What is curvature? In what relation is it with affine connection? What is the physical meaning of the curvature field when describing the world as space-time?

Curvature seems to be a purely geometric concept. General Theory of Relativity made the concept of the curvature of space-time quite widely known and associated with physics., or rather, with gravity. But this connection remains rather unclear., rather abstract. The reason for this, in my opinion, is, that in general relativity a distorted interpretation of the physical meaning of the used mathematical concepts has been approved, as metrics, and curvature. The origins of this distortion lie in the inconsistent interpretation of coordinates by interpreters of the results of general relativity.. And she rises, naturally, to the authority of the creator of the theory, A. Einstein. Here i am going, not really going into mathematical details, discuss the meaning of curvature, as geometric, so physical. So there will be more words, than formulas, although the formulas cannot be completely avoided.

In geometry, the concept of the curvature of space arises, naturally, upon attempt parallel vector translation in a closed loop. It turns out, that even if this contour contracts to a point, not at all necessary, so that the result of the transfer coincides with the original vector. There may well be some difference between the vectors., original and transferred, which is proportional to the vector itself and some tensor, existing at this point, curvature tensor Rⁱ_jkl. Then, that the coefficients of proportionality constitute the tensor, means, that every point in space has some measurable property, the measurement results of which are the components of the tensor, and do not depend in any way on that vector, what is transferred. By the number of indices of the curvature tensor, one can judge, that the point property, he wrote down, by no means simple. but, something quite familiar to us from daily practice is connected with it.

Let's start by explaining the name itself.. It would seem, what is there to explain? What is the difference between a straight line and a curve is quite understandable without any knowledge of mathematics. This is something true. That's just such a funny thing – on the line, considered as a one-dimensional space, there is no curvature and in sight. From an internal point of view, as one-dimensional spaces, all lines are the same. Their property of being crooked (or direct) appears only then, when they are part of at least two-dimensional (or more measurements) space. Since such (and only such!) lines are usual for us, insofar as we perceive “curvature” lines intuitively. In mathematics,, as a language as accurate as possible, the concept of curvature appears only in spaces, starting from two dimensions. And it turns out that it is not connected with our intuitive idea of ​​the difference between straight and curved lines.. For example, on the plane, the curvature tensor at each point is equal to zero, although curved lines on the plane can be drawn as much as you want. In order not to go too far here in the direction of describing the differences between the curves, I will only say, that for this, the concept of geodetic, for which only affine connection. Curvature is the next element, necessary to complete the description of the geometry of spaces of more than one dimension. This element, of course, already contained in connectivity, is its property. But this is the property, which must be considered explicitly. Then, what is called curvature in mathematics, of course we also know in everyday life, but known as curvature of surfaces. The difference between all kinds of surfaces from a plane is described precisely in the language of the curvature tensor. Perhaps, the easiest way to form some figurative geometric representation of curvature – this is to attach a plane to a point on an arbitrary surface and see their difference.

Actually math does a very similar thing, when constructing the curvature tensor. No matter how many dimensions a given space has, at each point, two-dimensional surfaces are selected (naturally, all possible) and in them some combination is calculated, simple enough, from the connectivity coefficients and their first derivatives in two directions, and then these two values ​​are subtracted. And so for each pair of coordinates. Therefore, the curvature tensor is antisymmetric in one of the pairs of subscripts, namely, by indices k and l. Since the curvature tensor also has a superscript, then two more tensors with a smaller number of indices can be made from its components using the convolution. This will be a tensor Richie R_jk = Rⁱ_jki = R¹_jk1 + R²_jk2 + R³_jk3 + R⁴_jk4 and one more tensor (known not really in geometry, and in physics, like tensor Maxwell) F_at = Rⁱ_cl . What is convolution can be seen from the expression for the Ricci tensor. I mean, that our space is four-dimensional, therefore the sum consists of four components. Exactly the same summation over the same upper and lower indices (but this time the first index changes at the bottom) is meant to obtain the tensor F_at. Note, that, like the total curvature tensor in indices k and l, Maxwell's tensor is antisymmetric in its indices. A priori, nothing can be said about the symmetry of the Ricci tensor.

Now let's talk a little about, why do you still need to build all sorts of curvature tensors and others like them, if all geometric relations in the space can be extracted from the affine connection. After all, the curvature tensor is nothing else, as some combination of connectivity coefficients and their first derivatives. The answer to this question is simple. All such, tensor, structures secondary to connectivity just describe the same geometric (and in the case of space-time and physical) ratios, which we are interested in. Moreover, it is permissible for the theorist to define connectivity as a set of known functions. If we apply the mathematics of spaces with affine connection to the description of space-time (namely, this area of ​​application of the theory is the most important for us), then we do not have a known connection and cannot be. The only way to restore it – these are experiments, measuring properties of objects in the real world. The more different properties we can measure, the closer to reality our description of the world will be. The results of measurements of these properties are just grouped into tensors.

What properties of the physical world are expressed in the curvature tensor? By and large, the curvature tensor can be called an indicator of the presence of a force at a given point. This is the tensor of the strengths of the unified field. Words power, the field of forces is quite familiar and intuitive to us. Tension – less understandable word. Therefore, I will explain it a little.. Tension – this is the specific force. The substance is familiar to us, usually consists of many particles, which together make up some total mass. And in the case of electromagnetic forces and the total charge. Here is the power, per unit mass or charge and is called the strength of the field of forces. She is important to those, which characterizes a given point in space always in the same way, regardless of whether, which body, on which the force acts, at this point is. Then, that the curvature tensor is precisely the specific quantity that follows from its very definition.

Although the central structure of the theory of gravity, created by Einstein and is believed metric tesor g_I, but the basic equation of this theory

    R_I= -other (T_I – 1/2 T g_I)

is written precisely for the curvature, more precisely for one of its bundles, Ricci tensor. This already makes curvature the central concept of the theory.. The presence of gravitating bodies in a given area of ​​space-time, or generally energy in a different form, for example in the form of an electromagnetic field, which is equivalent to the nonzero energy-momentum tensor T_I, means the difference from zero and this curvature convolution, Ricci tensor. As a matter of fact, if you try to explain the physics of space-time without introducing quantities external to geometry (and in this equation the energy-momentum tensor is a purely non-geometric quantity), then the energy-momentum tensor itself can be (and need) define as some tensor, constructed from curvature tensor components :   T_I= – (R_I– 1/2 R g_I)/сother. Then the basic equation of the theory of gravity becomes a simple definition of one of the quantities of the theory. And gravitating matter will be nothing else, as a manifestation of the curvature of space-time in a given area.

© Gavryusev V.G.
The materials published on the site can be used subject to the citation rules.