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Course work

Paradoxes of special relativity

Introduction

3. Relativity of distances

4. Lorentz transformations

5. Paradoxes of SRT

5.2 Clock paradox

5.3 Transport paradox

5.4 Paradox of the wheel

5.5 Pole and barn paradox

5.6 Thin man on bars

Conclusion

Literature

Introduction

Paradoxes, i.e., unexpected consequences or conclusions of a theory that contradict previously established ideas, play a special role in the development of science. When resolving one or another theoretical paradox, one has to turn to the most fundamental provisions of the theory and sometimes revise or refine the ideas associated with it. Thus, theoretical paradoxes in the process of their resolution represent some internal reason for the development of the theory, contributing to its logical improvement, and sometimes even clarifying the limits of applicability and ways of further generalization.

Of course, the basic facts for the development of any theory are the facts obtained from experiments and observations. However, facts alone cannot confirm, refine, or change a theory unless they lead to confirmation and refinement or revision of the logical structure of the theory. Therefore, in order to develop the theory, great importance disclosure of internal contradictions and their resolution. Contradictions in the theory are most clearly revealed when they arise in the form of various paradoxes. Thus, the analysis of theoretical paradoxes is not an end in itself, but is only a means for clarifying the true content of the theory, clarifying its individual provisions and finding ways for its further development. Many contradictions arise in the theory of relativity because of the standard way of presenting it according to the classical model that was given by Einstein. Since Einstein's first work, the theory of relativity has grown big amount new performances. As a result of numerous applications, the main content of the theory became clear. It turned out that some of the ideas that were considered basic at the time of the birth of the theory turned out to be in reality only auxiliary means used to build the theory. It also turned out that the theory can be built on the basis of various postulates. It turned out, in other words, that Einstein's postulates cannot be identified with the very content of the theory of relativity.

A deep analysis of the content of the theory of relativity is important right now, when a new stage is outlined for a sharp break in theoretical concepts in connection with the penetration into themselves elementary particles and the discovery of fundamentally new physical processes in space, occurring in radio galaxies and superstars or quasars.

We will see that the analysis of the question of the limiting speed of signals in the theory of relativity will lead us to a revision of the content, the so-called principle of causality and to general conclusion about the fundamental possibility of the existence of particles having negative and even imaginary own masses. But if such particles really exist in nature, then their discovery will lead to a radical restructuring of the entire existing physical picture of the world. And this, in turn, will lead to new discoveries that increase the power of man over nature.

1. Postulates of the special theory of relativity (SRT)

Newton's classical mechanics perfectly describes the motion of macrobodies moving at low speeds (x<< c). В нерелятивистской физике принималось как очевидный факт существование единого мирового времени t, одинакового во всех системах отсчета. В основе классической механики лежит механический принцип относительности (или принцип относительности Галилея): законы динамики одинаковы во всех инерциальных системах отсчета. Этот принцип означает, что законы динамики инвариантны (т. е. неизменны) относительно преобразований Галилея, которые позволяют вычислить координаты движущегося тела в одной инерциальной системе (K), если заданы координаты этого тела в другой инерциальной системе (K"). В частном случае, когда система K" движется со скоростью х вдоль положительного направления оси x системы K (рис. 1.1), преобразования Галилея имеют вид:

x=x"+хt, y=y", z=z", t=t".

It is assumed that at the initial moment the coordinate axes of both systems coincide.

Figure 1.1 Two inertial reference frames K and K"

From Galileo's transformations follows the classical law of the transformation of velocities in the transition from one frame of reference to another:

ux \u003d u "x + x, uy \u003d u" y, uz \u003d u "z.

The accelerations of the body in all inertial frames are the same:

Consequently, the equation of motion of classical mechanics (Newton's second law) does not change its form when moving from one inertial frame to another.

By the end of the 19th century, experimental facts began to accumulate that came into conflict with the laws of classical mechanics. Great difficulties arose when trying to apply Newton's mechanics to explain the propagation of light. The assumption that light propagates in a special medium - ether - has been refuted by numerous experiments. The American physicist A. Michelson, first independently in 1881, and then together with E. Morley (also an American) in 1887, tried to detect the motion of the Earth relative to the ether (“aether wind”) using interference experience. A simplified diagram of the Michelson-Morley experiment is shown in fig. 1.2.

Figure 1.2 Simplified diagram of the Michelson-Morley interference experiment. - Earth's orbital speed

In this experiment, one of the arms of the Michelson interferometer was set parallel to the direction of the Earth's orbital velocity (x=30 km/s). Then the instrument was rotated by 90°, and the second arm turned out to be oriented in the direction of the orbital velocity. Calculations showed that if a fixed ether existed, then when turning the device, the interference fringes should have shifted by a distance proportional to (х/c)2. The Michelson-Morley experiment, which was subsequently repeated many times with increasing accuracy, gave a negative result. An analysis of the results of the Michelson-Morley experiment and a number of other experiments led to the conclusion that the concept of the ether as a medium in which light waves propagate is erroneous. Therefore, there is no chosen (absolute) frame of reference for light. The motion of the Earth in orbit does not affect the optical phenomena on Earth.

Maxwell's theory played an exceptional role in the development of ideas about space and time. By the beginning of the 20th century, this theory had become generally accepted. Electromagnetic waves predicted by Maxwell's theory, propagating at a finite speed, have already found practical application - in 1895, AS Popov invented radio. But from Maxwell's theory it follows that the speed of propagation of electromagnetic waves in any inertial frame of reference has the same value, equal to the speed of light in vacuum. This means that the equations describing the propagation of electromagnetic waves are not invariant under the Galilean transformations. If an electromagnetic wave (in particular, light) propagates in the reference frame K "(Fig. 1.1) in the positive direction of the x-axis", then in the frame K, the light should, according to the Galilean kinematics, propagate at a speed c + x, and not c.

So, at the turn of the 19th and 20th centuries, physics was going through a deep crisis. The way out was found by Einstein at the cost of abandoning the classical concepts of space and time. The most important step along this path was the revision of the concept of absolute time used in classical physics. Classical ideas, seemingly clear and obvious, in reality turned out to be untenable. Many concepts and quantities, which in non-relativistic physics were considered absolute, i.e., independent of the frame of reference, were transferred to the category of relative in Einstein's theory of relativity.

Since all physical phenomena occur in space and time, the new concept of space-time laws could not but affect the whole of physics as a result.

The special theory of relativity is based on two principles or postulates formulated by Einstein in 1905.

The principle of relativity: all laws of nature are invariant with respect to the transition from one inertial frame of reference to another. This means that in all inertial frames the physical laws (not just mechanical ones) have the same form. Thus, the principle of relativity of classical mechanics is generalized to all processes of nature, including electromagnetic ones. This generalized principle is called Einstein's principle of relativity.

The principle of the constancy of the speed of light: the speed of light in vacuum does not depend on the speed of the light source or the observer and is the same in all inertial frames of reference. The speed of light in SRT occupies a special position. This is the maximum speed of transmission of interactions and signals from one point in space to another.

These principles should be regarded as a generalization of the totality of experimental facts. The consequences of the theory created on the basis of these principles were confirmed by endless experimental tests. SRT made it possible to solve all the problems of “pre-Einsteinian” physics and to explain the “contradictory” results of experiments known by that time in the field of electrodynamics and optics. Subsequently, SRT was supported by experimental data obtained in the study of the motion of fast particles in accelerators, atomic processes, nuclear reactions, etc.

The postulates of SRT are in clear contradiction with classical concepts. Consider the following mental experiment: at the time t=0, when the coordinate axes of two inertial systems K and K" coincide, a short-term flash of light occurred at the common origin. During the time t, the systems will shift relative to each other by a distance xt, and the spherical wave front will each system will have a radius ct (Fig.1. 3), since the systems are equal and in each of them the speed of light is equal to c.

Figure 1.3 Seeming contradiction of SRT postulates

From the point of view of an observer in frame K, the center of the sphere is at point O, and from the point of view of an observer in frame K" it will be at point O". Therefore, the center of the spherical front is simultaneously located at two different points.

The reason for the resulting misunderstanding lies not in the contradiction between the two principles of SRT, but in the assumption that the position of the fronts of spherical waves for both systems refers to the same moment in time. This assumption is contained in the Galilean transformation formulas, according to which time flows in the same way in both systems: t=t". Therefore, Einstein's postulates are in conflict not with each other, but with the Galilean transformation formulas. Therefore, SRT proposed other transformation formulas to replace the Galilean transformations during the transition from one inertial frame to another - the so-called Lorentz transformations, which, at speeds close to the speed of light, make it possible to explain all relativistic effects, and at low speeds (x<< c) переходят в формулы преобразования Галилея. Таким образом, новая теория (СТО) не отвергла старую классическую механику Ньютона, а только уточнила пределы ее применимости. Такая взаимосвязь между старой и новой, более общей теорией, включающей старую теорию как предельный случай, носит название принципа соответствия .

2. Relativity of time intervals

When performing any physical measurements, the spatio-temporal relationships between events play an exceptional role. In SRT, an event is defined as a physical phenomenon occurring at some point in space at some point in time in a chosen frame of reference. Thus, in order to fully characterize an event, it is required not only to find out its physical content, but also to determine its place and time. To do this, it is necessary to use procedures for measuring distances and time intervals. Einstein showed that these procedures needed a strict definition.

In order to measure the time interval between two events (for example, the beginning and end of a process) occurring at the same point in space in the chosen reference frame, it is sufficient to have a reference clock. Clocks based on the use of natural vibrations of ammonia molecules (molecular clocks) or cesium atoms (atomic clocks) currently have the highest accuracy. The measurement of the time interval is based on the concept of simultaneity: the duration of a process is determined by comparison with the time interval separating the clock reading, simultaneous with the end of the process, from the reading of the same clock, simultaneous with the beginning of the process. If both events occur at different points of the reference system, then to measure the time intervals between them at these points, it is necessary to have synchronized clocks.

Einstein's definition of the clock synchronization procedure is based on the independence of the speed of light in vacuum from the direction of propagation. Let a short light pulse be sent from point A at time point A (Fig. 2.1). Let the time of arrival of the pulse at B and its reflection back at the clock B is t". Finally, let the reflected signal return to A at the time of the clock A. Then, by definition, the clocks in A and B are synchronous if t" = () / 2 .

Figure 2.1 Clock synchronization in the workshop

The existence of a unified world time independent of the frame of reference, which was taken as an obvious fact in classical physics, is equivalent to the implicit assumption that clocks can be synchronized using a signal propagating at an infinitely high speed.

So, synchronized clocks can be placed at different points of the chosen reference system. Now we can define the concept of simultaneity of events occurring at spatially separated points: these events are simultaneous if synchronized clocks show the same time.

Consider now a second inertial frame K" that moves at some speed x in the positive direction of the x-axis of the frame K. At different points of this new frame of reference, you can also place clocks and synchronize them with each other using the procedure described above. Now the time interval between two events can be measured both by the clock in the K system and by the clock in the K system. Will these intervals be the same? The answer to this question must be in agreement with the postulates of SRT.

Let both events in the system K" occur at the same point and the time interval between them is equal to the clock of the system K". This period of time is called proper time. What will be the time interval between the same events, if measured by the clock system K?

To answer this question, consider the following thought experiment. A flash lamp B is located at one end of a solid rod of some length, and a reflecting mirror M is located at the other end. The rod is located motionless in the K "system and is oriented parallel to the y" axis (Fig. 2.2). Event 1 - lamp flash, event 2 - return of a short light pulse to the lamp.

Figure 2.2.

Relativity of time intervals. The times of occurrence of events in the system K" are fixed by the same clock C, and in the system K - by two synchronized spatially separated clocks u. The system K" moves at a speed x in the positive direction of the x axis of the system K

In frame K "both events under consideration occur at the same point. The time interval between them (proper time) is equal. From the point of view of an observer located in frame K, the light pulse moves between the mirrors in a zigzag manner and travels a path 2L equal to

where φ is the time interval between the departure of a light pulse and its return, measured by synchronized clocks and located at different points of the K system. But according to the second postulate of SRT, the light pulse moved in the K system with the same speed c as in the K system " Therefore, f=2L/c.

From these relations, one can find the connection between φ and:

Thus, the time interval between two events depends on the frame of reference, that is, it is relative. Proper time is always less than the time interval between the same events, measured in any other frame of reference. This effect is called relativistic time dilation. Time dilation is a consequence of the invariance of the speed of light.

The effect of time dilation is mutual, in accordance with the postulate of the equality of inertial systems K and K": for any observer in K or K" the clock associated with the system moving relative to the observer runs slower. This SRT conclusion finds direct experimental confirmation. For example, when studying cosmic rays, m-mesons were found in their composition - elementary particles with a mass approximately 200 times greater than the mass of an electron. These particles are unstable, their average proper lifetime is the same. But in cosmic rays m-mesons move at a speed close to the speed of light. Without taking into account the relativistic effect of time dilation, they would, on average, fly in the atmosphere a path equal to c ? According to SRT, the average lifetime of mesons according to the clock of an earthly observer is

Because it's close to unity. Therefore, the average path traveled by the meson to the system turns out to be much more than 660 m.

The so-called “twin paradox” is associated with the relativistic effect of time dilation. It is assumed that one of the twins remains on Earth, and the second goes on a long space journey at subluminal speed. From the point of view of an earthly observer, time passes more slowly in a spacecraft, and when the astronaut returns to Earth, he will be much younger than his twin brother, who remained on Earth. The paradox lies in the fact that the second of the twins, setting off on a space journey, can make a similar conclusion. For him, time passes more slowly on Earth, and he can expect that upon his return from a long journey to Earth, he will find that his twin brother, who remained on Earth, is much younger than he is.

To resolve the "twin paradox", one should take into account the inequality of reference systems in which both twin brothers are located. The first of them, which remained on Earth, is always in the inertial frame of reference, while the frame of reference associated with the spacecraft is fundamentally non-inertial. The spacecraft experiences accelerations during acceleration during launch, when changing direction at the far point of the trajectory, and during deceleration before landing on Earth. Therefore, the astronaut brother's conclusion is wrong. SRT predicts that when he returns to Earth, he will indeed be younger than his brother, who remained on Earth.

The effects of time dilation are negligible if the speed of the spacecraft is much less than the speed of light c. Nevertheless, it was possible to obtain direct confirmation of this effect in experiments with macroscopic clocks. The most accurate clocks are atomic clocks operating on a beam of cesium atoms. This clock ticks 9192631770 times per second. American physicists in 1971 compared two such clocks, one of which was in flight around the Earth on a conventional jet liner, while the other remained on Earth at the US Naval Observatory. In accordance with SRT predictions, the clocks traveling on the liners should have lagged behind the clocks on the Earth by (184±23)·10-9 s. The observed lag was (203±10)·10-9 s, i.e., within the limits of measurement errors. A few years later, the experiment was repeated and gave a result consistent with SRT with an accuracy of 1%.

At present, it is already necessary to take into account the relativistic effect of slowing down the clock when transporting atomic clocks over long distances.

3. Relativity of distances

Let a solid rod be at rest in a frame of reference K" moving at a speed x with respect to the frame of reference K (Fig. 3.1). The rod is oriented parallel to the x-axis". Its length, measured using a standard ruler in the K system, is equal to. It is called its own length. What will be the length of this rod, measured by an observer in the K system? To answer this question, it is necessary to define the procedure for measuring the length of a moving rod.

Under the length of the rod in the system K, relative to which the rod moves, is understood the distance between the coordinates of the ends of the rod, fixed simultaneously by the clock of this system. If the speed of the system K" relative to K is known, then measuring the length of a moving rod can be reduced to measuring time: the length of a rod moving at a speed x is equal to the product, where is the time interval in hours in the system K between the passage of the beginning of the rod and its end past some fixed points (for example, points A) in the system K (Fig. 3.1).Since in the system K both events (passage of the beginning and end of the rod past the fixed point A) occur at the same point, the time interval in the system K is proper time. the length of the moving rod is

Figure 3.1 Measuring the length of a moving rod

Let's find the connection between and. From the point of view of an observer in the system K", point A, belonging to the system K, moves along the fixed rod to the left with a speed x, so we can write =xf,

where φ is the time interval between the moments of passage of point A past the ends of the rod, measured by synchronized clocks in K". Using the relationship between the time intervals φ and

Thus, the length of the rod depends on the reference frame in which it is measured, that is, it is a relative value. The length of the rod is greatest in the frame of reference in which the rod is at rest. Bodies moving relative to the observer contract in the direction of their movement. This relativistic effect is called the Lorentz length contraction.

Distance is not an absolute value, it depends on the speed of the body relative to a given frame of reference. The reduction in length is not associated with any processes occurring in the bodies themselves. Lorentz contraction characterizes the change in the size of a moving body in the direction of its movement. If the rod in Fig. 3.1 is placed perpendicular to the x-axis along which the system K" moves, then the length of the rod turns out to be the same for observers in both systems K and K". This statement is in accordance with the postulate of the equality of all inertial systems. For proof, consider the following thought experiment. Let us place two rigid rods in the systems K and K" along the y and y" axes. The rods have the same proper lengths, measured by observers fixed with respect to each of the rods in K and K", and one of the ends of each rod coincides with the origin O or O". At some point, the rods are next to each other, and it is possible to compare them directly: the end of each rod can make a mark on the other rod. If these marks did not coincide with the ends of the rods, then one of them would be longer than the other from the point of view of both reference systems. This would be contrary to the principle of relativity.

It should be noted that at low speeds (x<< c) формулы СТО переходят в классические соотношения: и. Таким образом, классические представления, лежащие в основе механики Ньютона и сформировавшиеся на основе многовекового опыта наблюдения над медленными движениями, в специальной теории относительности соответствуют предельному переходу при в=х/c>0. This is the principle of correspondence.

4. Lorentz transformations

The classical transformations of Galileo are incompatible with the postulates of SRT and, therefore, must be replaced. These new transformations should establish a relationship between the coordinates (x, y, z) and the time t of an event observed in the reference frame K, and the coordinates (x", y", z") and the time t" of the same event observed in reference system K".

Kinematic formulas for the transformation of coordinates and time in SRT are called Lorentz transformations. They were proposed in 1904 even before the advent of SRT as transformations with respect to which the equations of electrodynamics are invariant. For the case when the system K" is moving relative to K with a speed x along the x axis, the Lorentz transformations have the form:

A number of consequences follow from the Lorentz transformations. In particular, the relativistic effect of time dilation and the Lorentz contraction of length follow from them. Let, for example, at some point x "of the system K" a process of duration (proper time) occurs, where and are the clock readings in the system K "at the beginning and end of the process. The duration φ of this process in the system K will be equal to

Similarly, it can be shown that the relativistic length contraction follows from the Lorentz transformations. One of the most important consequences of the Lorentz transformations is the conclusion about the relativity of simultaneity. Let, for example, at two different points of the reference frame K" () two events occur simultaneously from the point of view of the observer in K" (). According to the Lorentz transformations, an observer in frame K will have

Consequently, in the system K these events, while remaining spatially separated, turn out to be non-simultaneous. Moreover, the sign of the difference is determined by the sign of the expression, so in some frames of reference the first event may precede the second, while in other frames of reference, on the contrary, the second event precedes the first. This SRT conclusion does not apply to events connected by cause-and-effect relationships, when one of the events is a physical consequence of the other. It can be shown that the principle of causality is not violated in SRT, and the sequence of causal events is the same in all inertial frames of reference.

The relativity of the simultaneity of spatially separated events can be illustrated by the following example.

Let a long rigid rod be fixed in the reference frame K" along the x" axis. At the center of the rod is a flash lamp B, and at its ends are two synchronized clocks (Fig. 4.1 (a)), the system K" moves along the x axis of the system K with speed x. At some point in time, the lamp sends short light pulses in the direction of the ends rod. Due to the equality of both directions, the light in the system K "will reach the ends of the rod at the same time, and the clock fixed at the ends of the rod will show the same time t". Relative to the system K, the ends of the rod move with speed x so that one end moves towards the light pulse, and the other end of the world has to catch up.Since the speeds of propagation of light pulses in both directions are the same and equal to c, then, from the point of view of an observer in frame K, the light will reach the left end of the rod earlier than the right (Fig. 4.1 (b)).

Figure 4.1.

The relativity of simultaneity. The light pulse reaches the ends of the solid rod simultaneously in the reference frame K" (a) and not simultaneously in the reference frame K (b)

Lorentz transformations express the relative nature of time intervals and distances. However, in SRT, along with the assertion of the relative nature of space and time, an important role is played by the establishment of invariant physical quantities that do not change when moving from one frame of reference to another. One of these quantities is the speed of light in vacuum c, which becomes absolute in SRT. Another important invariant quantity, reflecting the absolute nature of space-time connections, is the interval between events.

The space-time interval is defined in SRT by the following relation:

where is the time interval between events in a certain frame of reference, and is the distance between the points at which the events under consideration occur in the same frame of reference. In a particular case, when one of the events occurs at the origin of the reference system at a time, and the second - at a point with coordinates x, y, z at a time t, the space-time interval between these events is written as

Using the Lorentz transformations, one can prove that the space-time interval between two events does not change when moving from one inertial frame to another. The invariance of the interval means that, despite the relativity of distances and time intervals, the course of physical processes is of an objective nature and does not depend on the frame of reference.

If one of the events is a flash of light at the origin of the reference frame at t=0, and the second is the arrival of a light front at a point with coordinates x, y, z at time t (Fig. 1.3), then

and hence the interval for this pair of events is s=0. In another frame of reference, the coordinates and time of the second event will be different, but in this frame, the space-time interval s "will be equal to zero, since

For any two events linked by a light signal, the interval is zero.

From the Lorentz transformations for coordinates and time, one can obtain the relativistic law of addition of velocities. Let, for example, in the reference frame K" a particle move along the x" axis with a velocity

The particle velocity components u "x and u" z are equal to zero. The speed of this particle in the frame K will be equal to

Using the operation of differentiation from the Lorentz transformation formulas, you can find:

These relations express the relativistic law of velocity addition for the case when the particle moves parallel to the relative velocity of the reference frames K and K".

At x<< c релятивистские формулы переходят в формулы классической механики: ux=u"x+х, uy=0, uz=0.

If in the system K "along the x" axis a light pulse propagates with a speed u "x \u003d c, then for the speed ux of the pulse in the system K we get

Thus, in the reference frame K, the light pulse also propagates along the x-axis with a speed c, which is consistent with the postulate of the invariance of the speed of light.

5. Paradoxes of SRT

5.1 Einstein's train paradox

Let three people (A, O and B) ride on a train moving at a speed close to one. A rides at the head of the train, O in the middle, and B in the tail (Fig. 1).

Figure 1. Who gave the signal first - traveler A or traveler B?

On the ground near the railway track stands a fourth person, O. "At the very moment when O passes O," the flash lamp signals from A and B reach O and O. Who first sent the signal? Using only the fact that the speed of light is finite and does not depend on the speed of its source.

Observers A and B are at rest with respect to observer O. In addition, they are at equal distances from O, which the latter can slowly check using his ruler. Therefore, the signals from A and B take the same time to reach O. These signals are received by the observer O at the same time. Therefore, observer O concludes that observers A and B sent their signals at the same moment: .

Observer O, standing next to the railroad tracks, draws completely different conclusions. His reasoning is as follows: “Two flashes came to me when the middle of the train passed me. Therefore, both of these flashes must have been emitted before the middle of the train caught up with me And until that moment, observer A was closer to me than observer B. Therefore, the light from B had to travel a longer distance to me and spend more time than the light from A. But both signals arrived at me at the same time. Therefore, observer B should have sent his signal before observer A" (<0). Итак, наблюдатель О", стоящий рядом с железнодорожными путями, делает заключение, что сначала послал свои сигнал В, а потом уже А, тогда как едущий на поезде наблюдатель О заключает, что оба наблюдателя, А и В, послали сигналы в одно и то же время.

What is the time interval between the sending of signals by observers A and B? In the unprimed reference frame (train) these signals were sent at the same time, so. The distance between the points of sending signals is, where L is the length of the train. Therefore, in a primed frame of reference (moving to the right with respect to an unprimed frame, that is, a train, as is usually the case when using primed and unprimed notation), the time interval between the sending of signals A and B can be found using the Lorentz transformation formulas:

The minus sign shows that observer B, who is on the positive part of the x / axis, sent his signal earlier in rocket time (more negative time!) than observer A.

5.2 Clock paradox

Let clock A be at point I in a fixed inertial frame of reference, and the same

Figure 2

with them, clock B, which was also at point I at the initial moment, moves towards point II with a speed v. Then, having passed the path I to point II, clock B slows down and, acquiring the opposite speed - . return to point I (Fig. 2).

If the time required to change the speed of clock B to reverse is sufficiently small compared to the time of rectilinear and uniform movement from point I to point II, then the time measured by clock A and the time measured by clock B can be calculated according to

formulas

where is a possible small correction for the time of the accelerated movement of clock B. Consequently, clock B, having returned to point I, will actually lag behind clock A by a time

Since the distance can be made arbitrarily large, the correction may not be taken into account at all. The peculiarity of this kinematic consequence of the Lorentz transformations is that here the lag of the moving clock is a very real effect.

In reality, all processes associated with the system should lag behind the processes running in the system. Including the biological processes of organisms that are together with clock B should lag behind. Physiological processes in the human body traveling in the system should slow down, as a result of which the organism that was in the system at the time it returns to point I will be less aged than the body remaining in the system.

It seems paradoxical here that some of the clocks actually lag behind the others. After all, this seems to contradict the very principle of relativity, since, according to the latter, any of the systems can be considered motionless. But then it seems that only depending on our choice, any of the clocks A and B can actually become lagging behind. But the latter is clearly absurd, since clocks B actually lag behind clocks A.

The fallacy of the last reasoning is that the systems and are not physically equal, since the system is inertial all the time, while the system is a certain period of time when its speed is changed to the opposite, non-inertial one. Consequently, the second of the formulas (1) for the system is incorrect, since during the acceleration the course of the remote

hours can vary greatly due to the inertial gravitational field.

However, even this perfectly correct explanation is quite striking. Indeed, over a long period of time, both systems move relative to each other in a straight line and uniformly. Therefore, from the point of view of the system, clock A, which is in, lags behind (and does not go ahead) in full accordance with formula (1). And only for a short period of time, when inertial forces act in the system, clock A quickly goes forward for a period of time twice as long as that calculated by formula (2). Moreover, the greater the acceleration experienced by the system, the faster time runs on clock A.

The essence of the obtained conclusions can be visually explained on the Minkowski plane (Fig. 3).

Figure 3

The segment Оb in fig. 3, a depicts a clock A at rest, a broken line Oab - a moving clock B. At point a, forces act that accelerate the clock system B and change its speed to the opposite. The points placed on the Ob axis separate the unit time intervals in the fixed system associated with the clock A.

The points on the polyline Oab mark equal unit time intervals, measured by clocks B located in the system. It can be seen from the figure that the number of unit segments that fit on the line Ob is greater than the number of the same, but related to the system of segments, that fit on the broken line Oab. Therefore, clock B is behind clock A.

According to the figure, the “stationary” clock A also lags behind the clock B up to the moment represented by the point a. Simultaneous with this moment is the moment, but until then, clock B is still moving with speed. But after a short period of time required to slow down the clock B and give it speed - , the same moment A will practically remain on the clock B, but the moment will become the moment in the system simultaneously with it, i.e., almost instantly, the time of the system will, as it were, jump to end interval.

This time jump is not, however, a really observable effect. Indeed, if light signals are regularly sent from the system, at unit intervals, then they will be received by the system quite regularly, at first more rarely, and then, after changing the speed to the reverse, more often. No break in the readings of clock A in the system will be observed, as can be seen from Fig. 3 b,

Thus, the “paradox of the clock” is also only a consequence of the pseudo-Euclidean geometry of the four-dimensional space-time manifold, unusual for ordinary ideas about space and time.

5.3 Transport paradox

The conveyor is an endless belt of flexible material, which moves along the guides with the help of two pulleys mounted on the frame AB (Fig. 4). Let us put this conveyor into action in such a way that the speed of the belt approaches the speed of light. Then the length of its horizontal parts will decrease by a factor of K, although the distance between the centers of the pulleys will remain unchanged. If at first the tape sagged freely, it will stretch. A

Figure 4

if there is not enough length, the tape material will be stretched. In this case, corresponding stresses will arise in it, which, in principle, could be detected by a dynamometer and even lead to a break. Conversely, the frame AB, under the influence of the belt tension, undergoes compression deformation, which can also be detected by a dynamometer.

This is how the phenomena in the Stanina system will be described. If, however, the reference system is connected not with the bed, but with the tape, then the tape will have to be considered at rest, and the bed - moving at high speed. Then it is not the tape that should be reduced, but the bed, the result of which will be no longer a tight tension, but a free sagging of the tape.

But this conclusion clearly contradicts the principle of relativity: arguments concerning the same phenomenon in two different frames of reference lead to mutually exclusive results. Having made an appropriate experiment, it will be possible to refute one of them and confirm the other. And this will allow you to determine which of the two objects (tape or bed) is in the "true", and which is only in the "apparent" movement.

Thus, we are faced with a paradox: in this particular case, the application of the theory of relativity leads to the denial of one of its own foundations - Einstein's principle of relativity.

True, this paradox could be dismissed: after all, sections of the tape sliding along the pulleys move curvilinearly, and the special theory of relativity requires that all frames of reference be inertial.

But this is not an answer to the paradox, but only an attempt to evade its real analysis (like the following “explanation”: “Of course, it will not be possible to obtain a perpetual motion machine by connecting an electric motor to a dynamo machine with a belt and wires, because the belt will definitely fray” ).

It can, of course, be assumed that the curvilinear sections of the tape do not shorten, but lengthen just enough to compensate for the main effect. But it is enough to increase the distance between the axes of the pulleys, for example, 10 times, so that the compensation is violated: the main effect of shortening the straight sections increases tenfold, while the supposed masking effect of the curved parts remains the same.

The real explanation of the paradox is the impossibility of relating the inertial frame of reference to the entire tape. And if the system is connected with only one of its sections, it is not inertial: after all, each section of the tape (you can imagine it painted in a special color) periodically changes the direction of its movement to the opposite.

You can, of course, use an inertial frame of reference, which is constantly moving relative to the bed in the same direction and at the same speed as the lower part of the tape. In this system, the bed moves at a speed to the left, the lower part of the belt is, of course, stationary, and the upper part moves in the same direction as the bed, but with a relativistically doubled speed.

In this case, the frame is shortened by K times, the lower part of the tape retains its natural length, but the upper part is reduced much more than by K times (approximately times). As a result, the total length of the belt is reduced so much that, despite the shortening of the bed, it is stretched rather than sagging (the quantitative side of the matter is discussed in Appendix E).

As expected, consideration in any truly inertial reference frame leads to the same result (tape tension). Thus, the paradox is completely removed: in this experiment, the frame and the tape are physically unequal, since, unlike the frame, the tape cannot be considered to be at rest in any inertial frame (because its parts move relative to each other). For this reason, the tape is shortened in comparison with the bed, and not vice versa.

Let us consider one more argument that could be put forward in support of the paradox by opponents of the theory of relativity. Exactly half of the conveyor belt that is not yet working is painted black. Let us choose a moment in time when the colored part of the tape is at the bottom, and the uncolored part is at the top (Fig. 5).

Figure 5

In the "Bed" system, both parts of the tape, being reduced by the same number of times, will always remain equal in length, as shown in Fig. 5.

In contrast to this, in the inertial system "Lower section of the belt", the reduction in the total length of the belt occurs only due to its upper part, while the lower part of the belt, compared with the bed, even lengthens by K times. Therefore, some part of the painted "half" will inevitably go up, so that the location of the tape on the pulleys will correspond to Fig. 5, and fig. 6.

Figure 6

It would seem that it is enough to look at a working transporter in order to establish which of the two contradictory conclusions corresponds to reality, and thereby highlight the predominant system!

But that's not the case at all. To establish which of the two figures 5 or 6) is confirmed by experience, it is necessary to determine whether both boundaries of the colored "half" of the tape pass simultaneously through the extreme right and extreme left positions. But in each frame of reference the concept of simultaneity is its own! Therefore, there is nothing impossible in the fact that in one reference frame the picture shown in Fig. 5 will be “observed”, and in the other - shown in Fig. 6.

5.4 Paradox of the wheel

Imagine a large wheel that can rotate relative to the "Star" system (Fig. 7).

Figure 7

At first, the wheel is stationary, and then it is brought into such a rapid rotation that the linear speed of its edges approaches the speed of light. In this case, the sections of the rim AB, BC, etc. are reduced by K times, while the radial "spokes" OA, OB, OS, etc. retain their length (after all, only the longitudinal dimensions experience relativistic shortening, i.e., the dimensions in the direction of travel).

It turns out that with a constant diameter, the circumference will decrease by a factor of K. If K=10, then the circle will become approximately three times shorter than its diameter - the straight line will no longer serve as the shortest distance between points!

How will the theory of relativity cope with such a geometrical inconsistency?

To better understand the details of the physical processes involved in rapid rotation, let us first imagine that we are rapidly cooling a stationary wheel. Let us assume that its rim is made of a material with a large coefficient of thermal expansion and contraction, while the length of the spokes hardly changes with temperature. Then, as a result of cooling, mechanical stresses will arise in the wheel: the arc rods, trying to contract, will press on the spokes.

Depending on the mechanical strength and elastic properties, after cooling the wheel, either its rim will remain in a stretched state, or the spokes will be shortened (or rather, both effects will always take place to some extent). In any case, there will be no shortening of the circle with a constant diameter. Such a stressed state of the wheel is mechanically unstable: the slightest deviation to the side - and it will take the form of a spherical segment (Fig. 8).

Figure 8

Then indeed the circumference of the rim will be less than, where r is the length of the curved spoke. However, bending of the wheel can be prevented by giving it sufficient bending stiffness or by placing it between two strong plates.

Something similar happens when the wheel, which is initially stationary, is brought into rapid rotation: its rim tends to shorten, and the spokes to maintain a constant length. Which of these tendencies prevails depends entirely on the mechanical properties of the rim and spokes; but there will be no shortening of the rim without a proportional shortening of the spokes (unless the wheel takes the form of a spherical segment). Obviously, from a fundamental point of view, nothing will change even if the spoked wheel is replaced by a solid disk.

So, no irresolvable contradiction with geometry arises. You just need to keep in mind that in the theory of relativity, even when considering purely kinematic issues, it is not always permissible to use the abstraction of an absolutely non-deformable body (however, the idea of an absolutely rigid rod is unacceptable even because it could be used to instantly transmit a signal: due to the invariance of the length, both its ends would be displaced simultaneously).

However, suppose now that the wheel is manufactured (for example, cast) inside a rapidly revolving workshop. This means that it is precisely in the state of rapid rotation relative to the Zvezda system that it is free from internal stresses. If you stop it, the rim will tend to lengthen, and the spokes will tend to maintain their length. In this case, stresses of an opposite nature arise in comparison with the previous case: in particular, the wheel will not show any tendency to become a spherical segment; on the contrary, it will form folds along the edges.

Let us now consider the same phenomena in the "Revolving Workshop" system. Then we will have to assume that the wheel cast in this workshop, which was just discussed, was at first at rest, and then came into rapid rotation. But at the same time, internal stresses arose in it, leading to the formation of marginal folds, and not a spherical segment. There is a sharp discrepancy with the result of the same experiment in the "Stars" system, which makes it possible to distinguish it from the "Revolving Workshop" system.

This time, the ability to distinguish one frame of reference from another is not imaginary, but real. However, it does not contradict the private theory of relativity, because only one of these systems is inertial. In this case, the non-inertiality of a reference frame rotating relative to fixed stars could be even more easily detected from other, non-relativistic effects (for example, centrifugal).

In the so-called general theory of relativity, Einstein made an attempt to formulate the principle of relativity in such a way that it covers not only inertial, but also non-inertial systems. However, as Acad. V. A. Fok, this could only be achieved by emasculating all of its physical content from the very principle of relativity. In reality (as the existence of centrifugal forces already shows), there is no physically meaningful "general principle of relativity", and the so-called "general theory of relativity" is in fact not an extension of the particular one, but the theory of universal gravitation.

This does not mean, of course, that it is impossible to use rotating and generally non-inertial frames of reference. It is only necessary to remember that they are not equal with inertial ones, and the physical phenomena in them are subject to other laws.

A more detailed study shows that the originality of non-inertial systems extends not only to physical, but even to geometric relationships. When an experimenter using a rotating frame of reference measures the length of a circle, he places the meter in the direction of motion. Therefore, from the point of view of a stationary observer, he receives an exaggerated value of the circumference, because he uses an abbreviated meter. When a rotating observer measures the diameter, he places his meter perpendicular to the direction of motion and therefore obtains a result with which the stationary observer will also unreservedly agree. But with the correct length of the diameter and the exaggerated circumference, their ratio can no longer be equal.

5.5 Pole and barn paradox

Let's take a pole 20 m long and move it in the direction of its length with such a speed that in the laboratory reference system it turns out to be only 10 m long. Then at some moment this pole can be completely hidden in a shed, which is also 10 m long. the same in the pole runner's frame of reference. For him, half the length is reduced to a barn. How can you hide a 20-meter pole in a 5-meter barn?!

The resolution of this "paradox" is that, in the runner's frame of reference, the front end of the pole leaves the barn before the rear end of the pole enters the barn. Therefore, from the point of view of the runner, the pole is not at all in the barn at all at any time. The sequence of events can be illustrated in more detail by two space-time diagrams (Fig. 9 and 10),

Figure 9. Spatio-temporal Figure 10 Spatio-temporal diagram in the frame of the barn diagram in the frame of reference of the runner

numerical values of lengths and moments of time at which can be obtained from the following considerations. Since the factor describing the Lorentz contraction is equal to 2 by the condition of the problem, then

Therefore, from the identity

follows that

Hence the relative speed of the two frames of reference is

To find the numerical values shown in Figures 9 and 10, it is enough to use these data, as well as the fact that the length of the pole in the runner's reference system is 20 m, and in the laboratory system 10 m.

The main idea of SRT

The paradox (twin paradox) says that a "stationary" observer perceives the processes of moving objects as slowing down. In accordance with the same theory, inertial frames of reference (frames in which the motion of free bodies occurs in a straight line and uniformly, or they are at rest) are equal relative to each other.

The twin paradox in brief

Taking into account the second postulate, an assumption about inconsistency arises. To solve this problem visually, it was proposed to consider the situation with two twin brothers. One (conditionally - a traveler) is sent on a space flight, and the other (a homebody) is left on planet Earth.

The formulation of the twin paradox under such conditions usually sounds like this: according to the stay-at-home, the time on the clock that the traveler has is moving more slowly, which means that when he returns, his (the traveler's) clock will lag behind. The traveler, on the contrary, sees that the Earth is moving relative to him (on which there is a homebody with his watch), and, from his point of view, it is his brother who will pass the time more slowly.

In reality, both brothers are on an equal footing, which means that when they are together, the time on their clocks will be the same. At the same time, according to the theory of relativity, it is the brother-traveler's watch that should fall behind. Such a violation of the apparent symmetry was considered as an inconsistency in the provisions of the theory.

Twin paradox from Einstein's theory of relativity

In 1905, Albert Einstein derived a theorem that states that when a pair of clocks synchronized with each other is at point A, one of them can be moved along a curved closed trajectory at a constant speed until they again reach point A (and on this will be spent, for example, t seconds), but at the time of arrival they will show less time than the clock that remained motionless.

Six years later, Paul Langevin gave this theory the status of a paradox. "Wrapped" in a visual story, it soon gained popularity even among people far from science. According to Langevin himself, the inconsistencies in the theory were due to the fact that, returning to Earth, the traveler moved at an accelerated rate.

Two years later, Max von Laue put forward a version that it is not the moments of acceleration of an object that are significant, but the fact that it falls into a different inertial frame of reference when it finds itself on Earth.

Finally, in 1918, Einstein himself was able to explain the paradox of two twins through the influence of the gravitational field on the passage of time.

Explanation of the paradox

The twin paradox has a rather simple explanation: the initial assumption of equality between the two frames of reference is incorrect. The traveler did not stay in the inertial frame of reference all the time (the same applies to the story with the clock).

As a consequence, many felt that special relativity could not be used to correctly formulate the twin paradox, otherwise incompatible predictions would result.

Everything was resolved when it was created. It gave an exact solution for the existing problem and was able to confirm that out of a pair of synchronized clocks, it was those that were in motion that would lag behind. So the initially paradoxical task received the status of an ordinary one.

controversial points

There are assumptions that the moment of acceleration is significant enough to change the speed of the clock. But in the course of numerous experimental tests, it was proved that under the influence of acceleration, the movement of time does not accelerate or slow down.

As a result, the segment of the trajectory, on which one of the brothers accelerated, demonstrates only some asymmetry that occurs between the traveler and the homebody.

But this statement cannot explain why time slows down for a moving object, and not for something that remains at rest.

Verification by practice

The formulas and theorems describe the twin paradox accurately, but this is quite difficult for an incompetent person. For those who are more inclined to trust practice, rather than theoretical calculations, numerous experiments have been carried out, the purpose of which was to prove or disprove the theory of relativity.

In one case, they were used. They are extremely accurate, and for a minimum desynchronization they will need more than one million years. Placed in a passenger plane, they circled the Earth several times and then showed quite a noticeable lag behind those watches that did not fly anywhere. And this despite the fact that the speed of movement of the first sample of the watch was far from light.

Another example: the life of muons (heavy electrons) is longer. These elementary particles are several hundred times heavier than ordinary particles, have a negative charge and are formed in the upper layer of the earth's atmosphere due to the action of cosmic rays. The speed of their movement towards the Earth is only slightly inferior to the speed of light. With their true lifespan (2 microseconds), they would have decayed before they touched the surface of the planet. But during the flight, they live 15 times longer (30 microseconds) and still reach the goal.

The physical cause of the paradox and the exchange of signals

Physics also explains the twin paradox in a more accessible language. During the flight, both twin brothers are out of range for each other and cannot practically make sure that their clocks move in sync. It is possible to determine exactly how much the movement of the traveler’s clocks slows down if we analyze the signals that they will send to each other. These are conventional signals of "exact time", expressed as light pulses or video transmission of the clock face.

You need to understand that the signal will not be transmitted in the present time, but already in the past, since the signal propagates at a certain speed and it takes a certain time to pass from the source to the receiver.

It is possible to correctly evaluate the result of the signal dialogue only taking into account the Doppler effect: when the source moves away from the receiver, the signal frequency will decrease, and when approached, it will increase.

Formulation of an explanation in paradoxical situations

There are two main ways to explain the paradoxes of these twin stories:

Careful consideration of existing logical constructions for contradictions and identification of logical errors in the chain of reasoning.
Implementation of detailed calculations in order to assess the fact of time deceleration from the point of view of each of the brothers.

The first group includes computational expressions based on SRT and inscribed in Here it is understood that the moments associated with the acceleration of movement are so small in relation to the total flight length that they can be neglected. In some cases, they can introduce a third inertial frame of reference, which moves in the opposite direction in relation to the traveler and is used to transmit data from his watch to the Earth.

The second group includes calculations built taking into account the fact that moments of accelerated motion are still present. This group itself is also divided into two subgroups: one uses the gravitational theory (GR), and the other does not. If general relativity is involved, then it is understood that the gravitational field appears in the equation, which corresponds to the acceleration of the system, and the change in the speed of time is taken into account.

Conclusion

All discussions connected with an imaginary paradox are due only to an apparent logical error. No matter how the conditions of the problem are formulated, it is impossible to ensure that the brothers find themselves in completely symmetrical conditions. It is important to take into account that time slows down precisely on moving clocks, which had to go through a change in reference systems, because the simultaneity of events is relative.

There are two ways to calculate how much time has slowed down from the point of view of each of the brothers: using the simplest actions within the framework of the special theory of relativity or focusing on non-inertial frames of reference. The results of both chains of calculation can be mutually agreed upon and equally serve to confirm that time passes more slowly on a moving clock.

On this basis, it can be assumed that when the thought experiment is transferred to reality, the one who takes the place of a homebody will indeed grow old faster than the traveler.

The issue tells that relativistic effects are not limited to time dilation and length contraction at all - in fact, there is a complex comparison between times and places for a moving observer and times and places for a stationary one, which is called the Lorentz transformation.

Editor column

Hello dear readers!

Initially, I planned to derive the Lorentz transformations in the traditional way - by mathematical selection of linear transformations of spatial coordinates and time. But one of the forum participants, named Arkady, began to ask me how it turns out that flying after the light, we get the same value of its speed as staying in place?

Indeed, I thought, we derive everything from the constancy of the speed of light, but we have not presented it visually. So I decided to draw a cartoon that shows all this. As a result, it turned out that Arkady helped me get a visual derivation of the Lorentz transformations, which I had not realized before.

In this regard, I recommend everyone to participate in discussions on the forum. For those who do not know the theory of relativity, this will help to voice their troubles and get fair criticism on them, and for those who know, it will help to more clearly understand what they understood before.

I repeat, do not neglect the benefits of communication! :-)

Introduction

So, let's go back a little and remember what three phenomena should be observed if the speed of light does not depend on the speed of its (speed) of the observer:

Movement along the light beam

Now let's put together the work of all three phenomena and see what they give together.

Imagine that someone is flying after the light at such a speed that its time and its lengths are halved (the first two phenomena work). To do this, the observer must fly at a speed of about 260 thousand kilometers per second (obtained from the formula from which found v, at which the whole expression becomes equal to 0.5). Let it flies out of the starting point together with a light pulse, and we are watching all this - a motionless observer.

We will see that at the moment when 1 second passes by our clock and light travels 300 thousand kilometers, only 1/2 second will pass by the clock of a moving observer. The beginning of its ruler will be at a distance of 260 thousand kilometers from us, and the light at a distance of 300. Since the ruler is shortened, our difference in distances of 300 - 260 = 40 thousand km means that the light will be opposite the division 40 * 2 = 80 lines of the moving observer.

If we wait until exactly one second has passed on the clock of a moving observer, then at that moment two seconds will have passed on our clock. The light will fly away from us at a distance of 600 thousand km, and the beginning of the ruler of the moving observer - by 260 * 2 = 520. The difference in the distances between the beginning of the ruler and the position of the light 600 - 520 = 80 will be displayed on the shortened ruler of the moving observer as 80 * 2 = 160 thousand kilometers - it is opposite this division at this moment that the light will be.

Does this mean that a moving observer will get the speed of light equal to 160,000 km/s? Of course not!

Since the constancy of the speed of light is a postulate, we are now let's force a moving observer to contrive and get 300 thousand km / s! :-)

To do this, we use the third phenomenon - the relativity of simultaneity.

Obviously, those two events ( event 1- readings of the clock of the moving observer, equal to 1 sec and event 2- the position of the light pulse opposite the mark of 160 thousand km), which are simultaneous for us - are not simultaneous for a moving observer. And in order to Right measure the speed, he must find the position of the light opposite the ruler, simultaneous with the moment showing 1 s on his watch.

Finding the Right Moment

So, our task is to find an event that, from the point of view of a moving observer, is simultaneous event 1(reading 1 sec on his watch).

What is this event?

It is clear that we have no alternative - this is an event when the light is opposite the mark of 300 thousand km along the ruler of a moving observer. After all, only in this case, that is, only if this particular event is simultaneous event 1, a moving observer will receive the speed of light equal to 300 thousand km / s.

Consider the mark of 300 thousand km on the ruler of a moving observer. Initially, it was (according to our ruler) at a distance of 150 thousand km from the beginning, then it began to move at a speed of 260 thousand km / s to the right.

Light did the same, it just started from the 0 mark (according to our ruler) and traveled 300 thousand km per second.

Here is the table:

It can be seen that the distances will coincide somewhere around 4 seconds according to our watch.

Many clocks of the moving observer

What time will we find on the clock of a moving observer? It is clear that since his clock is twice as slow as ours, then about 2 seconds.

What does it turn out!? There are 2 seconds on the watch of the moving observer, and he considers this moment to be the same time as when his watch had only one second!?

Yes exactly. It is clear that the point here is the relativity of simultaneity, but how to put it in your head?

It's very simple - we observe the ruler of a moving observer at different (for him or her, the ruler) points in time. When our clock shows 4 seconds, then we observe the ruler of the moving observer as it was at different moments of its life. We observe the neighborhood of its zero division at the moment of 2 seconds, and the neighborhood of the division of 300 thousand km - at the moment of 1 second.

It is clear that such a representation fully explains what is happening.

It is also clear that since the observed lifetimes of the two divisions of the ruler are different, then in fact this is true for all intermediate divisions.

How are times distributed along the line? Evenly, like this:

Attention! If you do not see the picture, then try to open the link http://www.relativity.ru/media/media07.shtml#multiplicity

This representation, which we have just received, is called the Lorentz transformation. So far, we have received it only on the fingers, in the form of a semi-quantitative image in the imagination. We will get the exact expression for the transformation later.

But even now it is clear that there is some correspondence between the coordinates obtained by a moving observer and the coordinates obtained by a stationary one. Moreover, together with the spatial coordinates, the clock readings are also converted.

Take another look at the cartoon and think that the bottom half of it is wrong because there is only one clock. Instead of one clock, a whole measuring system should have been drawn there, which would be a combination of a ruler and a chain of clocks. Between this unit and our fixed ruler and fixed clock there would be a correspondence: such and such indications of the moving clock and such and such division of the moving ruler correspond to such and such indications of the fixed clock and the ruler. This correspondence is the Lorentz transformation.

conclusions

We have found that the experience of watching a moving observer fly in pursuit of light is a very rich source of various patterns.

In trying to understand how both observers get the same speed, we had to invoke all three of the phenomena we discovered earlier: time dilation, length contraction, and the relativity of simultaneity.

As a result, we saw that a constant speed of light is quite possible, provided that we observe the world of a moving observer not only oblate and slowed down, but also divided into many adjacent points in time. We must observe an extended object at different moments of its "life".

A complex correspondence between distances and times for a moving observer and distances and times for us (for a stationary observer) is called the Lorentz transformation.

Origin of the name "theory of relativity"

The name “theory of relativity” arose from the name of the basic principle (postulate) put by Poincaré and Einstein as the basis of all theoretical constructions of the new theory of space and time.

The name “principle of relativity” or “postulate of relativity” arose as negation ideas about the absolute fixed frame of reference associated with motionless ether, introduced to explain optical and electrodynamic phenomena.

The fact is that by the beginning of the twentieth century, physicists, who built the theory of optical and electromagnetic phenomena by analogy with the theory of elasticity, had a false idea of the need for the existence of an absolute fixed frame of reference associated with the electromagnetic ether. Thus, the concept of absolute motion with respect to the system associated with the ether was born, a concept that contradicts the earlier views of classical mechanics (the principle of relativity of Galileo). The experiments of Michelson and other physicists refuted this theory of the “fixed ether” and gave grounds for formulating the opposite statement, which was called the “principle of relativity”. So this name is introduced and substantiated in the first works of Poincaré and Einstein.

Einstein writes: “... failed attempts to detect the motion of the Earth relative to the “light-bearing medium” lead to the assumption that not only in mechanics, but also in electrodynamics, no properties of phenomena correspond to the concept of absolute rest, and even more, to the assumption that for of all coordinate systems for which the equations of mechanics are valid, the same electrodynamic and optical laws hold, as has already been proven for first-order quantities. We intend to turn this provision (the content of which will be referred to as the “principle of relativity”) into a premise... “Here is what Poincaré writes: “This impossibility to show by experience the absolute motion of the Earth represents the law of nature; we come to pass this law, which we shall call postulate of relativity and accept it without reservation.”

But the greatest Soviet theorist L. I. Mandelstam explained in his lectures on the theory of relativity: “The name “principle of relativity” is one of the most unfortunate. The independence of phenomena from the unaccelerated motion of a closed system is affirmed. This misleads many minds” One of the creators of the theory of relativity, who revealed its content in a four-dimensional geometric form, Herman Minkowski, also pointed out the failure of the name. In 1908 he stated: “... the term 'the postulate of relativity' for the requirement of invariance with respect to a group seems to me too poor. Since the meaning of the postulate boils down to the fact that in phenomena we are given only a four-dimensional world in space and time, but that the projections of this world onto space and time can be taken with some arbitrariness, I would like to give this statement a name: postulate of absolute peace”

Thus, we see that the names "principle of relativity" and "theory of relativity" do not reflect the true content of the theory.

The theory of relativity as a modern theory of space-time.

The main difference between the concepts of space and time in the theory of relativity and the concepts of Newtonian physics is the limited relationship of space and time. This relationship is revealed in the formulas for the transformation of coordinates and time when moving from one reference system to another (Lorentz transformation)

In general, every physical phenomenon takes place in space and time and cannot be depicted in our consciousness otherwise than in space and time. Space and time are the forms of the existence of matter. No matter exists outside of space and time. A concrete representation of space and time is frame of reference, i.e. coordinate-time manifold of numbers

constituting an imaginary grid and temporal sequence of all possible spatial and temporal points. The same space and time can be represented by different coordinate-time grids (reference systems).

Instead of numbers

space-time can be represented by numbers, and these numbers are not arbitrary, but are associated with the previous absolutely certain type of transformation formulas, which express the properties of space-time.

So, each possible image of space and time can be associated with a specific reference system, the reference system - with a real body, coordinates - with specific points of the body, moments of time

with the readings of specific clocks placed in different reference systems. Reference body necessary for carrying out specific measurements of spatio-temporal relations.

However, one should not identify the reference frame with the reference body, as physicists assume. When depicting phenomena, physicists use any reference systems, including those with which it is impossible to associate any real body. The basis for this choice is the idea of complete equality of all conceivable frames of reference. Consequently, the choice of a reference system is only the choice of a way to represent space and time to display the phenomenon under study.

If two reference systems are chosen

and , each of which similarly depicts the same space-time, then, as established in the theory of relativity, the coordinates in systems and are related so that interval, defined for two separate events as (a)

remains the same when going from E to E', i.e.

(b)

In other words, it is an invariant of the Lorentz transformations relating coordinates and time in

and : , (c)

From (c), as well as from (a) and (b), spatially separated events follow, i.e. for two events

in a system moving with speed , we will have (d)

These properties of space-time coordinates reflect the essence of new ideas about space and time, connected in a single geometric type manifold, a manifold with a special, defined by (a) and (b) four-dimensional pseudo-Euclidean geometry, a geometry in which time is closely connected with space and cannot be considered independently of the latter, as can be seen from (d).

The most important consequences for the laws of nature follow from these ideas, expressed in the requirement covariance(i.e., immutability of form) of any physical processes in relation to transformations of four-dimensional space-time coordinates. The requirement also reflects the idea of space-time as a single four-dimensional manifold. This is how physicists who concretely apply the theory of relativity imagine its real content. At the same time, the concept of relativity acquires only the meaning of the possible multiplicity of space-time images of phenomena with the absoluteness of the content, i.e. laws of nature.

Einstein's postulates.

Lorentz transformations, reflecting the properties of space-time, were derived by Einstein based on 2 postulates: the principle of relativity and the principle of constancy of the speed of light.

1. The laws according to which the states of physical systems change do not depend on which of the two coordinate systems that are in uniform translational motion relative to each other, these changes in state refer to.

2. Each ray of light moves in a “resting” coordinate system with a certain speed

, regardless of whether this ray of light is emitted by a body at rest or a moving body.

The significance of these postulates for the further development of the theory of space-time consisted in the fact that their acceptance, first of all, meant the rejection of the old ideas about space and time as manifolds that are not organically connected with each other.

The principle of relativity in itself did not represent anything absolutely new, since it was also contained in Newtonian physics, built on the basis of classical mechanics. The principle of the constancy of the speed of light was also not something absolutely unacceptable from the point of view of Newton's ideas about space and time.

However, these two principles taken together led to a contradiction with the specific ideas about space and time associated with Newtonian mechanics. This contradiction can be illustrated by the following paradox.

Let in the reference system

at the initial moment, a flash of light occurred at a point coinciding with the origin of coordinates. At the next moment of time, the front of the light wave, due to the law of the constancy of the speed of light, propagated to a sphere of radius centered at the origin of the system coordinates . However, in accordance with Einstein's postulates, we can also consider the same phenomenon from the point of view of a reference system moving uniformly and rectilinearly along the axis, so that its origin and directions of all axes coincide at the time with the origin and directions of the axes of the original system. In this moving system, according to Einstein's postulates, in time the light will also propagate up to a sphere of radius

radius , however, unlike the previous sphere, it must lie at the origin of the system and not . The discrepancy between these spheres, i.e. of the same physical phenomenon seems to be something completely paradoxical and unacceptable from the point of view of existing ideas. It seems that in order to resolve the paradox, it is necessary to abandon the principle of relativity, or the principle of the constancy of the speed of light. The theory of relativity offers, however, a completely different resolution of the paradox, consisting in the fact that events that are simultaneous in one frame of reference are not simultaneous in another, moving frame, and vice versa. Then the simultaneous events consisting in the light front reaching the sphere defined by the equation

, are not simultaneous from the point of view of the system , where other events are simultaneous, consisting in reaching by the same light front the points of the sphere defined by the equation

Thus, the simultaneity of spatially separated events ceases to be something absolute, as it is commonly believed in everyday macroscopic experience, but becomes dependent on the choice of a frame of reference and the distance between the points at which events occur. This relativity of simultaneity spatially separated events indicates that space and time are closely related to each other, because when passing from one frame of reference to another, physically equivalent, the time intervals between events become dependent on distances (the zero interval becomes finite and vice versa).

So, Einstein's postulates helped us to come to a new fundamental position in the physical theory of space and time, the position of close interconnections space and time and their inseparability, this is the main meaning of Einstein's postulates.

The main content of the theory of relativity is the postulate of the constancy of the speed of light. The main argument in favor of this is the role that Einstein assigned to light signals, with the help of which the simultaneity of spatially separated events is established. The light signal, which always propagates only at the speed of light, is thus equated with some tool that establishes a connection between temporal relationships in different reference frames, without which the supposedly concepts of simultaneity of separated events and time lose their meaning. The need for such an interpretation of the content of the theory of relativity is easily proved if we turn to one of the possible conclusions of the Lorentz transformations, based on the postulate of relativity and instead of the postulate of the constancy of the speed of light, using only the assumption that the mass of the body depends on the speed.

Derivation of the Lorentz transformations without the postulate of the constancy of the speed of light.

To derive the Lorentz transformations, we will rely only on the “natural” assumptions about the properties of space and time, which were contained in classical physics, which was based on general ideas related to classical mechanics:

1. Isotropy of space, i.e. all spatial directions are equal.

2. Homogeneity of space and time, i.e. independence of the properties of space and time from the choice of initial reference points (the origin of coordinates and the origin of time).

3. The principle of relativity, i.e. complete equality of all inertial frames of reference.

Different reference systems depict the same space and time in different ways as universal forms of existence of matter. Each of these images has the same properties. Consequently, the transformation formulas expressing the relationship between coordinates and time in one - “fixed” system

with coordinates and time in another - “moving” system , cannot be arbitrary. Let us establish the restrictions that impose “natural” requirements on the form of transformation functions:

1. Due to homogeneity space and time transformations must be linear.

Indeed, if the derivatives of functions

would not be constants, but would depend on then and the difference , expressing the projections of the distances between points 1 and 2 in the “moving” frame, would depend not only on the corresponding projections , in the “fixed” frame, but also on the values of the coordinates themselves, which would contradict the requirement that the properties of space be independent of the choice of initial reference points. If we assume that distance projections of the form x ‘ = =

depend only on the projections of distances in the fixed system, i.e. on x = but does not depend on , then for i.e.

or .

Similarly, one can prove that the derivatives

in all other coordinates are also equal to constants, and therefore, in general, all derivatives

basically a constant.

2. Choose a "moving" system

in such a way that at the initial moment the point representing its origin, i.e.

coincided with the point depicting the origin of the "fixed" system, i.e. , and the velocity of the system would be directed only along

If we also take into account the requirement of space isotropy, then linear transformations for the reference frame chosen in this way will be written in the form On the same basis, in the expressions for and there are no terms proportional to, respectively, and , and the coefficients at and are the same. The terms containing and are absent in the expressions for and due to the fact that the axis always coincides with the axis. The latter would be impossible if and depended on and .

3. Isotropy also implies the symmetry of space. By virtue of symmetry, nothing should change in the transformation formulas if the signs are changed

and , i.e. simultaneously change the direction of the axis and the direction of movement of the system. Therefore, (d) Comparing these equations with the previous ones () we get: . Instead, it is convenient to introduce another function , so that it is expressed in terms of and by means of the relation

According to this relation, is a symmetric function. Using this relation, transformations (d) can be written in the form (e), and all the coefficients included in these formulas

the essence of the symmetry of the function.

4. By virtue of principle relativity both systems, "moving" and "stationary", are absolutely equivalent, and therefore the inverse transformations from the system

k must be identically direct from k. the system moves relative to the system to the right with a speed , and the system moves relative to the system (if the latter is considered stationary), to the left with a speed . Therefore, the inverse transformations must be of the form . (f) Comparing these transformations with (e), we get

. But due to symmetry, we obtain that , i.e. . Obviously, only the (+) sign makes sense, because the sign (-) would give at an inverted po and system. Hence . Noting that the coefficients are also symmetric functions, the first and last equations from (e) and (f) can be written as: A) , a) , B)

, V)

. Multiplying A) by , B) by and adding, we get

. Comparing this expression with a), we get

. Where do we get

Therefore, extracting the square root and noticing that the sign (-) is the same as for

, does not make sense, we get

. So the transformations take the form:

(g) or, more specifically:

,(h) where is the yet unknown function .

5. To determine the type

let's go back to principle of relativity. It is obvious that the transformations (g) must be universal and applicable in any transitions from one system to another. Thus, if we pass twice from the system to and from to, then the resulting formulas relating the coordinates and time in the system with the coordinates and time in must also have the form of transformations (g). This is a requirement arising from the principle of relativity, in conjunction with the previous requirements of reversibility, symmetry, etc. means that the transformations must be group.

Let us use this requirement of groupness of transformations. Let

- speed of the system relative - speed of the system relative to the system

Then according to (g)

and through and , we get

According to the requirement formulated above, the same transformations must be written in the form (g), i.e.

(k) The coefficients at at in the first of these formulas and at in the second are the same. Therefore, due to the identity of the previous formulas and these, the coefficients in the first of the previous formulas and in the second of the formulas (h) must be the same, i.e.

. The last equality can only be satisfied if

6. So, in the transformations (h) h is a constant having the dimension of the square of the speed. The value and even the sign of this constant cannot be determined without involving any new assumptions based on experimental facts.

If we put

, then the transformations (h) turn into the well-known Galileo transformations. These transformations, which are valid in low-velocity mechanics (), cannot be accepted as exact transformations that are valid at any speed of bodies, when a change in the mass of bodies with speed becomes noticeable. Indeed, taking into account the change in mass with speed leads to the need to accept the position on the relativity of the simultaneity of separated events. The latter is incompatible with Galilean transformations. Thus, the constant h must be chosen to be finite.

It is known from experience that at high speeds comparable to the speed of light, the equations of mechanics have the form

(i), where is its own mass, coinciding with the mass of the particle at low speeds (), c is a constant that has the dimension of speed and is numerically equal to cm/sec, i.e. equal to the speed of light in a vacuum. This experimental fact is interpreted as the dependence of mass on velocity, if mass is defined as the ratio of the body's momentum to its velocity.

Constant

has the same dimension as h , which is included in the coordinate and time transformation formulas (h). Therefore, it is natural to put (j), since the experimentally obtained dependence of the mass on the velocity does not include any other constant that has the square of the velocity. Taking this equality, the transformations (h) are written as

(l).

Poincaré called these transformations of coordinates and time Lorentz transformations.

By virtue of reversibility, the inverse Lorentz transformations must obviously be written in the form

The considerations of dimension applied by us for the choice of the constant h are, however, not completely unambiguous, since instead of relation (j), one could just as well choose

(k)

It turns out, however, that the equations of mechanics (i) coinciding with experience can only be obtained as a consequence of the Lorentz transformations and cannot be combined with the transformations resulting from the assumption (k). Indeed, it is known that the equations of mechanics based on the Lorentz transformations are the Minkowski equations, according to which the mass increases with speed according to the formula

. If, however, we choose as coordinate transformations

, then the corresponding Minkowski equations will give a mass m decreasing with velocity, which contradicts experiment.

So, without resorting to the postulate of the constancy of the speed of light in a vacuum, without referring to electrodynamics and without using the properties of light signals to determine simultaneity, we derived the Lorentz transformations using only the idea of the homogeneity and isotropy of space and time, the principle of relativity and the formula for the dependence of mass on speed.

Usually, following the path outlined in the first work of Einstein, instead of the formula for the dependence of mass on speed, the postulate of the constancy of the speed of light in vacuum is used. According to this postulate, when passing from the system

the equation must remain invariant to the system

, describing the front of a light wave propagating from the origin of the coordinate system . It is easy to verify that the equation

after substitution of transformation formulas (k) does not change its form, i.e. this equation goes into the previous one only if .

We have applied a different derivation, which does not use the postulate of the constancy of the speed of light, in order to show that Lorentz transformations can be obtained regardless of the signaling method chosen to synchronize clocks measuring time. Physicists might not know anything at all about the speed of light and the laws of electrodynamics, but they could get Lorentz transformations by analyzing the fact that mass depends on speed and proceeding from the mechanical principle of relativity.

Thus, Lorentz transformations express the general properties of space and time for any physical processes. These transformations, as it turned out in the course of the proof, form a continuous group called Lorentz group. In this fact, in the most general form, the properties of space and time, revealed by the theory of relativity, are displayed.

Depiction of Lorentz transformations on the Minkowski plane.

The first most striking consequences of Lorentz's transformations are: the reduction of moving scales in the direction of motion and the slowing down of the moving clock. From the point of view of everyday ideas about space and time, these consequences seem paradoxical.

An exhaustive, but always somewhat formal, explanation of these kinematic phenomena is given on the x, ct plane, if, in accordance with the rules of Minkowski's four-dimensional geometry, the grid of coordinates of the "fixed" and the grid of coordinates of the "moving" system are depicted on it.

Lorentz transformations leave invariant (unchanged) the interval

between any two events, determined according to (a), as can be easily seen by substituting (l) into (b).

Combining the first event with the moment t=0 and the origin of the system

and introducing symmetric notation of coordinates and time, the interval between the second and the first event can be written in the form (o)

(m) or from a simple four-dimensional generalization of geometry, where the invariant is

(n) In Euclidean geometries defined by (m) or (n), the square of "distance" is always positive, and hence "distance" is a real quantity. But in the four-dimensional geometry defined by the interval (o), which is analogous to "distance", the square of the interval can be positive, negative or equal to zero. Accordingly, in this pseudo-Euclidean geometry interval can be valid or imaginary size. In a particular case, it can be equal to zero for mismatched events.

It sometimes seems that the qualitative difference between four-dimensional Euclidean geometry and four-dimensional pseudo-Euclidean geometry is erased if, using Minkowski's proposal, time is considered to be proportional to some imaginary fourth coordinate, i.e. put

In this case, the square of the interval will be written as

those. coincides with (n) up to sign. However, due to its imaginary nature, this expression, like (o), can have different signs and, thus, qualitatively differs from (n).

Due to the invariance of the interval, the qualitative difference in the connection between events does not depend on the choice of the frame of reference, and the real or timelike, interval (

) remains valid in all reference systems, imaginary, or spacelike, the interval () also remains imaginary in all frames of reference.

All these features of pseudo-Euclidean geometry can be clearly illustrated on the Minkowski plane

The segments 0a and 0b on this plane represent, respectively, the unit scales of the time axis

and the spatial axis. The curve coming out to the right from point a is the hyperbola described by the equation and the curve going up from point b is the hyperbola described by the equation

Thus, the point of origin and all points lying on the hyperbola emanating from the point a are separated by a unit timelike interval. The points lying on the hyperbola emanating from the point b are separated from the origin by a spacelike interval.

Dashed line extending parallel to the axis

from point a depicts points with coordinates , and the line extending from point b parallel to the axis represents points with coordinates .

Lines are drawn on the same plane.

and representing, respectively, points with coordinates and , as well as lines passing through and

and respectively representing points with coordinates

. These lines represent the coordinate grid of the system.

It can be seen from the figure that the transition from the system S to the system

corresponds to the transition from rectangular to oblique coordinates on the Minkowski plane. The latter also follows directly from the Lorentz transformations, which can also be written in the form

where or in the form (p) where

and obviously

But the transformations (p) are identical to the transformations of the transition from Cartesian to oblique coordinates. Under these transformations, the timelike vectors, i.e. vectors directed from the origin to points lying above the OO" line in any coordinate system will also remain timelike, since the ends of the vectors lie on hyperbolas. Therefore, spacelike vectors in all coordinate systems will remain spacelike.

It can be seen on the Minkowski plane that the "spatial" projection of the unit vector

per axis is 1, and per axis is , i.e. less than 1. Consequently, the scale at rest in the system, when measured from the system S, turned out to be shortened. But this statement is reversible, because the "spatial" projection of the vector Ob onto the axis is equal to Ob, i.e. in the system is less than, which is the unit vector.

The situation is similar with the "time" projections on the axis

and Segment , depicting in the system a process lasting a unit of time, in the system S will be projected as , i.e. as a process lasting less time than Oa=1. Consequently, the course of a clock at rest in the system, when measured from the system S, will be slowed down. It is easy to check that this phenomenon is also reversible, i.e. the rate of clocks at rest in the system S turns out to be slowed down in the system.

Reduction of moving scales.

If the length of a fixed scale can be measured by applying reference scales to it, without using any clock, then the length of a moving scale cannot be measured from a fixed reference system without using clocks or signals indicating the simultaneous passage of the ends of the measured scale relative to the points of the standard. Thus, the length of a moving scale must be understood as the distance between its ends, measured using a fixed standard at the same time for each end. Simultaneity measuring the positions of the ends is an essential condition for the experiment. It is easy to see that the violation of this condition can lead to the fact that the measured length can be anything, including negative or equal to zero.

Let

the length of a moving scale, previously measured by direct application to a standard placed in any coordinate system. Then if the moments and passages of the ends of the scale past the points and the fixed standard are the same (i.e. t1=t2), then is, by definition, the length of the moving scale. According to the Lorentz transformations, we have

, whence, by virtue of t1=t2, we obtain

.(r)

The paradox of this conclusion is that, due to the principle of relativity, exactly the same formula should be obtained for the length of the scale located in the S system and measured from the system

"Paradoxes"

general relativity

As in the special theory of relativity, in general relativity "paradoxes" allow not only to reject reasoning based on the so-called "common sense" (ordinary, everyday experience), but also to give a correct, scientific explanation of the "paradox", which, as a rule, is manifestation of a deeper understanding of nature. And this new understanding is given by a new theory, in particular, general relativity.

"Twin Paradox"

When studying SRT, it is noted that the "twin paradox" cannot be explained within the framework of this theory. Recall the essence of this "paradox". One of the twin brothers flies away on a spaceship and, having made a journey, returns to Earth. Depending on the magnitude of the accelerations that the astronaut will experience during launch, turn around and landing, his clock may be significantly behind the earth's clock. It is also possible that he will not find on Earth either his brother or the generation that he left on Earth at the start of the flight, since more than one tens (hundreds) of years will pass on Earth. This paradox cannot be resolved within the framework of SRT, since the considered SSs are not equal (as required in SRT): the spacecraft cannot be considered by the SRT, since it moves unevenly in certain parts of the trajectory.

Only within the framework of general relativity can we understand and explain the "twin paradox" in a natural way, based on the provisions of general relativity. This problem is related to the slowdown in the rate of the clock in moving

CO (or in an equivalent gravitational field).

Let two observers - "twins" are initially on the Earth, which we will consider as inertial CO. Let the observer "A" remain on the Earth, and the second observer - the "twin" "B" starts on a space ship, flies into the unknown expanses of the Cosmos, turns his ship around and returns to the Earth. Even if the movement in the Cosmos occurs evenly, then during takeoff, turnaround and landing, the twin "B" experiences overloads, as it moves with acceleration. These non-uniform movements of astronaut "B" can be likened to his state in some equivalent gravitational field. But under these conditions (in IFR without a gravitational field or in an equivalent gravitational field) there is a physical (and not kinematic, as in SRT) slowing down of the clock rate. In general relativity, a formula was obtained that received a specific expression through the gravitational potential:

from which it is clearly seen that the rate of the clock slows down in a gravitational field with a potential (the same is true for an equivalent rapidly moving CO, which in our problem is a spaceship with a "twin" "B").

Thus, the clock on Earth will show a longer time span than the clock on the spacecraft when it returns to Earth. It is possible to consider another version of the problem, assuming that "twin" "B" is fixed, then "twin" "A" together with the Earth will move away and approach "twin" "B". An analytical calculation in this case also leads to the result obtained above, although this should not seem to be the case. But the fact is that in order to keep the "spaceship" immobile, it is necessary to introduce holding fields, the presence of which will cause the expected result represented by formula (1).

We repeat once again that the "twin paradox" has no explanation in the special theory of relativity, which uses only equal inertial frames. According to SRT, the "twin" "B" must always move away uniformly and rectilinearly from the observer "A". Popular literature often bypasses the "acute" moment in explaining the paradox, replacing the physically ongoing turn of the spacecraft "back to Earth" with its instantaneous turn, which is impossible. But by this "deceptive maneuver" in reasoning, the accelerated movement of the ship on a turn is eliminated, and then both SOs ("Earth" and "Ship") turn out to be equal and inertial, in which the provisions of SRT can be applied. But such an approach cannot be considered scientific.

In conclusion, it should be noted that the "twin paradox" is, in fact, a kind of effect that is called a change in the frequency of radiation in a gravitational field (the period of the oscillatory process is inversely proportional to the frequency, if the period changes, the frequency also changes)

Deviation of light rays passing near the Sun

Thus, the results of our expedition leave little doubt that the rays of light are deflected near the Sun and that the deflection, if attributed to the action of the gravitational field of the Sun, is in magnitude consistent with the requirements of Einstein's general theory of relativity.

F. Dyson, A. Eddington, C. Davidson 1920

Above is a quotation from the report of scientists who observed the total solar eclipse on May 9, 1919 in order to detect the effect of deflection of light rays predicted by general relativity when they pass near gravitating bodies. But let's touch on a little history of this issue. As you know, thanks to the indisputable authority of the great Newton, in the XVIII century. his doctrine of the nature of light triumphed: unlike his contemporary and no less famous Dutch physicist Huygens, who considered light as a wave process, Newton proceeded from the corpuscular model, according to which particles of light, like material (real) particles, interact with a medium in which move and are attracted by bodies according to the laws of gravity, built by Newton himself. Therefore, light corpuscles near gravitating bodies must deviate from their rectilinear motion.

Newton's problem was theoretically solved in 1801 by the German scientist Seldner. A quantitative calculation predicted the angle of deflection of light rays passing near the Sun by 0.87".

A similar effect is also predicted in general relativity, but its nature is assumed to be different. Already with SRT, particles of light - photons - are massless particles, so the Newtonian explanation is completely unsuitable in this case. Einstein approached this problem from the general idea that a gravitating body changes the geometry of the surrounding space, making it non-Euclidean. In curved space-time, free movement (which is the movement of light) occurs along geodesic lines, which will not be straight lines in the Euclidean sense, but will be the shortest lines in curved space-time. Theoretical calculations gave a result twice as large as that obtained according to the Newtonian hypothesis. So the experimental observation of the deflection of light rays near the surface of the Sun could also solve the problem of the physical reliability of the entire general relativity.

It is possible to check the effect of general relativity by the deflection of light rays by the gravitational field only in the case when the light from the star passes near the surface of the Sun, where this field is strong enough to significantly affect the space-time geometry. But under normal conditions, it is impossible to observe a star near the disk of the Sun because of the brighter light from the Sun. That is why scientists used the phenomenon of a total solar eclipse, when the disk of the Sun is covered by the disk of the Moon. Einstein suggested taking pictures of space around the Sun during the minutes of a total solar eclipse. Then photograph the same part of the sky again when the Sun is far from it. Comparing both photographs will reveal the shift in the positions of the stars. Einstein's theory gives the following expression for the magnitude of this angle:

Where M is the mass of the sun. R- radius of the Sun, G-gravitational constant, WITH- the speed of light.

Already the first observations of this effect (1919) gave a completely satisfactory result: with an error of 20%, the angle turned out to be 1.75". the fact that eclipses occur several times a year, but not always where there are conditions for observation, and the weather (clouds) was not always favorable for scientists.In addition, the accuracy of observations was affected by the diffraction of light, which distorted the image of the star. it was possible to increase the accuracy and reduce the error to 10%. The situation changed significantly when radio interferometers were created, thanks to the use of which the observation error was reduced to 0.01 "(i.e. 0.5% of 1.75").

In the 70s. the deviation of radio beams from quasars (stellar formations, the nature of which is not well understood) 3C273 and 3C279 was measured.

The measurements gave the values 1",82±0",26 and 1",77 ±0",20, which is in good agreement with the predictions of general relativity.

Thus, the observation of the deviation of light (electromagnetic) waves from straightness (in the sense of Euclidean geometry) when passing near massive celestial bodies unequivocally testifies in favor of the physical reliability of general relativity.

Rotation of Mercury's perihelion

A. Einstein, developing general relativity, predicted three effects, the explanation of which and their quantitative estimates did not coincide with what could be obtained on the basis of the Newtonian theory of gravitation. Two of these effects (the redshift of the spectral lines emitted by massive stars and the deflection of light rays as they pass near the surface of the Sun and other celestial bodies) have been discussed above. Consider the third gravitational effect predicted by Einstein - the rotation of the perihelion of the planets of the solar system. Based on the observations of Tycho Brahe and Kepler's laws, Newton established that the planets revolve around the Sun in elliptical orbits. Einstein's theory made it possible to discover a more subtle effect - the rotation of the ellipses of orbits in their plane.

Without going into rigorous mathematical calculations, we show how to estimate the expected values of orbital rotations. To do this, we use the so-called method of dimensions. In this method, on the basis of theoretical considerations or experimental data, the quantities that determine the process under consideration are established. From these values, an algebraic expression is compiled, which has the dimension of the desired value, to which the latter is equated. In our problem, we choose as the defining quantities:

1) The so-called gravitational radius of the Sun, which for the Sun (and other celestial bodies) is calculated by the formula

2) The average distance of the planet to the Sun

(for Mercury it is 0.58)

3) The average angular velocity of the planet around the Sun

According to the method of dimensions, we will compose the following value (it should be noted that the method of dimensions requires the intuition of the researcher, a good understanding of physics, which, as a rule, is given by repeated training and solving similar problems):

where determines the angular velocity of movement of the perihelion of the planet's orbit.

For Mercury (for Earth). To imagine the magnitude of the rotation angle of the planet's perihelion, let us recall that an arc second is the angle at which a penny coin is "visible" from a distance of 2 km!

The movement of the perihelion of the planet Mercury was first observed long before the creation of general relativity by the French astronomer Le Verrier (19th century), but only Einstein's theory gave a consistent explanation for this effect. Interestingly, scientists managed to "reproduce" this celestial phenomenon by observing the movement of artificial satellites of the Earth. Since the angle of rotation of the perihelion is proportional to the semi-major axis of the satellite's orbit, its eccentricity and inversely proportional to the period of revolution of the satellite, then, by selecting the appropriate values of these quantities, we can make = 1500 "for 100 years, and this is more than 30 times the angle of rotation of the orbit for Mercury However, the task becomes much more complicated, since the movement of an artificial satellite is influenced by air resistance, non-sphericity and heterogeneity of the Earth, attraction to the Moon, etc. And yet, observation of thousands of artificial satellites launched into near-Earth space over the past more than 30 years , unambiguously confirm the predictions of general relativity.

Calculation of the "radius" of the Universe

Among the various models of the Universe considered in general relativity, there is the so-called model of the stationary Universe, first considered by A. Einstein himself. The world turns out to be finite (but limitless!), it can be represented as a ball (the surface of the ball has no border!). Then it becomes possible to determine the "radius" of such a universe. To do this, we assume that the total energy of the spherical Universe is due exclusively to the gravitational interaction of particles, atoms, stars, galaxies, stellar formations. According to SRT, the total energy of a stationary body is, where M- the mass of the Universe, which can be related to its "radius" as , - the average density of matter distributed evenly in the volume of the World. The gravitational energy of a spherical body of radius can be calculated elementarily and is equal to:

Neglecting numerical coefficients of the order of unity, we equate both expressions for energy, we obtain the following expression for the "radius" of the Universe:

Accepting (which is consistent with observations)

we get the following value for the "radius" of the World:

This value determines the visible "horizon" of the World. Outside this sphere there is no substance and no electromagnetic field. But immediately new problems arise: what about space and time, do they exist outside the sphere? All these questions have not been resolved, science does not know an unambiguous answer to such questions.

The "finiteness" of the Universe in the model under consideration removes the so-called "photometric paradox": the night sky cannot be bright (as it should be if the Universe is infinite and the number of stars is also infinite), since the number of stars (according to the model under consideration) is finite due to finiteness of the volume of the World, and due to the absorption of the energy of electromagnetic waves in interstellar space, the illumination of the sky becomes small.

The model of the stationary Universe is the very first model of the World, as mentioned above, proposed by the creator of GR himself. However, already in the early 1920s Soviet physicist and mathematician gave a different solution to Einstein's equations in general relativity and received two development options for the so-called non-stationary Universe. A few years later, the American scientist Hubble confirmed Friedman's solutions by discovering the expansion of the universe. According to Friedman, depending on the value of the average density of matter in the Universe, the currently observed expansion will either continue forever, or after the slowdown and stop of galactic formations, the process of shrinking the World will begin. Within the scope of this book, we cannot discuss this topic further and refer the inquisitive reader to additional literature. We have touched upon this issue because the model of the expanding Universe also makes it possible to eliminate the photometric paradox discussed above, while relying on other grounds. Due to the expansion of the Universe and the removal of stars from the Earth, the Doppler effect should be observed (in this case, a decrease in the frequency of incoming light) - the so-called red shift in the frequency of light (not to be confused with a similar effect associated not with movement, but with its gravitational field). As a result of the Doppler effect, the energy of the light flux is significantly weakened and the contribution of stars located beyond a certain distance from the Earth is practically equal to zero. At present, it is generally recognized that the Universe cannot be stationary, but we have used such a model due to its "simplicity", and the resulting "radius" of the World does not contradict modern observations.

"Black holes"

Let's say right away that "black holes" in the Universe have not yet been experimentally discovered, although there are up to several dozen "candidates" for this name. This is due to the fact that a star that has turned into a "black hole" cannot be detected by its radiation (hence the name "black hole"), since, having a giant gravitational field, it does not allow either elementary particles or electromagnetic waves to leave its surface. Many theoretical studies have been written on "black holes", their physics can only be explained on the basis of general relativity. Such objects can arise at the final stage of the evolution of a star, when (at a certain mass, not less than 2-3 solar masses) the light pressure of the radiation cannot counteract gravitational contraction and the star experiences a "collapse", i.e., it turns into an exotic object - "black hole." Let us calculate the minimum radius of a star, starting from which its "collapse" is possible. In order for a material body to leave the surface of a star, it must overcome its attraction. This is possible if the body's own energy (rest energy) exceeds the potential energy of gravity, which is required by the law of conservation of total energy. You can make an inequality:

Based on the principle of equivalence, the same body mass is on the left and on the right. Therefore, up to a constant factor, we obtain the radius of a star that can turn into a "black hole":

For the first time this value was calculated by the German physicist Schwarzschild back in 1916, in honor of him this value is called the Schwarzschild radius, or gravitational radius. The sun could turn into a "black hole" with the same mass, having a radius of only 3 km; for a celestial body equal in mass to the Earth, this radius is only 0.44 cm.

Since the formula for , includes the speed of light, this celestial object has a purely relativistic nature. In particular, since GR states the physical deceleration of clocks in a strong gravitational field, this effect should be especially noticeable near a "black hole". Thus, for an observer who is outside the gravitational field of the "black hole", a stone freely falling into the "black hole" will reach the Schwarzschild sphere in an infinitely long period of time. While the watch of the "observer" falling along with the stone will show the final (proper) time. Calculations based on the provisions of general relativity lead to the fact that the gravitational field of the "black hole" is not only able to bend the trajectory of the light beam, but also capture the light flux and make it move around the "black hole" (this is possible if the light beam passes at a distance about 1.5, but such a movement is unstable).

If the collapsing star had an angular momentum, i.e., rotated, then the "black hole" must also retain this angular momentum. But then around this star the gravitational field should also have a vortex character, which will manifest itself in the peculiarity of the properties of space-time. This effect may make it possible to detect a "black hole".

In recent years, the possibility of "evaporation" of "black holes" has been discussed. This is due to the interaction of the gravitational field of such a star with the physical vacuum. Quantum effects should already have their effect in this process, i.e., general relativity turns out to be connected with the physics of the microworld. As we can see, the exotic object predicted by GR - the "black hole" - turns out to be a link between seemingly distant objects - the microworld and the Universe.

Literature for additional reading

1., Polnarev gravitation M., Mir, 1972.

2 Novikov of black holes M., Knowledge, 1986.

3. Novikov the Universe exploded M., B-ka "Quantum", 1988

4. Rozman to the general theory of relativity A. Einstein Pskov, ed. POIPKRO, 1998

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