectivity, and hence the curvature. In the standard interpretation of general relativity, the metric is considered as more important, formative theory, structure. And the affine connection and curvature turn out to be secondary., derived from the metric. This interpretation was laid down by Einstein, in those times, when mathematics has not yet developed a sufficiently advanced and consistent understanding of hierarchy in terms of the importance of structures, which define the properties of the family of spaces, leading to Euclidean. After the creation of the general relativity apparatus, primarily through the work Weyl and Shoutena (not one of them, of course), the mathematics of spaces with affine connection was developed. Actually, this work was stimulated by the appearance of general relativity. As you can see, the canonical interpretation of the importance of structures in general relativity does not coincide with the current view of mathematics on their relationship. This canonical interpretation is nothing more, as the identification of certain mathematical structures with physical fields. Giving them physical meaning.

In general relativity, there are two plans for describing space-time. The first one – space-time itself as a space of events. Events, continuously filling any region of space-time are characterized by four coordinates. Consequently, coordinate systems are assumed to be entered. The very name of the theory focuses on this. – Nature laws, occurring in such a space-time must be formulated in the same way with respect to any admissible coordinate system. This requirement is called the principle of general relativity.. Note, that this plan of the theory still does not say anything about the presence or absence of a metric in space-time, but already provides a basis for the existence of an affine connection in it (along with curvature and other derived mathematical structures). Naturally, already at this level, it becomes necessary to give a physical meaning to the mathematical objects of the theory. There he is. A point in time space depicts an event, on the one hand, characterized by position and moment of time, with another – four coordinates. Something strange? Isn't it the same? But no. In general relativity, they are not the same. Coordinates of the most general form, admissible in theory cannot be interpreted as positions and times. Such a possibility is postulated only for a very limited group of coordinates – locally inertial, which exist only in the vicinity of each point, but not in the whole area, covered by a common coordinate system. This is another postulate of the theory.. Here is such a hybrid. I will note, that it is here that many problems of general relativity are born, but I will not deal with them now.

The second plan of the theory can be considered that part of its postulates, which introduces into consideration the physical phenomenon in space-time – gravity, mutual attraction of massive bodies. Approved, that this physical phenomenon can, under certain conditions, be eliminated by a simple choice of a suitable frame of reference, namely, locally inertial. For all bodies, having the same acceleration (free fall) due to the presence in a small area of ​​the gravitational field of a distant massive body, this field is not observable in some reference frame. Formally, postulates end there, but in fact the basic equation of the theory, which introduces the metric, also applies to postulates, and as a mathematical statement, and how physical. Although I'm not going to go into the details of the equation (actually, systems of equations), but it's still useful to have it in front of your eyes:

R_I= -s (T_I – 1/2 T g_I)

Here on the left is the so-called tensor Richie, certain convolution (combination of constituent components) full curvature tensor. It can rightfully be called the curvature. On the right is a construction from the energy-momentum tensor (a purely physical quantity in general relativity, singular for massive bodies and external for space-time, which for the energy-momentum in this theory is simply the carrier) and metrics, which is implied by the existing. Moreover, this metric, as a scalar, produced by the metric tensor, is the same for all points of the area. There is also a dimensional constant with, proportional to the gravitational constant. This equation shows, what, by and large, curvature is mapped to energy-momentum and metric. The physical meaning of the metric is attributed in general relativity after obtaining the solution of these equations. Since in this solution the metric coefficients turn out to be linearly related to the potential of the gravitational field (calculated through it) then the meaning of the potentials of this field is attributed to the metric tensor. With this approach, the curvature should have a similar meaning. And the affine connection is interpreted as the field strength. This interpretation is wrong., its erroneousness is associated with the above-mentioned paradox in the interpretation of coordinates. Naturally, for theory this does not pass without leaving a trace and manifests itself in a number of well-known problems (nonlocalizability of the energy of the gravitational field, singularity treatment), which, when the geometric values ​​are given the correct physical meaning, simply do not arise. All this is discussed in more detail in the book. “Dimension and Properties of Space-Time“.

However, in general relativity, the metric involuntarily, in addition to the meaning artificially imposed on her, has another physical meaning. Let's remember, which characterizes the metric in the case of a Euclidean space? One very important thing for measurements in space-time – the ability to introduce in this space a rigid, uniformly filling the entire area with a rectangular grid. This grid is called in physics the inertial frame of reference.. Such a frame of reference (coordinate system) corresponds to one and only one standard form of the metric tensor. In frames of reference, arbitrarily moving relative to inertial, the form of the metric tensor is different from the standard. From a physical point of view, the role “reference grids” sufficiently transparent. If you have a rigid reference body, each point of which is equipped with the same clock, existing in time, then it just implements such a grid. For empty space, we simply conjecture such a reference body, supplying him (space) exactly the same metric. In this understanding, metric tensor, different from the standard Euclidean, He speaks, what is the frame of reference (coordinates) built with a non-solid body, and, may be, the clock also runs differently at its points. What do I want to say with this? And that, what the metric tensor is a mathematical image of some of the most important properties of the frame of reference for us. Those properties, which absolutely characterize the structure of the reference frame itself, allow to define, how much she “good”, how different from ideal – inertial system. Here is GR and uses the metric tensor just as such an image. how image of measuring devices distributed in the reference area, possibly changing its orientation from point to point, but having the same norm everywhere, common for all reference vectors. Metrics, considered as a scalar is this norm, scale value. The metric as a tensor allows one to consider arbitrary relative motion relative to each other of all scales, constituting the reference body. And general relativity describes such a situation, when in space-time it is possible to have such a reference body, real or imagined.

This view of the metric is certainly correct.. Moreover, he is also productive, since it immediately draws attention to the agreements remaining in GTR. Indeed, we have allowed the use of the frame of reference, in which the scales at different points can be oriented in different ways (in a four-dimensional world, orientation also includes movement). And we still demand, so that some absolute characteristic of the scale, his norm (interval) remained the same. Consequently, still the statement of general relativity, that she took into consideration all possible frames of reference excessively. She's not so common, relativity in this theory.

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