Key metric as a compass for startup development. The most important QA metrics

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

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

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

A metric space (“equipped with a metric”) in mathematics is a space in which for any two of its ordered points (that is, one of them is called the first, and the other is called the second) real number such that it is equal to zero if and only if the points coincide and the “triangle” inequality is satisfied - for any three points (x,y,z) this number for any pair (x,y) is equal to or less than the sum of these numbers for the other two pairs, (x,z) and (y,z). It also follows from the definition that this number is non-negative and does not change (the metric is symmetric) when the order of points in the pair changes.

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

In addition, the metric can also be determined not for all points in space, but only for infinitely close ones (locally). Such spaces are called Riemannian and in everyday life they are also called metric. Moreover, It was Riemannian spaces that made the metric so famous and attracting the attention of both mathematicians and physicists, and familiar even to many people who have little connection with these sciences.

Ultimately, here we will discuss the metric in relation specifically to Riemannian spaces, i.e. V local sense. And even locally signally indefinite.

The formal mathematical definition and its extensions are the result of understanding and clarifying the concept of metric. Let's see where this concept grew from and with what properties real world it was originally connected.

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

Distance on the Euclidean plane (on a piece of paper) seems to be an extremely simple and obvious thing. Indeed, 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 that this distance (for any two points on the plane) exactly satisfies the above definition. Actually, the definition was copied one-to-one from the properties of the Euclidean distance on a plane. And the word “metrics” is initially associated with measurement (using a meter), “metrization” of a plane.

Why was it necessary to measure distances, to carry out this very metrization of the plane? Well, why do they measure distances? real life Everyone probably has their own idea. And in geometry they really started thinking about this when they introduced coordinates in order to describe each point of the plane separately and uniquely from others. The coordinate system on the plane will clearly be more complex than just the distance between two points. Here is the origin, and the coordinate axes, and the distances (how can we do without them?) from the origin to the projections of the point on the axis. It seems clear why a coordinate system is needed - it is a continuous grid of lines perpendicular to each other (if the coordinates are Cartesian), completely filling the plane and thus problem solving addresses of any point on it.

It turns out that the metric is distance and coordinates are distances. Is there a difference? Entered coordinates. Why then a metric? There is a difference, and a very significant one. The choice of coordinate systems implies a certain freedom. In Cartesian systems we use straight lines as axes. But we can also use curves? Can. And all sorts of twisty ones too. Can we measure distance along such lines? Certainly. Measuring distance, length along a line is not related to what kind of line it is. The curved path also has a length and mileposts can be placed on it. But the metric in Euclidean space is not an arbitrary distance. This is the length of a straight line connecting two points. Straight. And what is it? Which line is straight and which is curved? IN school course straight lines are an axiom. We see them and get the idea. But in general geometry, straight lines (in itself this is a name, a label, nothing more!) can be defined as some special lines among all possible ones connecting two points. Namely, as shortest ones having the shortest length. (And in some cases, for some mathematical spaces, on the contrary, the longest, having the greatest length.) It would seem that we have grasped the difference between a metric and an arbitrary distance between two points. Not so. We took the wrong path. Yes, that’s right, straight lines are the shortest in Euclidean space. But the metric is not just the length of the shortest path. No. This is its secondary property. In Euclidean space, the metric is not only the distance between two points. The metric is, first of all, an image of the Pythagorean theorem. A theorem that allows you to calculate the distance between two points if you know their coordinates and two other distances. Moreover, it is calculated very specifically, as the square root of the sum of the squares of the coordinate distances. The Euclidean metric is not linear form coordinate distances, but quadratic! Only the specific properties of the Euclidean plane make the connection of the metric with the shortest paths connecting points so simple. Distances are always linear functions of displacement along the path. The metric is a quadratic function of these displacements. And here lies the fundamental difference between the metric and the intuitively understood distance, as linear function offsets from a point. Moreover, for us in general, distance is directly associated with the displacement itself.

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

Let's take a small step aside and talk in more detail about the properties of units of measurement. Let's ask ourselves: what should the rulers be like in order to be able to apply a coordinate grid on a sheet of paper? Solid, tough and unchanging, you say. And why “rulers”? One is enough! True, if it can be rotated as desired in the plane of the paper and moved along it. Did you notice the “if”? Yes, we have the opportunity to use such a ruler in relation to a plane. The ruler is on its own, the plane is on its own, but the plane allows us to “attach” our ruler to itself. What about a spherical surface? No matter how you apply it, everything sticks out beyond the surface. I just want to bend it, give up its hardness and rigidity. Let's leave this line of thought for now. What more do we want from the line? Hardness and rigidity actually imply something else, much more important for us when taking measurements - a guarantee of the invariability of the chosen ruler. We want to measure with the same scale. Why is this necessary? 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 we shift it, some of its properties, length, must be guaranteed to remain unchanged. Length is the distance between two points (in a straight line) on a ruler. Very similar to metric. But the metric is introduced (or exists) in the plane, for points on the plane, and what does the ruler have to do with it? And despite the fact that the metric is precisely the image of the constant length of an abstract ruler taken to its logical conclusion, torn off 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 a plane, we also think of them as internal scales belonging to the plane. Consequently, we are talking about a general property of both the external and internal rulers. And this property is one of the two main ones - magnitude, which is what makes scale a unit of measurement (the second property of scale is direction). For Euclidean space, this property seems to be independent of the direction of the ruler and its position (from a point in space). There are two ways to express this independence. The first method, a passive view of things, speaks of the invariance of a quantity, its sameness under an arbitrary choice of permissible coordinates. The second method, active gaze, speaks of invariance under translation and rotation, as a result of an explicit transition from point to point. These methods are not equivalent to each other. The first is simply a formalization of the statement that a quantity existing in this place(point) is the same regardless of point of view. The second also states that the values of quantities at different points are the same. Clearly this is a much stronger statement.

Let us dwell for now on the invariance of the scale value for an arbitrary choice of coordinates. Oops! Like this? To assign coordinates to points you already need to have scales. Those. this very line. What are other coordinates? Other lines? In fact, that's exactly it! But! The fact 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. It's an illusion, but such a pleasant illusion! How accustomed we are to it! We always say – a rotated coordinate system. And this illusion is based on a certain postulated property of scale in the Euclidean plane - the invariance of its “length” under arbitrary rotation at a point, i.e. with an arbitrary change in the second property of scale, direction. And this property takes place at any point of the Euclidean plane. The scale everywhere has a “length” independent of local choice directions of the coordinate axes. This is a postulate for Euclidean space. And how do we determine this length? In a coordinate system in which the selected scale is a unit of measurement along one of the axes, we define it very simply - this is that same unit. And in a coordinate system (rectangular), in which the selected scale does not coincide with any of the axes? Using the Pythagorean theorem. Theorems are theorems, but there is a little deception here. In fact, this theorem should replace some of the axioms formulated by Euclid. She is equivalent to them. And with further generalization of geometry (for arbitrary surfaces, for example), they rely precisely on the method of calculating the length of the scale. In fact, this method is being relegated to the category of axioms.

Let us now repeat something that underlies geometry, which allows us to assign coordinates to points in the plane.

We are talking about a unit of measurement, a scale. Scale exists at any point. It has magnitude – “length” and direction. The length is invariant (does not change) when the direction at a point changes. IN rectangular coordinates in Euclidean space, the square of the length of a scale directed arbitrarily from a point is equal to the sum of the squares of its projections on the axis. This geometric quantity is also called a vector. So the scale is a vector. And the “length” of the vector is also called the norm. Fine. But where is the metric here? A metrics with this approach there is a way to assign a norm to any vector at every point, a method for calculating this norm for an arbitrary position of this vector relative to the vectors that make up the base, reference point(those that determine the directions of the coordinate axes from a given point and have a unit norm by definition, i.e. units of measurement). It is very important that this method is defined for each point in space (plane in this case). Thus, it is a property of this space and its internal vectors, and not of objects external to the space.

Excuse me, but already at the very beginning we gave a definition of metric spaces. Why a new definition? And does it agree with the old? But why. Here we have indicated how exactly this real number is set and determined. Namely, the distance between points is equal to the “length”, the norm of the vector connecting these points (in Euclidean space). The fact that a vector has a certain norm, independent of the point of view on it (the choice of reference point) is the definition of a vector. The most important condition, which makes the space metric, is the requirement 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 it possible to define a metric on a certain space differently? In principle, it is possible. And even in many ways. Only these will be completely different classes of spaces that do not include Euclidean space even as a special case.

Why is Euclidean space special for us? Well, what is it like? At first glance, the very space in which we live has precisely these properties. Yes, upon closer examination, 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 can be very close to it. Consequently, we must choose from the family of spaces in which Euclidean space exists. That's what we do. But still, what is so special about Euclidean space that is expressed in certain properties of its metric? There are quite a lot of properties, most of them have already been mentioned above. I will try to formulate this feature quite compactly. Euclidean space is such that it is possible to choose scales (that is, enter coordinates) so that it is completely filled with a rectangular coordinate grid. Perhaps this is when the metric at each point in space is the same. Essentially, this means that the scales required for this exist at every point in space and they are all identical to one single one. For the entire space, one ruler is sufficient, which can be moved to any point (in the active sense) without changing both its magnitude and its direction.

Above I posed the question why the metric is quadratic function offsets. It remains unanswered for now. We will definitely come to this again. Now make a note for yourself for the future - the metric in the family of spaces we need is a quantity invariant under coordinate transformations. We have talked so far about Cartesian coordinates, but I will immediately emphasize here that this is true for any coordinate transformations that are permissible at a given point given space. A quantity that is invariant (not changing) during coordinate transformations has another special name in geometry - scalar. Look how many names there are for the same thing - constant, invariant, scalar... Maybe there is something else, it doesn’t immediately come to mind. This speaks to the importance of the concept itself. So, a metric is a scalar in a certain sense. Of course, there are other scalars in geometry.

Why in a “certain sense”? Because the concept of a metric includes two points and not one! And the vector is connected (defined) with only one point. It turns out I misled you? No, I just haven’t said everything that needs to be said. But it must be said that the metric is the norm not of an arbitrary vector, but only of a vector of infinitesimal displacement from a given point in an arbitrary direction. When this norm does not depend on the direction of displacement from a point, then its scalar value can be considered as a property of only this one point. At the same time, it still remains the rule for calculating the norm for any other vector. Like this.

Something doesn’t add up... The norms are different for different vectors! And the metric is scalar, the value is the same. Contradiction!

There is no contradiction. I said it clearly - the calculation rule. For all vectors. And the specific value itself, which is also called a metric, is calculated according to this rule only for one vector, the displacement. Our language is accustomed to liberties, omissions, abbreviations... So we are accustomed to calling both a scalar and the rule for calculating it a metric. In fact, it's almost the same thing. Almost, but not quite. It is still important to see the difference between a rule and the result obtained with its help. What is more important – the rule or the result? Oddly enough, in this case, the rule... Therefore, much more often in geometry and physics, when they talk about metrics, they mean the rule. Only very stubborn mathematicians prefer to talk strictly about the result. And there are reasons for this, but more on them elsewhere.

I would also like to note that with more the usual way presentation, when the concepts of vector spaces are taken as a basis, the metric is introduced as a scalar pairwise product of all basis and reference vectors. In this case, the scalar product of vectors must be defined in advance. And on the path that I followed here, it is the presence of a metric tensor in space that allows us to introduce and define the scalar product of vectors. Here the metric is primary, its presence allows us to introduce the scalar product as a kind of invariant connecting two different vectors. If a scalar is calculated using a metric for the same vector, then this is simply its norm. If this scalar is calculated for two different vectors, then it is their dot product. If this is also the norm of an infinitesimal vector, then it is quite acceptable to simply call it a metric at a given point.

And what can we say about the metric as a rule? Here we will have to use formulas. Let the coordinates along the axis number i be denoted as x i. And the displacement from a given point to the neighboring one dx i. Please note that coordinates are not a vector! And the displacement is just a vector! In such notation, the metric “distance” between a given point and the neighboring one, according to the Pythagorean theorem, will be calculated using the formula

ds 2 = g ik dx i dx k

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

So, we have written down a rule that determines the metric “distance” between two points in Euclidean space. This rule is written for two arbitrarily close points. In Euclidean space, i.e. in one 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, for example), where this difference is significant. So, we assume that the metric tensor is generally not diagonal and changes when moving from point to point in space. But the result of its application, ds 2, remains at each point independent of the choice of the direction of displacement and of the point itself. This is a very stringent condition (less stringent than the Euclidean condition) and it is when it is fulfilled that the space is called Riemannian.

You may have noticed that very often I put the words “length” and distance in quotation marks.” This is why I do this. In the case of the plane and three-dimensional Euclidean space, metric “distance” and “length” appear to be exactly the same as ordinary distances measured with rulers. Moreover, these concepts were introduced to formalize work with measurement results. Why then “seem to coincide”? It’s funny, but this is exactly the case when mathematicians, along with the dirty (they didn’t need) water, threw the child out of the bath. No, they left something, but what was left ceased to be a child (distance). This is easy to see even using the Euclidean plane as an example.

Let me remind you that the metric “distance” does not depend on the choice of Cartesian (and not only) coordinates, say, on a sheet of paper. Let in some coordinates this distance between two points on the coordinate axis be equal to 10. Is it possible to indicate other coordinates in which the distance between these same points will be equal to 1? No problem. Simply plot as a unit along the same axes a new unit equal to 10 previous ones. Has Euclidean space changed because of this? What's the matter? But the fact is that when we measure something, it is not enough for us to know the number. We also need to know what units were used to obtain this number. Mathematics in the form familiar to everyone today is not interested in this. She deals only with numbers. The choice of units of measurement was made before applying mathematics and should not change again! But our distances and lengths without indicating scales tell us nothing! Mathematics doesn't care. When it comes to metric “distance,” its formal application is indifferent to the choice of scale. Even meters, even fathoms. Only numbers matter. That's why I put quotation marks. Do you know what side effect this approach has in the mathematics of Riemannian spaces? Here's what it is. It makes no sense to consider the change in scale from point to point. Only a change in its direction. And this despite the fact that changing scales using coordinate transformations in such geometry is quite an ordinary thing. Is it possible to include in geometry a consistent consideration of the properties of scales in their entirety? Can. Only To do this, you will have to remove many conventions and learn to call things by their proper names. One of the first steps will be to realize the fact that no metric is essentially a distance and cannot be. She certainly has some physical meaning, and a very important one at that. But different.

In physics, attention to the role of metrics was drawn with the advent of theories of relativity - first special, then general, in which the metric became the central structure of the theory. The Special Theory of Relativity was formed on the basis of the fact that three-dimensional distance is not a scalar from the point of view of a set of inertial physical reference systems moving relative to each other uniformly and rectilinearly. Another quantity turned out to be a scalar, an invariant, which was called an 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 that the rule for calculating the metric (and the interval immediately began to be considered as a metric in the unified space-time, the space of events) is different from the usual Euclidean rule in three-dimensional space. Similar, but a little different. The corresponding metric space of four dimensions introduced Herman Minkowski, began to be called. It was Minkowski's work that drew the attention of physicists, including Einstein, to the importance of the concept of metric as a physical quantity, and not just a mathematical one.

The General Theory of Relativity also included into consideration physical reference systems accelerated relative to each other. And thus, she 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 meaning to the physical field specifically to the metric - both the value and the rule, the metric tensor. At the same time, it uses the mathematical construction of Riemannian space as an image of space-time. We won't go too far into the details of this theory. Among other things, this theory states that the world (space-time), in which there are massive bodies, that is, bodies that attract each other, has a metric that is different from the Euclidean metric that is so pleasant to us. All statements below are equivalent:

Physical statement. Point bodies with mass are attracted to each other.

In space-time, in which there are massive bodies, it is impossible to introduce a rigid rectangular grid everywhere. There are no such measuring instruments that allow you to do this. Always, no matter how small, the “cells” of the resulting grid will be curved quadrangles.

You can choose a scale with the same value (norm) for the entire space-time. Any such scale can be moved from its point to any other point and compared with what already exists there. BUT! Even if the displacement is infinitesimal, the directions of the compared scales will generally not coincide. The stronger the closer the scale is to the body with mass and the larger this same mass. Only where there are no masses (though, here’s a question for you - what about the scales themselves?) will the directions coincide.

In the region of space-time containing massive bodies, there is no coordinate system in which the metric tensor at each point is represented by a matrix that is zero everywhere except for the diagonal on which the ones are located.

The difference between the metric and the Euclidean one is a manifestation of the presence of a gravitational field (gravitational field). Moreover, the field of the metric tensor is the gravitational field.

Many more similar statements could be cited, but now I would like to draw your attention to the last one. Curvature. This is something we haven't discussed yet. What does it have to do with metrics? By by and large- none! is a more general concept than metric. In what sense?

The family of Riemannian spaces, which also includes Euclidean spaces, is itself part of the more general family. 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 on curvature (or connectivity) does a metric exist in such spaces. Then these spaces are called Riemannian. Any Riemannian space has connectivity and curvature. But not the other way around.

But it also cannot be said that the metric is secondary to connectivity or curvature. No. The existence of a metric is a statement of certain properties of connectivity, and therefore curvature. In the standard interpretation of general relativity, the metric is considered as a more important structure that forms the form of the theory. And affine connection and curvature turn out to be secondary, derived from the metric. This interpretation was laid down by Einstein, at a time when mathematics had not yet developed a sufficiently advanced and consistent understanding of the hierarchy of importance of structures that determine the properties of the family of spaces leading to Euclidean ones. After the creation of the GTR apparatus, primarily through the works of Weyl and Schouten (not only them, of course), the mathematics of spaces of affine connection was developed. Actually, this work was stimulated by the emergence 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 than 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 of them is space-time itself as a space of events. Events that continuously fill any region of space-time are characterized using four coordinates. Therefore, the coordinate systems are assumed to be entered. The very name of the theory focuses attention precisely on this - the laws of nature that take place in such space-time must be formulated identically with respect to any admissible coordinate system. This requirement is called the principle of general relativity. Note that this plan of theory does not yet say anything about the presence or absence of a metric in space-time, but already provides the basis for the existence of affine connection in it (together with curvature and other derived mathematical structures). Naturally, already at this level there is a need to give physical meaning to the mathematical objects of the theory. Here he is. A point in space-time depicts an event, characterized on the one hand by position and moment of time, on the other by four coordinates. Something strange? Aren't they the same thing? But no. In general relativity it is not the same thing. Coordinates of the most general form, admissible in theory, cannot be interpreted as positions and moments of time. This possibility is postulated only for a very limited group of coordinates - locally inertial ones, which exist only in the vicinity of each point, but not in the entire region covered common system coordinates This is another postulate of the theory. This is such a hybrid. I will note that this is where many of the problems of general relativity arise, 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, the mutual attraction of massive bodies. It is argued that this physical phenomenon can be destroyed under certain conditions simple choice a suitable reference frame, namely, a locally inertial one. For all bodies that have the same acceleration (free fall) due to the presence in a small region of the gravitational field of a distant massive body, this field is not observable in a certain reference frame. Formally, the postulates end there, but in fact the main equation of the theory, which introduces the metric into consideration, also refers to the postulates, both as a mathematical statement and as a physical one. While I'm not going to go into detail about the equation (system of equations, really), it's still useful to have it in front of you:

R ik = -с (T ik – 1/2 T g ik)

Here on the left is the so-called Ricci tensor, a certain convolution (combination of constituent components) of the complete curvature tensor. It can rightfully also be called curvature. On the right is a construction of the energy-momentum tensor (purely physical quantity in GTR, singular for massive bodies and external for space-time, which for energy-momentum in this theory is simply a carrier) and the metric, which is assumed to exist. Moreover, this metric, as a scalar quantity produced by the metric tensor, is the same for all points in the region. There is also a dimensional constant c, proportional to the gravitational constant. From this equation it is clear that, by and large, curvature is compared with energy-momentum and metric. The physical meaning is assigned to the metric in General Relativity after obtaining a solution to these equations. Since in this solution the metric coefficients are linearly related to the potential of the gravitational field (calculated through it), the meaning of the potentials of this field is assigned to the metric tensor. With this approach, curvature should have a similar meaning. And affine connection is interpreted as field strength. This interpretation is incorrect; its fallacy is associated with the paradox noted above in the interpretation of coordinates. Naturally, this does not go unnoticed for the theory and manifests itself in a number of good ways. known issues(non-localizability of gravitational field energy, interpretation of singularities), which simply do not arise when geometric quantities are given the correct physical meaning. All this is discussed in more detail in the book ““.

However, even in general relativity, the metric inevitably, in addition to the meaning artificially imposed on it, has another physical meaning. Let us remember what characterizes the metric in the case of Euclidean space? One very important thing for measurements in space-time is the ability to introduce in this space a rigid rectangular coordinate grid that uniformly fills the entire area. This grid is called an inertial reference frame in physics. Such a reference system (coordinate system) corresponds to one and only one standard view metric tensor. In reference systems that move arbitrarily relative to the inertial one, the form of the metric tensor is different from the standard one. From a physical point of view, the role of the “reference grid” is quite transparent. If you have a rigid body of reference, 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 invent such a body of reference, providing it (space) with exactly the same metric. In this understanding, the metric tensor, different from the standard Euclidean one, says that the reference system (coordinates) is built using a non-rigid body, and perhaps the clock also runs differently at its points. What do I mean by this? But the fact that the metric tensor is a mathematical image of some of the most important properties of the reference system for us. Those properties that absolutely characterize the structure of the reference system itself allow us to determine how “good” it is, how different it is from the ideal – the inertial frame. So GTR uses the metric tensor precisely as such an image. How an image of measuring instruments distributed in a reference area, possibly changing its orientation from point to point, but having everywhere the same norm, common to all reference vectors. The metric, considered as a scalar, is this norm, the magnitude of the scale. The metric as a tensor allows us to consider arbitrary relative motion relative to each other of all scales that make up the body of reference. And General Relativity describes a situation where in space-time it is possible to have such a body of reference, real or imaginary.

This view of metrics is certainly correct. Moreover, it is also productive, since it immediately focuses attention on the remaining agreements in the GTR. Indeed, we have allowed for frames of reference in which scales at different points can be oriented differently (in a four-dimensional world, orientation also includes motion). And we still demand that some absolute characteristic of the scale, its norm (interval) remain the same. Consequently, the statement of General Relativity that it took into consideration all possible reference systems is excessive. It is not so general, relativity in this theory.

Metrics are bullshit, you say, and you’ll be right. In something.

Indeed, when it comes to metrics, the very first metric that comes to mind is traffic.

Many people like to meditate for hours looking at the traffic graph of their website.

How cool it is to watch the line jump - back and forth, back and forth... And it’s even cooler when site traffic grows continuously.

Then blissful warmth spreads throughout the body and the mind soars to heaven in anticipation of heavenly manna.

Ah, what joy, what bliss!

And even if the picture is sad...

You still can’t take your eyes off the chart, it’s so addictive.

It seems that there is a secret meaning hidden in the graphics. A little more, and the picture will reveal its secrets and tell an incredibly simple and effective method attracting huge amount clients. And then the money will definitely flow like a river.

But in fact, attendance is a typical “sweet (vanity) metric” that does not carry any useful meaning.

And these are the majority of metrics. Basically, all the metrics you see are sugary. And that is why metrics have a bad reputation as a pointless waste of time and effort.

But in reality this is not the case at all. The right metrics provide extremely important and sometimes invaluable information for a business and a project.

The main bonus and purpose of metrics is that they make it possible to manage your business or project.

How to determine if a metric is bad?

Let's look at a very simple example - the speed of a car.

Please tell me what speed means...

100 km/h?

So what does it mean?

I think you probably guessed it yourself that... does not mean anything!

OK. Now the second question:

Is 100 km/h good or bad?

Neither one nor the other?

Speed is a completely useless and stupid metric. Unless, of course, you use it on its own. Combined with other metrics, it can, of course, say something, but on its own, it certainly doesn’t.

Site traffic is exactly the same speed.

That is why there is absolutely no point in hanging out in front of the site traffic chart. He will not reveal to you the secret of life. Do you understand now?

What metrics are good then?

For example, Churn rate. This metric tells you how many customers have left the company/site forever over time.

Churn rate = 1% says that we only lose 1% of customers. Those. We hardly lose anyone.

If Churn rate = 90%, then this means that we are losing almost all of our clients. It's horrible!

Do you see the difference between this metric and speed?

Churn rate is a meaningful metric that answers the question of whether something is good or bad. And you don't have to guess what it means.

This is a metric that speaks for itself!

And now we are ready to take urgent action to reduce customer churn.

That is why such metrics are called actionable. Because they encourage action.

Criterion for “sweetness” of metrics

There is a very simple way to determine that a metric is “vanity”.

Most absolute metrics, such as traffic, number of downloads, number of retweets, number of emails/subscribers, number of likes, etc. are cheesy.

Relative, weighted metrics are often actionable. But not all!

As for quality metrics, there is no certainty here, because a qualitative assessment in itself cannot be accurate and unambiguous.

But on the other hand, it is possible and necessary to evaluate the convenience of a program precisely by the level of perception end users and nothing else.

How to approach metrics in general?

The first thing you need to do is turn your brain around.

No kidding.

Everyone(!) who comes across metrics, first of all begins to look for the reason for being in them. But they won't show it, unfortunately.

Metrics are just like an ordinary ruler with which we measure everything we want.

You're not looking for the reason for existence in an ordinary wooden ruler, right?

Finding the meaning of life in a line is what is called a “bottom-up approach.”

For proper operation With metrics, you need to change the paradigm and start working the other way around, from top to bottom.

Those. first do some action, and then use metrics to measure the effect resulting from it.

Metrics should be used as an ordinary subject for measurement and nothing more.

Think about these words.

Measure the effect of your actions using metrics, rather than inventing actions based on the readings of a wooden ruler.

This approach is also called “Hypothesis->Measurement”.

Ok, this is clear.

Question No. 2: “What exactly to measure? How to find the right metrics?

How to create your own set of metrics?

Having surfed the Internet, you will probably find dozens, or even hundreds of different metrics on the same topic.

For example, you can find about a hundred software quality metrics. These include GOSTR-ISO standards, metrics calculated in SonarQube, some self-written options, and even “quality” metrics based on user reviews.

So which ones are worth using and which ones are not?

The best approach is to be guided by a “core value.”

OMTM (One Metric That Matters)

Let's look at an example.

It is clear that if you want to improve the quality of your software product, then you can measure this quality in different ways.

Quality is not just about the number of errors. If you look at quality as a whole, then this is:

Number of incidents in the industry,
ease of use and ease of perception,
speed of work,
completeness and timeliness of implementation of the planned functionality,
safety.

There are many criteria and it is impossible to work with all of them at once. They do it very simply: they choose one, the most important criterion at the moment, and work only with it.

This approach is called OMTM (One Metric That Matters).

It is logical for the software quality OMTM to select the number of serious (important and critical) incidents in an industrial environment.

For online stores, you don’t need to think about OMTM at all - it’s sales volume or profit (depending on your decision).

This One Important Metric will be the core value for your set of metrics. And their final set will depend on it.

Value Inside

They often start compiling a set of metrics “out of the blue,” by scouring the Internet and choosing the best options from what they found according to the principle: “Oh! This will suit us!”

As you understand, this is not The best way, right?

But how do you decide which metric to take and which not to?

For example, various types of user conversions are often measured.

But why do they measure users and not something else? Have you thought about this question?

Naturally, there is an answer.

Let's look at an online store as the easiest example to understand.

Let's say you want to increase your sales. What metrics will you need for this? How to approach this?

There is one simple, logical and working way. Everything falls into place when you answer the question:

WHO PRODUCES VALUE?

We work based on sales volume, right? We want to increase it, right?

Who and what needs to be influenced to increase sales?

Certainly,

need to influence the cause -
on the one who “produces” value.

Who makes money in an online store? Where does the money come from?

Very simple: from clients.

Where exactly in an online store can you influence customers?

Yes, anywhere!
Right. At every stage life cycle client.

To represent the life cycle, it is convenient to build the so-called. “funnel” of the client’s movement through the process.

An example of an online store funnel:

Why is this so? Because customers get lost precisely when moving from one step of the funnel to another.

By increasing the number of clients at any level of the funnel, we automatically increase the resulting sales volume.

A simple example.

The “Cart Abandonment Rate” metric essentially shows the conversion rate from a shopping cart to a completed order.

Let’s say that during the first measurement you discovered that 90% of the baskets are lost, i.e. Out of 10 baskets, only 1 order is made.

There's clearly something wrong with the shopping cart, right?

As a result of the cart improvements, the percentage of abandoned carts decreased by 10% to 80%. What does this look like in numbers?

Out of 10 baskets, 2 orders began to be placed. 100 rubles * 2 = 200 rubles.

But this is an increase in sales volume by 100%! Bingo!

By increasing your step conversion by just 10%, you have increased your sales volume by 100%.

Fantastic!

But that's exactly how it works.

Do you understand now what is the beauty of correctly constructed metrics?

With their help you can achieve a fantastic impact on your processes.

With an online store, everything is quite simple, but how can all this be transferred, for example, to the quality of the software product? Yes exactly the same:

We choose the core value we are working on. For example, we are reducing the number of incidents in the industry.
We understand who and what produces this value. For example, source code.
Building a life cycle funnel source code and set up metrics at each step of the funnel. All.

Here, for example, what quality metrics could be obtained (off the top of my head)…

Value indicator:

Density of industrial defects per 1000 lines of code

Metrics based on the source code life cycle:

proportion of unsuccessful compilations,
autotest coverage,
percentage of unsuccessful autotests,
failure rate of deployments.

Metrics based on the defect life cycle:

dynamics of defect detection,
dynamics of correction,
dynamics of rediscoveries,
dynamics of defect deviations,
average waiting time for a fix,
average time to fix.

As you can see, the topic of metrics is really very important, necessary and interesting.

How to choose the right metrics:

Choose an OMTM, think about its core value, and measure the producers of that value.

Build metrics based on the manufacturer’s life cycle funnel.

Avoid using absolute metrics.

The topic of metrics became popular in the wake of the Lean Startup movement, so it’s best to start reading from the primary sources - the books “Lean Startup” (translation into Russian - “Business from Scratch. The Lean Startup Method” on Ozon) and “Lean Analytics” (there is no translation, but the book in English is sold on Ozon).

Some information can be found on the Internet even in Russian, but, unfortunately, a comprehensive textbook has not yet been found even in the Western segment.

By the way, now there are even individual specialists “productologists” whose task is to build correct system metrics for your product and suggesting ways to improve them.

That's all.

If the article helped you better understand the essence of the issue, the author would be grateful for a “like” and repost.

You can't manage what you can't measure. This phrase is found in many guides on IT management, and indeed on any management. It is impossible to manage production without measuring instruments that allow you to make the right decisions in a timely manner and respond to a changing situation. The IT department is the same production, comparable in complexity to an average plant, and it requires appropriate tools to manage it. Process metrics are management tools. How effectively a manager can manage an organization depends on how the metrics system is built.

Difference between metrics and indicators

Before moving further, let's agree on terms. A metric is a measurable parameter. An indicator is a measurable parameter for achieving a certain goal. A target value and a desired trend must be defined for the indicator.

Metric classification

The activities of any IT organization can be divided into three segments

Services
Processes
Infrastructure

Each of these segments should be managed and therefore measured.

Service metrics

Shows how our services are provided. These metrics correspond to the service parameters agreed upon in the SLA. It is the change in these metrics that the customer first feels. They are formulated in terms understandable to the customer and must correlate with the subjective perception of the customer. Examples of such metrics: time of report generation, number of clients served per unit of time, etc. It is for the values of service metrics that the IT organization is responsible to the customer. It is obvious that the meaning of service metrics depends on both the operation of processes and the infrastructure. Big time downtime (service metric) can be caused by both excessive load on the communication channel (technological metric) and insufficient speed incident resolution (process metric).

Technology metrics

Technology metrics reflect the health of the infrastructure. These include the current load of communication channels, free disk space, number of failures in the disk array, etc. Control of these metrics is most often assigned to monitoring and event management systems.

Process metrics and their classification

Process metrics show the efficiency of the internal processes of an IT organization. Any process has an input and an output, in addition, the process uses resources and is subject to control influences.

When building a system of metrics, you need to remember these four components and measure each of them. Input metrics - measure the load on the process. For example, for an incident management process, the number of incidents is an input metric. It is important that for a process manager, input metrics are a purely informational indicator; they cannot be influenced, but only reacted. Output metrics, or performance metrics, show how much the process achieves its goal. Resource metrics show the load and sufficiency of resources used by the process. Metrics controls show how controllable the process is, and how effective control actions are. CobiT proposes its own classification of metrics: performance indicators, controllability indicators. In addition, a maturity model is proposed for each process. The decomposition of metrics into four components is comparable to the classification of metrics proposed in CobiT. Performance indicators are output metrics, rationality indicators are resource metrics, process maturity is a controllability metric. It is obvious that when planning target values for different metrics, you should balance them with each other. Because the process can be very effective and completely irrational and vice versa. For example, we can resolve all incidents in half an hour, with the help of a thousand people, or we can cope with a dozen people, but resolve incidents in a month.

Reporting corrective actions

Indicators are not interesting in themselves; they are needed to implement management influences on the process. Therefore, responsibility for achieving target indicators should be assigned to the management of the process and to employees assigned to roles in the process. A reporting system for indicators should be built. It is important that appropriate indicators are presented at each level of management. It is unlikely that the IT director will be interested in analyzing the failure statistics of one of the disk arrays. The reporting system should be structured in such a way that at each level, each manager controls and is responsible for 3-9 indicators. Larger quantities are difficult to keep under constant control.

Motivation issues

Process performance indicators are often used to set personal goals and motivate employees. At first glance, this is logical and correct solution. However, a person whose salary depends on meeting target values will be too tempted to “tweak” the indicators in his favor. Such an “edit” leads to a violation of the main principle of using indicators - objectivity and reliability, as if a doctor made a diagnosis and prescribed treatment based on incorrect information about the patient’s temperature. How to avoid this? It is possible to create a developed internal audit service that would guarantee the relevance of indicators. But, often, the costs of maintaining such a service are unjustified. There is no clear recipe here. We have to look for a reasonable balance between trust and control.

Building a system of IT indicators

Processes do not operate in a vacuum. Each process is aimed at achieving a specific goal, is connected to other processes and contributes to the provision of services. The goal of each process should be set based on what aspect of the service it supports. Thus, the primary goals of the services are the parameters recorded in the SLA. The objectives of the processes and the indicators that measure them must be determined by decomposing these parameters. For example, the SLA may specify the target availability of a service. It is obvious that service availability is influenced by such indicators as time to resolve incidents, number of incidents, success of changes, etc. The target values of these indicators should be determined from the target value of service availability. It is also clear that the significance of each of these indicators is not the same and may vary depending on the conditions. Based on the significance of each indicator, it should be assigned an appropriate weight. By decomposing it into several stages, it is possible to build a system of goals and indicators of the organization. Such a system provides the organization’s management with a tool for operational control and quick diagnostics, the basis for decision-making at both the operational and tactical and even strategic levels.

On what basis should we build a system of indicators?

What should you be guided by when building a measurement system for IT processes? First of all, of course, CobiT. This methodology proposes a process model consisting of thirty-four processes. It is believed that any activity within an IT organization can be classified into one of these processes. For each process, a set of performance and efficiency indicators is provided. The problem is the superficial and general description of metrics. When using CobiT recommendations, methods for measuring target values and algorithms for calculating indicators will have to be thought out independently. As for ITIL®, it is more specific. You can find enough in it detailed list indicators for each process with ways to measure them and desired trends. However, ITIL®, as you know, does not describe all possible IT processes. For example, for indicators of the Software Development process, you will have to turn to other sources. In the topic of process measurement, it is impossible not to touch upon the Balanced Scorecard (BSC). This tool, originally developed for enterprise management, can be successfully applied to an IT management system. The main postulate of BSC is that when building a system of organizational goals, it is dangerous to allow a bias in one direction or another. For example, paying all attention to financial efficiency, without paying attention to customer loyalty or the organization of internal processes. The same is true for IT. For example, a skew to the side technological aspect infrastructure can lead to disruption of internal processes or deterioration of relationships with the customer. A bias in the service aspect is also harmful. Thus, when building a system of IT goals and measurements, it is necessary to highlight several perspectives that should be given balanced attention. For example, you can offer the following perspectives:

Services
Internal processes
Infrastructure
Finance
Staff.

In each specific case, the weight of goals for these perspectives may change. You can also use a different perspective system.

Very often in many startups this situation arises: everyone is working, the creation process is going well, and then it turns out that there are few users, and those who exist are dissatisfied with your product. Many startups go through this situation, and the perseverance and perseverance of the founders, as well as the ability to influence the team, maintaining a “fighting spirit” are of no small importance for overcoming the crisis. But this situation can be avoided if you take advantage of the experience of previous teams.

In a startup at any stage, you need clear metrics that will tell you in which direction you are moving and whether you are moving at all.

IN English language there is a word " traction". Translated it means:

craving; traction force;
support, chances of success; a situation where someone or something has its own supporters.

Note that the second meaning is related in meaning to the first. The project leader needs to monitor " traction". He is described as significant feedback from your clients. One of the classic definitions of the concept " traction"in business is the following: " quantitative evidence of market demand", i.e. clear, quantitative proof that the product or service your startup offers actually has a market demand.

Simply put, this is an actual, and mass, and not isolated, confirmation of the need for the final (target) consumer of the project you are creating.

How to measure" traction"? There are different metrics. Let's start with the wrong ones. Here are some examples:

What's wrong with these metrics? These are the so-called “vanity metrics” - a term coined by Eric Ries. They have a good effect on our self-esteem, but, in essence, are no different from “likes” on social networks. Note:

these metrics will grow in any case;
they do not allow us to understand what led to the change in indicators and, therefore, cannot help startups make decisions.

The right metrics need to be accurate. They can be comparative (we compare indicator X1 with indicator X2) and/or relative, i.e. expressed as a percentage. They should also be understandable to a wide range of people, at least to your team. Based on the CDM (Customer Development Methodology) and Lean Startup methodology, metrics should influence the behavior of startup founders and serve as a call to action.

AARRR - Startup Metrics for Pirates

Now about the right metrics. Dave McClure, one of the founders of the accelerator-incubator (there are about 450 companies in the portfolio) of startups "500 Startups", proposed a system of metrics AARRR - Startup Metrics for Pirates.

AARRR is an acronym formed from the names of the stages a user goes through in your sales funnel:

Acquisition (acquiring users, attracting them to the resource). These are the visitors to your site who came from different places. Some through advertising channels, some through search, some through a link on Twitter.
Activation. Visitors enjoyed their first visit. They didn’t immediately close your site, they read something on it, understood what the point was and took the necessary action - subscribed to the newsletter, registered, ordered back call etc.
Retention. Your visitors return to your site because they like something about it and take action there.
Referral (transfer). Visitors like your site so much that they talk about it, attracting new users, who in turn also go through the activation stage (this is important).
Revenue. Your visitors have turned into clients who pay you money.

In different sources, the 4th and 5th stages are often interchanged. This is logical, because In the classic sales scheme, users first become customers and then start talking about the product.

Now let's talk about each stage in more detail.

Acquisition

First stage. People come to you from different channels: social media, SEO ( search engine optimization, from English search engine optimization), SEM ( search marketing, from English search engine marketing), E-mail, blogs, contextual advertising, offline (events not online), partnership programs and others. The main indicator is the ratio of transitions to the site and money spent/ad views.

Activation

Users have landed on your site. How to understand that activation has occurred?

User:

spent 10-30 seconds or more on the site.
looked at 2-3 or more pages.
made 3-5 clicks.
took advantage of one key function.

What might be the key function or action that is considered activation?

In order for activation to be successful, it is necessary to create many " landing page"("landing" pages). Startups usually use from 4 to 16 "landing pages", but more are possible. After this, it is necessary to conduct A/B testing, i.e. different pages and track conversion. This needs to be done quickly.

Key metrics of this stage:

Number of pages viewed per visit.
Time on site.
Conversion.