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The laws of electrodynamics and the principle of relativity brief. The principle of relativity. Postulates of the theory of relativity. Work program of the educational subject of a municipal educational institution of a secondary school in the village. Bereznyak

The laws of electrodynamics and the principle of relativity brief. The principle of relativity. Postulates of the theory of relativity. Work program of the educational subject of a municipal educational institution of a secondary school in the village. Bereznyak

Ideas about space and time have changed. According to the classical concepts of space and time, considered unshakable for centuries, movement has no effect on the passage of time (time is absolute), and the linear dimensions of any body do not depend on whether the body is at rest or moving (length is absolute).

Einstein's special theory of relativity is a new doctrine of space and time, which replaced the old (classical) ideas.

§ 75 LAWS OF ELECTRODYNAMICS AND THE PRINCIPLE OF RELATIVITY

The principle of relativity in mechanics and electrodynamics. After in the second half of the 19th century. Maxwell formulated the basic laws of electrodynamics; the question arose: does the principle of relativity, which is valid for mechanical phenomena, also apply to electromagnetic phenomena? In other words, do electromagnetic processes (the interaction of charges and currents, the propagation of electromagnetic waves, etc.) proceed the same way in all inertial frames of reference? Or perhaps uniform straight motion, without affecting mechanical phenomena, has some effect on electromagnetic processes?

To answer these questions, it was necessary to find out whether the basic laws of electrodynamics change when moving from one inertial frame of reference to another, or, like Newton's laws, they remain unchanged. Only in the latter case can we cast aside doubts about the validity of the principle of relativity in relation to electromagnetic processes and consider this principle as a general law of nature.

The laws of electrodynamics are complex, and a rigorous solution to this problem is not an easy task. However, simple considerations would seem to allow us to find the correct answer. According to the laws of electrodynamics, the speed of propagation of electromagnetic waves in a vacuum is the same in all directions and is equal to c = 3 10 8 m/s. But in accordance with the law of addition of velocities of Newtonian mechanics, the speed can be equal to the speed of light only in one selected frame of reference. In any other frame of reference, moving with respect to this chosen frame of reference with speed , the speed of light must already be equal to -. This means that if the usual law of addition of velocities is valid, then when moving from one inertial frame of reference to another, the laws of electrodynamics must change so that in this new system reference, the speed of light was no longer equal to , but - .

Thus, certain contradictions were discovered between electrodynamics and Newtonian mechanics, the laws of which are consistent with the principle of relativity. They tried to overcome the difficulties that arose in three different ways.

First way: declare the principle of relativity invalid as applied to electromagnetic phenomena. This point of view was shared by the great Dutch physicist, founder of the electronic theory X. . Since the time of Faraday, electromagnetic phenomena have been considered as processes occurring in a special, all-pervasive medium that fills all space - the world ether. The inertial frame of reference, at rest relative to the ether, is, according to Lorentz, a special, preferential frame of reference. In it, Maxwell's laws of electrodynamics are valid and most simple in form. Only in this reference frame is the speed of light in vacuum the same in all directions.

Second way: consider Maxwell's equations incorrect and try to change them in such a way that when moving from one inertial reference system to another (in accordance with the usual, classical ideas about space and time) did not change. Such an attempt, in particular, was made by G. Hertz. According to Hertz, the ether is completely entrained by moving bodies and therefore electromagnetic phenomena proceed in the same way regardless of whether the body is at rest or moving. The principle of relativity remains valid.

Finally, the third way: abandon the classical concepts of space and time in order to preserve both the principle of relativity and Maxwell's laws. This is the most revolutionary path, because it means a revision of the deepest, most basic concepts in physics. From this point of view, it is not the equations that turn out to be inaccurate electromagnetic field, and Newton's laws of mechanics, consistent with old ideas about space and time. It is the laws of mechanics that need to be changed, not Maxwell's laws of electrodynamics.

The third method turned out to be the only correct one. Consistently developing it, A. Einstein came to new ideas about space and time. The first two ways, as it turns out, are refuted by experiment.

Lorentz's point of view, according to which there must be a chosen frame of reference associated with the world ether, which is at absolute rest, was refuted by direct experiments.

If the speed of light were equal to 300,000 m/s only in the reference frame associated with the ether, then by measuring the speed of light in an arbitrary inertial reference frame, it would be possible to detect the movement of this reference system relative to the ether and determine the speed of this movement.

Einstein Albert (1879-1955)- great physicist XX century He created a new theory of space and time - the special theory of relativity. Generalizing this theory for non-inertial reference systems, he developed general theory relativity, which is modern theory gravity For the first time he introduced the concept of particles of light - photons. His work on the theory of Brownian motion led to the final victory of the molecular kinetic theory of the structure of matter.

Just as wind arises in a frame of reference moving relative to air, when moving relative to the ether (if, of course, the ether exists), an “etheric wind” should be detected. An experiment to detect the “ethereal wind” was carried out in 1881 by American scientists A. Michelson and E. Morley, based on an idea expressed 12 years earlier by Maxwell.

This experiment compared the speed of light in the direction of the Earth's motion and in the perpendicular direction. The measurements were carried out very accurately using a special device - a Michelson interferometer. The experiments were carried out in different time days and different seasons. But the result was always negative: the movement of the Earth in relation to the ether could not be detected.

Thus, the idea of the existence of a preferential frame of reference did not stand up to experimental testing. In turn, this meant that no special medium, the “luminiferous ether,” with which such a preferential frame of reference could be associated, existed.

When Hertz tried to change Maxwell's laws of electrodynamics, it turned out that the new equations were unable to explain a number of observed facts. Thus, according to Hertz’s theory, moving water should completely entrain the light propagating in it, since it entrains the ether in which the light propagates. Experience has shown that in reality this is not the case.

It turned out to be possible to reconcile the principle of relativity with Maxwell's electrodynamics only by abandoning the classical concepts of space and time, according to which distances and the passage of time do not depend on the reference system.

Myakishev G. Ya., Physics. 11th grade: educational. for general education institutions: basic and profile. levels / G. Ya. Myakishev, B. V. Bukhovtsev, V. M. Charugin; edited by V. I. Nikolaeva, N. A. Parfentieva. - 17th ed., revised. and additional - M.: Education, 2008. - 399 p.: ill.

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In the second half of the 19th century, D. Maxwell formulated the basic laws of electrodynamics. At the same time, doubts arose about the validity of Galileo’s mechanical principle of relativity in relation to electromagnetic phenomena. Let us remember the essence of the mechanical principle of relativity.
If the reference systems move relative to each other uniformly and rectilinearly and in one of them Newton’s laws of dynamics are valid, then these systems are inertial. In all inertial frames of reference, the laws of classical dynamics have the same form (invariant); this is the essence of the mechanical principle of relativity or Galileo's principle of relativity.
To prove this principle, consider two reference systems: the inertial frame TO(with coordinates x, y, z), which we will conventionally consider stationary and a moving system K"(with coordinates x", y", z"), moving relative to TO evenly and straightly at speed u= const. Let us assume that at the initial moment of time t= 0 start O And O" both coordinate systems coincide. Location of coordinate systems at an arbitrary point in time t has the form shown in Fig. 5.1. Speed u directed along a straight line OO", and the radius vector drawn from the point O exactly O", is equal r 0 =ut.
Arbitrary coordinates material point A in stationary and moving reference systems are determined by radius vectors r And r", and

In projections on the coordinate axes vector equation(5.1) is written in the form called Galilean transformations:

(5.2)

In the special case when the system K" moves at speed v along the positive axis direction x systems K, Galilean coordinate transformations have the following form:

In classical mechanics it is assumed that the passage of time does not depend on relative motion reference systems. Therefore, the system of equations (5.2) is supplemented with one more relation:

(5.3)

Relations (5.2) – (5.3) are valid only in the case u. At speeds comparable to the speed of light, Galilean transformations are replaced by more general Lorentz transformations.
Let us differentiate equation (5.1) with respect to time and taking into account that u= const, let's find the relationship between the velocities and accelerations of the point A relative to both reference systems:

where
(5.4)

And

(5.5)

If on point A other bodies do not act, then a= 0 and according to (5.5) a"= 0, i.e. mobile system K" is inertial - an isolated material point either moves relative to it uniformly and rectilinearly, or is at rest.
From expression (5.5) it follows that

those. Newton's equations (equations of dynamics) for a material point are the same in all inertial frames of reference or are invariant with respect to Galilean transformations. This result is often formulated as follows: the uniform and linear motion of the system as a whole does not affect the course of the mechanical processes occurring in it.
Classical Newtonian mechanics reliably describes the motion of macroscopic bodies moving at speeds much lower than the speed of light. IN late XIX V. it was found that the findings classical mechanics contradict some experimental data. In particular, when studying the movement of fast charged particles, it turned out that their movement does not obey Newton’s laws. Further difficulties arose when trying to apply classical mechanics to explain the propagation of light. According to the laws of electrodynamics, the speed of propagation of electromagnetic waves in a vacuum is the same in all directions and is approximately equal to With= 3*10 8 m/s. But in accordance with the laws of classical physics, the speed of light can be equal to With only in one selected frame of reference. In any other reference system moving relative to the chosen system with speed v, it should already be equal With-v, or With+v. This means that if the law of addition of velocities of classical mechanics is valid (formula (5.4)), then when moving from one inertial system to another, the laws of electrodynamics must change, since the speed of light must change. Thus, contradictions were discovered between electrodynamics and Newtonian mechanics, the laws of which are consistent with Galileo’s principle of relativity. Various methods have been proposed to overcome the difficulties encountered:

Accept the inconsistency of the principle of relativity in relation to electromagnetic phenomena. Since the time of Faraday, electromagnetic phenomena have been considered as processes in a special, all-pervasive medium that fills all space - on air. According to H. Lorentz, an inertial frame of reference, at rest relative to the ether, is a special system in which Maxwell's laws of electrodynamics are valid. Only in this reference frame is the speed of light in vacuum the same in all directions.
Consider Maxwell's equations of electrodynamics to be erroneous and try to change them in such a way that they do not change when moving from one inertial system to another (in accordance with classical ideas about space and time). Such an attempt, in particular, was made by G. Hertz, who believed that the ether is completely carried away by moving bodies, therefore electromagnetic phenomena proceed in the same way, regardless of whether the body is at rest or moving. The principle of relativity is correct.
Abandon the classical concepts of space and time in order to preserve both the principle of relativity and Maxwell's laws. From this point of view, it is not the equations of the electromagnetic field that turn out to be inaccurate, but Newton’s laws of mechanics, consistent with the old ideas about space and time. Thus, it is necessary to change the laws of classical mechanics, and not Maxwell’s laws of electrodynamics.

Let us recall how space and time were interpreted in classical physics. Space was considered as an infinite empty extension, containing all bodies and independent of matter. Time was seen as the absolute factor of the uniform flow of duration in which everything arises and disappears. At the same time, time does not depend on any processes in the world.
The development of natural science has refuted these ideas. There is no absolute space and time. The Universe is filled with matter in the form of matter and field, and space acts as a universal property of matter. Time is always associated with the movement and development of matter. Thus, space– this is the form of existence of matter, which expresses its extension and structure; time– this is a form of existence of matter, characterizing the duration of existence of all objects, fields and the sequence of changes in events.
The main properties of space and time are: a) unity and inextricable connection of matter, space and time; b) absolute continuity and relative discontinuity of space and time. Continuity is manifested in the distribution of material fields in the space of all bodies and systems, in the endless succession of elements of length when a body moves between two points. The discontinuity of space is relative and manifests itself in the separate existence of material objects and systems, each of which has certain dimensions and boundaries. The discontinuity of time is characterized only by the time of existence of qualitative states of matter, each of which appears and disappears, passing into other forms; c) time has duration, unidirectionality, irreversibility.
Consistently developing new, different from classical, ideas about space and time, A. Einstein at the beginning of the 20th century. created special theory of relativity(ONE HUNDRED). Within the framework of this theory, it was possible to reconcile the principle of relativity with Maxwell's electrodynamics. Wherein new theory did not cancel the old (Newtonian mechanics), but included it as a special, limiting case.

Purpose of the lesson: to form students’ understanding of how the concepts of space and time have changed under the influence of provisions special theory Einstein's relativity.

During the classes

1. Analysis of test work.

2. Learning new material.

At the end of the 19th century, the basic principles of electrodynamics were formulated. A question arose about the validity of Galileo's principle of relativity in relation to electromagnetic phenomena. Do electromagnetic phenomena occur in the same way in different inertial systems: how do electromagnetic waves propagate, how do charges and currents interact when moving from one inertial system to another?

Inertial is a reference system relative to which free bodies move at a constant speed. Does uniform rectilinear motion have an effect on electromagnetic processes (does it not affect mechanical phenomena)?

When moving from one inertial frame to another, do the laws of electrodynamics change or do Newton's laws remain constant?

For example, according to the laws of addition of velocities in mechanics, the speed can be equal to c = 3·108 m/s in only one reference system. In another frame of reference, which itself moves with speed V, the speed of light should be equal to с̄-V̄. But according to the laws of electrodynamics, the speed of electromagnetic waves in vacuum in different directions is equal to c = 3 108 m/s

Contradictions arose between electrodynamics and Newtonian mechanics.

To resolve the contradictions that arose, three different methods were proposed.

First way The idea was to abandon the principle of relativity as applied to electromagnetic phenomena. This possibility was supported by the founder of electronic theory, H. Lorenz (Dutch). Then it was believed that electromagnetic phenomena occur in the “world ether” - this is an all-pervasive medium that fills the entire world space. The inertial reference system was considered by Lorentz as a system at rest relative to the ether. In this system, the laws of electrodynamics are strictly observed and in this reference system the speed of light in vacuum is the same in all directions.

Second way was to declare Maxwell's equations incorrect.

G. Hertz tried to rewrite them in such a way that they did not change during the transition from one inertial system to another, i.e., like the laws of mechanics. Hertz believed that the ether moves together with moving bodies and therefore electromagnetic processes occur in the same way regardless of the movement or rest of the bodies. That is, G. Hertz retained the principle of relativity.

The third way was to abandon traditional ideas about space and time. Maxwell's equations and the principle of relativity were preserved, but the most obvious, most basic concepts of classical mechanics had to be abandoned.

This method of resolving contradictions turned out to be correct in the end.

The experiment refuted both the first and second attempts to correct the contradictions that arose between electrodynamics and mechanics, leaving the principle of relativity unchanged.

Developing the third way to solve the problem, A. Einstein proved that ideas about space and time were outdated and replaced them with new ones.

Maxwell's equations, corrected by Hertz, could not explain the observed phenomena. Experience has shown that the medium cannot carry away light, since it will carry away the ether in which the light propagates.

The experiments of American scientists A. Michelson and E. Morley proved that no medium like the “luminiferous ether” exists

It turned out to be possible to combine Maxwell's electrodynamics and the principle of relativity by abandoning traditional ideas about space and time, i.e. neither distance nor the flow of time depend on the reference system.

At the end of the 19th century, experimental data were obtained that could not be explained from the standpoint of Newtonian physics. In particular, if the source and receiver of light move towards each other uniformly and rectilinearly, then their Newtonian velocities must add up. However, the American physicist Michelson and others, conducting experiments using a sensitive interferometer, showed that the speed of light in a vacuum does not depend on the speed of movement of the source and receiver and is the same in all inertial frames of reference. Einstein came to the conclusion that constancy of the speed of light- a fundamental law of nature. This conclusion was used by Einstein as the basis for his special theory of relativity (see section 2.5). The invariance of Maxwell's equations (see Section 3.5) under Lorentz transformations was also proven, whereas they are not invariant under Galilean transformations (see 2.4). From Einstein's theory it followed that electromagnetic interactions (for example, charges) are transmitted in a vacuum at a speed limited by the speed of light through a field (the concept of short-range action) in all reference frames.

Division of the electromagnetic field into electric and magnetic field relatively – in nature there is a single electromagnetic field. Light also has an electromagnetic nature (Fig. 3.27).

Based on the special theory of relativity, the patterns were explained Doppler effect for electromagnetic waves. When a light source moves away from the observer with a speed V, a change in frequency (or wavelength by an amount Δλ) occurs in the radiation spectrum of the source with a radiation wavelength λ ( redshift):

The Doppler effect has found application in radar for measuring the speed V and distance to a moving object, in astrophysics for measuring the speed of receding galaxies, etc.

The change in the apparent position of stars on the celestial sphere due to the finite speed of light is called aberrations of light.

3.7. Quasi-stationary magnetic field

The displacement current is fundamentally different from the conduction current - it is not associated with the movement of charges. It is caused only by changes over time electric field(see 3.5). Even in a vacuum, a change in the electric field leads to the emergence of a magnetic field in the surrounding space. It is on this basis that the displacement current is identical to the conduction current and this makes it possible to conditionally call it “current”.

The displacement current j cm occurs not only in vacuum or dielectrics, but also in conductors when an alternating conduction current j pr passes through them. However, it is small compared to j pr (because of this, it is neglected).

In massive conductors placed in an alternating magnetic field, induction currents can be induced in accordance with the law (3.70). These currents are eddy currents in the volume of conductors and are known as currents Foucault.

Foucault currents create their own magnetic field, which, in accordance with Lenz's rule (see 3.73), prevent the change in the magnetic flux that caused them. High-frequency Foucault currents lead to heating of conductors, which allows them to be used for melting metals in induction furnaces, in microwave ovens for heating current-conducting products, in physiotherapy (the human body is a conductor), etc. In other cases, to reduce heat losses in electrical machines and transformers, the resistance to Foucault currents is increased, making their cores not solid, but from thin plates isolated from each other.

In circuits with alternating electric current, the electrical resistance of conductors increases with increasing frequency of the current. This is explained by the fact that the distribution of current density over the cross section of the conductor becomes non-uniform taking into account Foucault currents: the current density increases near the surface (the so-called skin - effect). This also allows you to make conductors hollow (tubular). Methods of high-frequency hardening of the surface of parts are based on the skin effect.

The strength of the alternating current turns out to be different at the same moment in time in different parts of the conductor. This is due to the finite speed of propagation of a changing electromagnetic field along the conductor. However, if we take into account the low speed of movement of charge carriers compared to the speed of propagation of the field, then the currents can be considered quasi-stationary as well as the magnetic fields they excite.

Alternating currents are produced using generators. When the circuit rotates in a uniform magnetic field with angular velocity through the area limited by the contour, changes periodically magnetic flux(see 3.67).

where Ф 0 is the maximum value of the flow through the area S of the contour.

The electromotive force arising in this case (see 3.70) will be
change according to a sinusoidal law. ε 0 =ωФ 0 is the amplitude of the emf. If the circuit is closed, then alternating current will flow in it:

In general, any conductor, in addition to the ohmic resistance R, has inductance L and capacitance C. They provide additional resistance to the current due to the appearance of self-inductive emf (see 3.73) and the inertia of recharging the capacitance. Then the amplitude value of the alternating current is:

(3.90)

Magnitude
has the character of total resistance ( impedance). It depends on the values of R, L, C and frequency. When  satisfies the condition:

the total resistance has a minimum value equal to R, and the amplitude of the alternating current reaches its maximum value:

Frequency
- called resonant.R L =Land
- are called inductive and capacitive reactances in an alternating current circuit.

Alternating electric current has great practical application. It can be transmitted with low losses over long distances and, with the help of transformers, its strength and voltage can be varied over a wide range.

To characterize action alternating current in comparison with direct current, the concept is introduced effective values of current and voltage. The effective value of the current is the quantity I associated with the amplitude I 0 as follows:

voltage is the same
. They determine the AC power. You can also give another definition: the effective value of the alternating current force is equal to this force direct current, which releases the same amount of heat in the circuit as alternating current.

« Physics - 11th grade"

Laws of electrodynamics and the principle of relativity

According to the classical concepts of space and time, considered unshakable for centuries, movement has no effect on the passage of time (time is absolute), and the linear dimensions of any body do not depend on whether the body is at rest or moving (length is absolute).

Einstein's special theory of relativity is a new doctrine of space and time, which replaced the old (classical) ideas.

The principle of relativity in mechanics and electrodynamics

After in the second half of the 19th century. Maxwell formulated the basic laws of electrodynamics; the question arose: does the principle of relativity, which is valid for mechanical phenomena, also apply to electromagnetic phenomena? In other words, do electromagnetic processes (the interaction of charges and currents, the propagation of electromagnetic waves, etc.) proceed the same way in all inertial frames of reference? Or, perhaps, uniform rectilinear motion, without affecting mechanical phenomena, has some effect on electromagnetic processes?

The laws of electrodynamics are complex, and a rigorous solution to this problem is not an easy task. However, simple considerations would seem to allow us to find the correct answer. According to the laws of electrodynamics, the speed of propagation of electromagnetic waves in a vacuum is the same in all directions and is equal to s = 3 10 8 m/s. But in accordance with the law of addition of velocities of Newtonian mechanics, the speed can be equal to the speed of light only in one selected frame of reference. In any other frame of reference, moving with respect to this chosen frame of reference with speed , the speed of light must already be equal to - . This means that if the usual law of addition of velocities is valid, then when moving from one inertial frame of reference to another, the laws of electrodynamics must change so that in this new frame of reference the speed of light is no longer equal to , but - .

First way:
declare the principle of relativity invalid as applied to electromagnetic phenomena. This point of view was shared by the great Dutch physicist, founder of electronic theory, X. Lorentz. Since the time of Faraday, electromagnetic phenomena have been considered as processes occurring in a special, all-pervasive medium that fills all space - the world ether. The inertial frame of reference, at rest relative to the ether, is, according to Lorentz, a special, preferential frame of reference. In it, Maxwell's laws of electrodynamics are valid and most simple in form. Only in this reference frame is the speed of light in vacuum the same in all directions.

Second way:
consider Maxwell's equations incorrect and try to change them in such a way that they do not change when moving from one inertial reference system to another (in accordance with the usual, classical concepts of space and time). Such an attempt, in particular, was made by G. Hertz. According to Hertz, the ether is completely entrained by moving bodies and therefore electromagnetic phenomena proceed in the same way regardless of whether the body is at rest or moving. The principle of relativity remains valid.

Third way:
abandon the classical concepts of space and time in order to preserve both the principle of relativity and Maxwell's laws. This is the most revolutionary path, because it means a revision of the deepest, most basic concepts in physics. From this point of view, it is not the equations of the electromagnetic field that turn out to be inaccurate, but Newton’s laws of mechanics, consistent with the old ideas about space and time. It is the laws of mechanics that need to be changed, not Maxwell's laws of electrodynamics.

The third method turned out to be correct. Consistently developing it, A. Einstein came to new ideas about space and time. The first two ways, as it turns out, are refuted by experiment.

Lorentz's point of view, according to which there must be a chosen frame of reference associated with the world ether, which is at absolute rest, was refuted by direct experiments.

If the speed of light were equal to 300,000 km/s only in the reference frame associated with the ether, then by measuring the speed of light in an arbitrary inertial reference frame, it would be possible to detect the movement of this reference system relative to the ether and determine the speed of this movement. Just as wind arises in a frame of reference moving relative to air, when moving relative to the ether (if, of course, the ether exists), an “etheric wind” should be detected. The experiment to detect the “ethereal wind” was carried out in 1881 by American scientists A. Michelson and E. Morl and based on an idea expressed 12 years earlier by Maxwell.

This experiment compared the speed of light in the direction of the Earth's motion and in the perpendicular direction. The measurements were carried out very accurately using a special device - a Michelson interferometer. The experiments were carried out at different times of the day and different seasons. But the result was always negative: the movement of the Earth in relation to the ether could not be detected.

So,
It turned out to be possible to reconcile the principle of relativity with Maxwell's electrodynamics only by abandoning the classical concepts of space and time, according to which distances and the passage of time do not depend on the reference system.

Postulates of the theory of relativity

The theory of relativity is based on two postulates.

What is a postulate?

Postulate in physical theory plays the same role as an axiom in mathematics.
This is a basic proposition that cannot be logically proven.
In physics, a postulate is the result of a generalization of experimental facts.

1.
All processes in nature proceed identically in all inertial frames of reference.

This means that in all inertial frames of reference the physical laws have the same form.
Thus, the principle of relativity of classical mechanics applies to all processes in nature, including electromagnetic ones.

2.
The speed of light in vacuum is the same in all inertial frames of reference and does not depend on either the speed of the source or the speed of the receiver of the light signal.

The speed of light thus occupies a special position.
Moreover, as follows from the postulates of the theory of relativity, the speed of light in a vacuum is the maximum possible speed of transmission of interactions in nature.

In order to formulate the postulates of the theory of relativity, great scientific courage was needed, since they contradicted the classical ideas about space and time.

In fact, let us assume that at the moment of time when the origin of coordinates of the inertial reference systems TO And K 1, moving relative to each other at a speed , coincide, a short-term flash of light occurs at the origin of coordinates.
During t the reference systems will shift relative to each other by a distance υt, and the spherical wave surface will have a radius υt.
Reference systems TO And K 1 are equal, and the speed of light is the same in both frames of reference.

Therefore, from the point of view of the observer associated with the reference frame TO, the center of the sphere will be at the point ABOUT, and from the point of view of the observer associated with the reference system K 1, - at the point O 1.

But the same spherical surface cannot have centers at points O and O 1.
This obvious contradiction follows from reasoning based on the postulates of the theory of relativity.

So,
there is a contradiction with classical ideas about space and time, which are unfair at high speeds of movement.
However, the theory of relativity itself does not contain contradictions and is absolutely logical.