Geometry and physics - History, philosophy and methodology of natural sciences

Geometry and physics

The theory of gravity of Einstein has sharply raised the question, as many of the leading figures of science have expressed, about the relationship between geometry and physics. In electrodynamics, the pseudo-Euclidean geometry is used to express the relationship between space and time. In the theory of gravitation, non-Euclidean geometry is used. Why does this difference occur?

Of course, one should keep in mind that geometry is not a physical, but a mathematical theory. The reference to it means that physicists realize the potential of physical and mathematical modeling. Mathematics is used in the interests of physics. This means that all geometric concepts begin to act as symbols of the concepts of physical theories. Mathematical geometry gives way to physical geometry. By definition, it must express the conceptual structure of the physical theory. Mathematical theories that do not have this quality are physically unpromising. With this conclusion it is difficult not to agree. Nevertheless, different opinions are expressed on the question of the relationship between physical geometry and physics.

According to A. Poincaré, mathematical geometry can not be tested empirically, for it is guided by the idea of ​​an absolutely rigid body, which physicists do not have. But if there are no physical criteria for choosing a mathematical geometry (for example, Euclidean geometry or one of non-Euclidean geometries), then it is permissible to be guided by any of them. For the sake of economy of effort, it is better to take the simplest geometry, i.e. Euclidean geometry. To it, physical laws can be adjusted in such a way as not to contradict the experimental data.

A fundamentally different position was held by G. Reichenbach. In a laconic form, R. Carnap introduced: "Whenever universal effects are discovered in physics, Reichenbach declares, they can always be eliminated by a suitable transformation of the theory."

Translated into geometric language, this means the following: in theory it is expedient to preserve the differential forces that are responsible for the dynamic effects. Universal forces must be excluded from theory, for they do not express the specifics of physical interactions in any way. In the theory of gravitation, with proper choice of the appropriate type of non-Euclidean physical geometry, there are no universal forces. Consequently, all parts of the theory are brought into harmony with each other. If we rely on the Minkowski space-time concept, then in theory the harmonic coexistence of its parts will be violated. Under these conditions, judgments about the true structure of the physical theory will prove difficult. No one has yet proved that the content of a physical theory can be adequately expressed in surrogate concepts.

Thus, the corresponding choice of geometry is determined by the status of the physical theory. It is not arbitrary. The same applies to any mathematical concept used in physics.

It is interesting that in connection with the physical geometry of the theory of gravitation, a fierce polemic between two academicians, VL Ginzburg and AA Logunov has been going on for decades. Ginzburg invariably spoke of Einstein's theory in rapturous tones, avoiding any criticism of it. AA Logunov, on the other hand, subjected Einstein's views to sharp criticism. He builds a relativistic theory of gravity differently than Einstein. Logunov postulates that the laws of conservation of energy and momentum must be satisfied for gravitational phenomena (in closed systems) . To fulfill this condition, Minkowski's pseudo-Euclidean space-time is introduced. Curvilinear Riemannian space-time appears as appendage to the Minkowski space, it is defined by the Minkowski space plus the gravitational field: "Thus, we can state that Minkowski space is still universal to this day, valid for all forms of matter, including the gravitational field".

The concept of A. A. Logunov, apparently, is not without serious weaknesses. The Minkowski space-time adjoining to gravity is adhered to it from the outside, it does not represent its specificity. In fact, Logunov seeks to develop a quantum field theory of gravity. However, as an exemplary physical theory, he considers electrodynamics with its concept of space-time. It is possible that in the future such a theory will be created. But while it exists only in projects.

The absolute majority of physicists are inclined in geometrization of the theory of gravitation, conducted by Einstein, to see one of the models of productive physical thinking. Geometrization of physical theories attests to the complexities associated with mathematical modeling of physical phenomena.

With the purpose of isolating the dynamic side of gravity, let us once again write down the gravitational equations:


Concentration on it allows us to clarify the idea of ​​active factors of gravity. The equation has a left and a right side. The left side is nothing more than space-time. The right side represents the energy and impulse response.

The scientific community takes the most clear position on the question of the conditionality of the metric (gik) by pulse-energy parameters. To think differently is to leave the variability of gik without any explanation. However, the frequently used expression "the bodies move in curved space-time" is superimposed on recognition of the secondary nature of gik. This expression is full of incongruities.

First, it does not take into account that space-time is not something ontical, objective-real. It is impossible to move in an abstract concept. One can talk about objective time and objective space. But in them it is impossible to move. You can move only in some environment, for example, in the field. Every now and then they say that bodies move along geodesic lines of space-time. These lines (trajectories) are signs of not a space-time, but a gravitational field.

Secondly, if the bodies move in space-time, and even along curved lines, then it is necessary to recognize its effect on space-time. But even this thesis is incorrect. The dynamic side of gravity is its pulse-energy characteristics. It seems that it is permissible to reason this way: pulse-energy parameters determine space-time, and it acts on the bodies moving in it. But at the same time, in fact, the interaction between bodies is considered. And in bodily form the gravitational field is not a physical object in its entirety.

In order to avoid misunderstandings, it is necessary to distinguish between object interactions and conditionality of the metric by pulse-energy characteristics. It is perfectly legitimate to assert that gravitational fields and objects interact as special condensed media. The conditionality of the space-time attributes of objects by their pulse-energy characteristics is just as valid. But the statement is false that space and time or space-time act on bodies and particles.

The idea of ​​the conditionality of space-time characteristics by pulse-energy parameters in the preliminary plan was formulated by B. Riemann and V. Clifford.

Scientists' opinions

Riemann admitted (1854) that "we should try to explain the appearance of metric relations by something external - the communication forces acting on this real". Clifford, relying on the ideas of Riemann, in 1876 argued that "the change in the curvature of space is what really happens in a phenomenon that we call" the motion of matter, "whether it be significant or ethereal."

Einstein, unlike his predecessors, was the first to manage the philosophical idea of ​​the unity of electromagnetic and gravitational phenomena, including spatio-temporal properties, connected with certain physical theories, respectively. The conclusions were rather unexpected. Both Riemann and Clifford believed that Euclidean space is an abstraction that deviates from physical reality. If they lived to the epoch-making discoveries of Einstein, they would have every reason to welcome the theory of gravity. It would be more difficult for them to accept the electrodynamics of Einstein, according to which the pseudo-Euclidean space-time adequately expresses the actual spatial and temporal signs of physical phenomena.


1. The main difficulty of the relativistic theory of gravity is reduced to an insufficiently clarified mechanism for realizing the activity of the gravitational field.

2. The content of the relativistic theory of gravitation testifies to the conditionality of the space-time characteristics by the pulse-energy parameters.

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