Einstein's twin paradox. Imaginary paradoxes of SRT

The so-called “clock paradox” was formulated (1912, Paul Langevin) 7 years after the creation of the special theory of relativity and indicates some “contradictions” in the use of the relativistic effect of time dilation. For ease of speech and for “greater clarity” the clock paradox also formulated as the "twin paradox". I also use this wording. Initially, the paradox was actively discussed in scientific literature and especially much in popular literature. Currently, the twin paradox is considered completely resolved, does not contain any unexplained problems, and has practically disappeared from the pages of scientific and even popular literature.

I draw your attention to the twin paradox because, contrary to what was said above, it “still contains” unexplained problems and is not only “unsolved”, but in principle cannot be resolved within the framework of Einstein’s theory of relativity, i.e. This paradox is not so much “the paradox of the twins in the theory of relativity”, but rather “the paradox of Einstein’s theory of relativity itself.”

The essence of the twin paradox is as follows. Let P(traveler) and D(homebody) twin brothers. P goes on a long space journey, and D stays at home. Over time P returns. Most of the way P moves by inertia, at a constant speed (the time for acceleration, braking, stopping is negligible compared to the total travel time and we neglect it). Movement at constant speed is relative, i.e. If P moves away (approaches, is at rest) relative to D, then D also moving away (approaching, at rest) relative to P let's call it symmetry twins. Further, in accordance with SRT, the time for P, from point of view D, flows slower than proper time D, i.e. own travel time P less waiting time D. In this case they say that upon return P younger D . This statement, in itself, is not a paradox, it is a consequence of relativistic time dilation. The paradox is that D, due to symmetry, maybe with the same right , consider yourself a traveler, and P homebody, and then D younger P .

The generally accepted (canonical) resolution of the paradox today boils down to the fact that accelerations P cannot be neglected, i.e. its reference system is not inertial; inertial forces sometimes arise in its reference system, and therefore there is no symmetry. Moreover, in the reference system P acceleration is equivalent to the appearance of a gravitational field, in which time also slows down (this is based on the general theory of relativity). So the time P slows down as in the reference system D(according to service station, when P moves by inertia) and in the reference system P(according to general relativity, when it accelerates), i.e. time dilation P becomes absolute. Final conclusion : P, upon return, younger D, and this is not a paradox!

This, we repeat, is the canonical solution to the twin paradox. However, in all such reasoning known to us, one “small” nuance is not taken into account - the relativistic effect of time dilation is the KINEMATIC EFFECT (in Einstein’s article, the first part, where the effect of time dilation is derived, is called the “Kinematic part”). In relation to our twins, this means that, firstly, there are only two twins and THERE IS NOTHING ELSE, in particular, there is no absolute space, and secondly, twins (read Einstein's clocks) have no mass. This necessary and sufficient conditions formulations of the twin paradox. Any additional conditions lead to "another twin paradox." Of course, it is possible to formulate and then resolve “other twin paradoxes”, but then it is necessary, accordingly, to use “other relativistic effects of time dilation”, for example, to formulate and prove that the relativistic effect of time dilation occurs only in absolute space, or only under the condition that the clock has mass, etc. As is known, there is nothing like this in Einstein’s theory.

Let's go through the canonical proofs again. P accelerates from time to time... Accelerates relative to what? Only relative to the other twin(there is simply nothing else. However, in all canonical reasoning default the existence of another “actor” is assumed, which is not present either in the formulation of the paradox or in Einstein’s theory, absolute space, and then P accelerates relative to this absolute space, whereas D is at rest relative to the same absolute space; there is a violation of symmetry). But kinematically acceleration is relatively the same as speed, i.e. if the traveler twin is accelerating (removing, approaching or at rest) relative to his brother, then the stay-at-home brother, in the same way, is accelerating (removing, approaching or at rest) relative to his traveler brother, symmetry is not broken in this case either (!). No inertial forces or gravitational fields arise in the frame of reference of the accelerated brother also due to the lack of mass in the twins. For the same reason, the general theory of relativity is not applicable here. Thus, the symmetry of the twins is not broken, and The twin paradox remains unresolved . within the framework of Einstein's theory of relativity. A purely philosophical argument can be made in defense of this conclusion: kinematic paradox must be resolved kinematically , and it is not appropriate to involve other, dynamic theories to resolve it, as is done in canonical proofs. Let me note in conclusion that the twin paradox is not a physical paradox, but a paradox of our logic ( aporia type of Zeno's aporia) applied to the analysis of a specific pseudophysical situation. This, in turn, means that any arguments such as the possibility or impossibility of the technical implementation of such a trip, possible communication between twins through the exchange of light signals taking into account the Doppler effect, etc., should also not be used to resolve the paradox (in particular, without sinning against logic , we can calculate the acceleration time P from zero to cruising speed, turn time, braking time when approaching the Earth, as small as desired, even “instantaneous”).

On the other hand, Einstein's theory of relativity itself points to another, completely different aspect of the twin paradox. In the same first article on the theory of relativity (SNT, vol. 1, p. 8), Einstein writes: “We must pay attention to the fact that all our judgments in which time plays any role are always judgments about simultaneous events(Einstein's italics)." (We, in a certain sense, go further than Einstein, believing the simultaneity of events a necessary condition reality events.) In relation to our twins, this means the following: regarding each of them, his brother always simultaneous with him (i.e. really exists), no matter what happens to him. This does not mean that the time elapsed from the beginning of the journey is the same for them when they are at different points in space, but it absolutely must be the same when they are at the same point in space. The latter means that their ages were the same at the start of the journey (they are twins), when they were at the same point in space, then their ages changed mutually during the journey of one of them, depending on its speed (the theory of relativity has not been canceled), when they were at different points in space, and again became the same at the end of the journey, when they again found themselves at the same point in space.. Of course, they both grew old, but the aging process could take place differently for them, from the point of view of one or the other, but ultimately, they aged equally. Note that this new situation for twins is still symmetrical. Now, taking into account the last remarks, the twin paradox becomes qualitatively different fundamentally unsolvable within the framework of Einstein's special theory of relativity.

The latter (together with a number of similar “claims” to Einstein’s SRT, see Chapter XI of our book or the annotation to it in the article “Mathematical principles of modern natural philosophy” on this site) inevitably leads to the need to revise the special theory of relativity. I do not consider my work as a refutation of SRT and, moreover, I do not call for abandoning it altogether, but I propose its further development, I propose a new one "Special theory of relativity(SRT* new edition)", in which, in particular, there is simply no "twin paradox" as such (for those who have not yet become acquainted with the article ""Special" theories of relativity", I inform you that in the new special theory of relativity time slows down, only when the moving inertial system approaching to motionless, and time accelerates, when the moving frame of reference deleted from motionless, and as a result, the acceleration of time in the first half of the journey (moving away from the Earth) is compensated by the slowdown of time in the second half (approaching the Earth), and there is no slow aging of the traveler twin, no paradoxes. Travelers of the future need not fear that upon their return they will find themselves in the distant future of the Earth!). Two fundamentally new theories of relativity have also been constructed, which have no analogues, "Special general" theory of relativity(SOTO)" and "Quatern Universe"(model of the Universe as an “independent theory of relativity”). The article "Special" Theories of Relativity" was published on this site. I dedicated this article to the upcoming 100th anniversary of the theory of relativity . I invite you to comment on my ideas, as well as on the theory of relativity in connection with its 100th anniversary.

Myasnikov Vladimir Makarovich [email protected]
September 2004

Addendum (Added October 2007)

"Paradox" of twins in SRT*. No paradoxes!

So, the symmetry of twins is irremovable in the problem of twins, which in Einstein’s SRT leads to an unsolvable paradox: it becomes obvious that the modified SRT without the twin paradox should give the result T (P) = T (D) which, by the way, fully corresponds to our common sense. These are the conclusions reached in STO* - new edition.

Let me remind you that in STR*, unlike Einstein’s STR, time slows down only when the moving reference system approaches the stationary one, and accelerates when the moving reference system moves away from the stationary one. It is formulated as follows (see formulas (7) and (8)):

Let us further clarify the concept of an inertial reference system, which takes into account the inextricable unity of space and time in SRT*. I define an inertial reference system (see Theory of relativity, new approaches, new ideas. or Space and ether in mathematics and physics.) as a reference point and its neighborhood, all points of which are determined from the reference point and the space of which is homogeneous and isotropic. But the inextricable unity of space and time necessarily requires that the reference point fixed in space should also be fixed in time, in other words, the reference point in space must also be the reference point of time.

So, I consider two fixed frames of reference associated with D: stationary reference system at the moment of launch (reference system mourner D) and a stationary reference system at the moment of finish (reference system greeter D). A distinctive feature of these reference systems is that in the reference system mourner D time flows from the starting point into the future, and the path traveled by the rocket with P grows, no matter where and how it moves, i.e. in this frame of reference P moving away from D both in space and time. In the reference system greeter D- time flows from the past to the starting point and the moment of meeting is approaching, and the path of the rocket with P decreases to the reference point, i.e. in this frame of reference P approaching D both in space and time.

Let's return to our twins. As a reminder, I view the twins problem as a logic problem ( aporia type of Zeno's aporia) in pseudophysical conditions of kinematics, i.e. I believe, that P moves all the time at a constant speed, relying on time for acceleration during acceleration, braking, etc. negligible (zero).

Two twins P(traveler) and D(homebodies) discussing the upcoming flight on Earth P to the star Z, located at a distance L from the Earth and back, at a constant speed V. Estimated flight time, from start on Earth to finish on Earth, for P V his frame of reference equals T=2L/V. But in reference system mourner D P is removed and, therefore, its flight time (the time it waits on Earth) is equal to (see (!!)), and this time is significantly less T, i.e. Waiting time is less than flight time! Paradox? Of course not, since this completely fair conclusion “remained” in reference system mourner D . Now D meets P already in another reference system greeter D , and in this reference system P is approaching, and its waiting time is equal, in accordance with (!!!), i.e. own flight time P and own waiting time D match up. No contradictions!

I propose to consider a specific (of course, mental) “experiment”, scheduled in time for each twin, and in any frame of reference. To be specific, let the star Z removed from the Earth at a distance L= 6 light years. Let it go P flies back and forth on a rocket at a constant speed V = 0,6 c. Then its own flight time T = 2L/V= 20 years. Let us also calculate and (see (!!) and (!!!)). Let us also agree that at intervals of 2 years, at control points in time, P will send a signal (at the speed of light) to Earth. The “experiment” consists of recording the time of reception of signals on Earth, analyzing them and comparing them with theory.

All measurement data for moments in time are shown in the table:

In columns with numbers 1 - 7 are given: 1. Reference points in time (in years) in the rocket's reference frame. These moments record the time intervals from the moment of launch, or the readings of the clock on the rocket, which is set to “zero” at the moment of launch. Control points of time determine on the rocket the moments of sending a signal to Earth. 2. The same control points in time, but in the reference system mourner twin(where “zero” is also set at the moment of rocket launch). They are determined by (!!) taking into account . 3. Distances from the rocket to the Earth in light years at control points in time or the propagation time of the corresponding signal (in years) from the rocket to the Earth 4. in the reference system mourner twin. Defined as a control point in time in the reference frame of the accompanying twin (column 2 3 ). 5. The same control points in time, but now in the reference system greeter twin. The peculiarity of this reference system is that now “zero” time is determined at the moment of the rocket’s finish, and all control moments of time are in the past. We assign them a minus sign, and taking into account the invariance of the direction of time (from past to future), we change their sequence in the column to the opposite. The absolute values ​​of these times are found from the corresponding values in the reference system mourner twin(column 2 ) multiplication by (see (!!!)). 6. Moment of reception of the corresponding signal on Earth in the reference system greeter twin. Defined as reference point in time in the reference system greeter twin(column 5 ) plus the corresponding propagation time of the signal from the rocket to the Earth (column 3 ). 7. Real times of signal reception on Earth. The fact is that D motionless in space (on Earth), but moves in real time, and at the moment of receiving the signal it is no longer located in the reference system mourner twin, But in the reference system point in time signal reception. How to determine this moment in real time? The signal, according to the condition, propagates at the speed of light, which means that two events A = (Earth at the moment the signal is received) and B = (the point in space at which the rocket is located at the moment the signal is sent) (I remind you that an event in space - time is called a point at a certain point in time) are simultaneous, because Δx = cΔt, where Δx is the spatial distance between events, and Δt is the temporal distance, i.e. time of signal propagation from the rocket to the Earth (see the definition of simultaneity in the “Special” theories of relativity, formula (5)). And this, in turn, means that D, with equal right, can consider itself both in the reference frame of event A and in the reference frame of event B. In the latter case, the rocket is approaching, and in accordance with (!!!), all time intervals (up to this control moment) in the reference system mourner twin(column 2 ) should be multiplied by and then added the corresponding signal propagation time (column 3 ). The above is true for any control point in time, including the final one, i.e. the end of the journey P. This is how the column is calculated 7 . Naturally, the actual moments of signal reception do not depend on the method of their calculation; this is what the actual coincidence of the columns indicates 6 And 7 .

The considered “experiment” only confirms the main conclusion that the traveler twin’s own flight time (his age) and the stay-at-home twin’s own waiting time (his age) coincide and there are no contradictions! "Contradictions" arise only in some reference systems, for example, in the reference system mourner twin, but this does not in any way affect the final result, since in this frame of reference the twins, in principle, cannot meet, whereas in the reference system greeter twin, where twins actually meet, there are no longer any contradictions. I repeat: Travelers of the future need not fear that upon returning to Earth they will find themselves in its distant future!

October 2007

First, let's understand what twins are and who twins are. Both are born to the same mother almost simultaneously. But while twins may have different heights, weights, facial features and personalities, twins are virtually indistinguishable. And there is a strict scientific explanation for this.

The fact is that at the birth of twins, the fertilization process could go in two ways: either the egg was fertilized by two sperm at the same time, or the already fertilized egg split in two, and each half began to develop into an independent fetus. In the first case, which is not difficult to guess, twins that are different from each other are born, in the second - monozygotic twins that are absolutely similar to each other. And although these facts have been known to scientists for a long time, the reasons that provoke the appearance of twins have not yet been fully elucidated.

True, it has been noted that any stress can lead to spontaneous division of the egg and the appearance of two identical embryos. This explains the increase in the number of births of twins during periods of war or epidemics, when a woman’s body experiences constant anxiety. In addition, the geological features of the area also affect the statistics of twins. For example, they are born more often in places with increased biopathogenic activity or in areas of ore deposits...

Many people describe a vague but persistent feeling that they once had a twin who has disappeared. Researchers believe this statement is not as strange as it might seem at first glance. It has now been proven that during conception, many more twins develop - both identical and just twins - than are born. Researchers estimate that 25 to 85% of pregnancies begin with two embryos but end with one child.

Here are just two of the hundreds and thousands of examples known to doctors that confirm this conclusion...

Thirty-year-old Maurice Tomkins, who complained of frequent headaches, was given a disappointing diagnosis: a brain tumor. It was decided to carry out the operation. When the tumor was opened, the surgeons were dumbfounded: it turned out to be not a malignant tumor, as previously thought, but not the dissolved remains of the twin brother’s body. This was evidenced by hair, bones, muscle tissue found in the brain...

A similar formation, only in the liver, was found in a nine-year-old schoolgirl from Ukraine. When the tumor, which had grown to the size of a football, was cut, a terrible picture appeared before the eyes of the surprised doctors: bones, long hair, teeth, cartilage, fatty tissue, pieces of skin were sticking out from the inside...

The fact that a significant part of fertilized eggs actually begin their development with two embryos was confirmed by ultrasound studies of the course of pregnancy in tens and hundreds of women. Thus, in 1973, the American doctor Lewis Helman reported that out of 140 high-risk pregnancies he examined, 22 began with two embryonic sacs - 25% more than expected. In 1976, Dr. Salvator Levy of the University of Brussels published his startling statistics on ultrasound examinations of 7,000 pregnant women. Observations carried out in the first 10 weeks of pregnancy showed that in 71% of cases there were two embryos, but only one child was born. According to Levy, the second embryo usually disappeared without a trace by the third month of pregnancy. In most cases, the scientist believes, it is absorbed by the mother's body. Some scientists have suggested that perhaps this is a natural way of removing a damaged embryo, thereby preserving a healthy one.

Proponents of another hypothesis explain this phenomenon by the fact that multiple pregnancies are inherent in the nature of all mammals. But in large representatives of the class, due to the fact that they give birth to larger cubs, at the stage of embryo formation it becomes singleton. Scientists have gone even further in their theoretical constructions, stating the following: “yes, indeed, a fertilized egg always forms two embryos, of which only one, the strongest, survives. But the other embryo does not dissolve at all, but is absorbed by its surviving brother.” That is, in the first stages of pregnancy, real embryonic cannibalism occurs in the womb of a woman. The main argument in favor of this hypothesis is the fact that in the early stages of pregnancy, twin embryos are recorded much more often than in later periods. Previously it was believed that these were early diagnostic errors. Now, judging by the above facts, this discrepancy in statistical data has been fully explained.

Sometimes the missing twin makes itself known in a very original way. When Patricia McDonell from England became pregnant, she learned that she had not one blood type, but two: 7% type A blood and 93% type 0. Type A blood was hers. But most of the blood circulating through Patricia's body came from the unborn twin brother she had absorbed in her mother's womb. However, decades later, his remains continued to produce their own blood.

Twins also demonstrate a lot of interesting features in adulthood. You can verify this using the following example.

The "Jim Twins" were separated at birth, grew up separately and became sensations when they found each other. Both had the same name, both were married to women named Linda, from whom they divorced. When both married for the second time, their wives also had the same name - Betty. Everyone had a dog named Toy. Both worked as sheriff's deputies and at McDonald's and gas stations. They spent their vacations on the beach in St. Petersburg (Florida) and drove a Chevrolet. The two bit their nails and drank Miller beer and set up white benches near a tree in their gardens.

Psychologist Thomas J. Bochard Jr. devoted his entire life to the similarities and differences in the behavior of twins. Based on observations of twins, who were raised in different families and in different environments from early childhood, he came to the conclusion that heredity plays a much larger role than previously thought in the formation of personality traits, its intellect and psyche, and susceptibility to certain diseases . Many of the twins he examined, despite significant differences in upbringing, showed very similar behavioral traits.

For example, Jack Yuf and Oscar Storch, born in 1933 in Trinidad, were separated immediately after their birth. They only met once in their early 20s. They were 45 when they met again at Bochard's in 1979. Both turned out to have mustaches, identical glasses with thin metal frames, and blue shirts with double pockets and shoulder straps. Oscar, raised by his German mother and her family in the Catholic faith, joined the Hitler Youth during the time of fascism. Jack was raised in Trinidad by his Jewish father and later lived in Israel, where he worked on a kibbutz and served in the Israeli Navy. Jack and Oscar discovered that despite their different living conditions, they had the same habits. For example, they both liked to read out loud in the elevator just to see how others would react. They both read magazines back to back, had stern dispositions, wore rubber bands around their wrists, and flushed the toilet before using it. Other pairs of twins studied showed strikingly similar behavior. Bridget Harrison and Dorothy Lowe, born in 1945 and separated when they were a week old, came to Bochard with a watch and bracelets on one hand, two bracelets and seven rings on the other. It later turned out that each of the sisters had a cat named Tiger, that Dorothy's son was named Richard Andrew, and Bridget's son was Andrew Richard. But more impressive was the fact that both, when they were fifteen years old, kept a diary, and then, almost simultaneously, gave up this activity. Their diaries were of the same type and color. Moreover, although the content of the records differed, they were kept or omitted on the same days. When answering questions from psychologists, many couples finished their answers at the same time and often made the same mistakes when answering. The research revealed the similarity of the twins in the manner of speaking, gesticulating, and moving. It was also found that identical twins even sleep the same, and their sleep phases coincide. It is assumed that they may develop the same diseases.

We can conclude this study about twins with the words of Luigi Gelda, who said: “If one has a hole in his tooth, then the other has one in the same tooth or will soon appear.”

Imaginary paradoxes of SRT. Twin paradox

Putenikhin P.V.
[email protected]

There are still numerous discussions on this paradox in the literature and on the Internet. Many of its solutions (explanations) have been proposed and continue to be proposed, from which conclusions are drawn both about the infallibility of STR and its falsity. The thesis that served as the basis for the formulation of the paradox was first stated by Einstein in his fundamental work on the special (particular) theory of relativity “On the electrodynamics of moving bodies” in 1905:

“If there are two synchronously running clocks at point A and we move one of them along a closed curve at a constant speed until they return to A (...), then these clocks, upon arrival at A, will lag behind compared to for hours, remaining motionless...”

Later this thesis received its own names: “clock paradox”, “Langevin paradox” and “twin paradox”. The latter name stuck, and nowadays the formulation is more often found not with watches, but with twins and space flights: if one of the twins flies on a spaceship to the stars, then upon return he turns out to be younger than his brother who remained on Earth.

Much less frequently discussed is another thesis, formulated by Einstein in the same work and immediately following the first, about the lag of the clocks at the equator from the clocks at the Earth's pole. The meanings of both theses coincide:

“... a clock with a balancer, located on the earth’s equator, should go somewhat slower than exactly the same clock placed at the pole, but otherwise placed in the same conditions.”

At first glance, this statement may seem strange, because the distance between the clocks is constant and there is no relative speed between them. But in fact, the change in the pace of the clock is influenced by the instantaneous speed, which, although it continuously changes its direction (tangential speed of the equator), but in total they give the expected lag of the clock.

A paradox, an apparent contradiction in the predictions of the theory of relativity, arises if the moving twin is considered to be the one that remained on Earth. In this case, the twin who has now flown into space should expect that the brother remaining on Earth will be younger than him. It’s the same with clocks: from the point of view of the clock at the equator, the clock at the pole should be considered moving. Thus, a contradiction arises: which of the twins will be younger? Which watch will show time with a lag?

Most often, a simple explanation is usually given to the paradox: the two reference systems under consideration are not actually equal. The twin that flew into space was not always in the inertial frame of reference during its flight; at these moments it cannot use the Lorentz equations. It's the same with watches.

Hence the conclusion should be drawn: the “clock paradox” cannot be correctly formulated in STR; the special theory does not make two mutually exclusive predictions. The problem received a complete solution after the creation of the general theory of relativity, which solved the problem exactly and showed that, indeed, in the described cases, moving clocks lag behind: the clock of the departing twin and the clock at the equator. The “paradox of twins” and clocks is thus an ordinary problem in the theory of relativity.

Clock lag problem at the equator

We rely on the definition of the concept of “paradox” in logic as a contradiction resulting from logically formally correct reasoning, leading to mutually contradictory conclusions (Enciplopedic Dictionary), or as two opposing statements, for each of which there are convincing arguments (Dictionary of Logic). From this position, the “twin, clock, Langevin paradox” is not a paradox, since there are no two mutually exclusive predictions of the theory.

First, let us show that the thesis in Einstein's work about the clock at the equator completely coincides with the thesis about the lag of moving clocks. The figure shows conventionally (top view) a clock at the pole T1 and a clock at the equator T2. We see that the distance between the clocks is unchanged, that is, between them, it would seem, there is no necessary relative speed that can be substituted into the Lorentz equations. However, let's add a third clock T3. They are located in the ISO pole, like the T1 clock, and therefore run synchronously with them. But now we see that clock T2 clearly has a relative speed with respect to clock T3: at first, clock T2 is close to clock T3, then it moves away and approaches again. Therefore, from the point of view of the stationary clock T3, the moving clock T2 lags:

Fig.1 A clock moving in a circle lags behind a clock located in the center of the circle. This becomes more obvious if you add stationary clocks close to the trajectory of moving ones.

Therefore, clock T2 also lags behind clock T1. Let us now move the clock T3 so close to the trajectory T2 that at some initial moment of time they will be nearby. In this case, we get a classic version of the twin paradox. In the following figure we see that at first the clocks T2 and T3 were at the same point, then the clocks at the equator T2 began to move away from the clocks T3 and after some time returned to the starting point along a closed curve:

Fig.2. The clock T2 moving in a circle is first located next to the stationary clock T3, then moves away and after some time approaches them again.

This is fully consistent with the formulation of the first thesis about clock lag, which served as the basis for the “twin paradox.” But clocks T1 and T3 are synchronous, therefore, clock T2 is also behind clock T1. Thus, both theses from Einstein's work can equally serve as the basis for the formulation of the “twin paradox”.

The amount of clock lag in this case is determined by the Lorentz equation, into which we must substitute the tangential speed of the moving clock. Indeed, at each point of the trajectory, clock T2 has speeds that are equal in magnitude, but different in direction:

Fig.3 A moving clock has a constantly changing direction of speed.

How do these different speeds fit into the equation? Very simple. Let's place our own fixed clock at each point of the trajectory of the clock T2. All of these new clocks are synchronized with clocks T1 and T3, since they are all located in the same fixed ISO. Clock T2, each time passing by the corresponding clock, experiences a lag caused by the relative speed just past these clocks. During an instantaneous time interval according to this clock, clock T2 will also lag behind by an instantaneously small time, which can be calculated using the Lorentz equation. Here and further we will use the same notation for the clock and its readings:

Obviously, the upper limit of integration is the readings of clock T3 at the moment when clocks T2 and T3 meet again. As you can see, the readings of the T2 clock< T3 = T1 = T. Лоренцев множитель мы выносим из-под знака интеграла, поскольку он является константой для всех часов. Введённое множество часов можно рассматривать как одни часы - «распределённые в пространстве часы». Это «пространство часов», в котором часы в каждой точке пространства идут синхронно и обязательно некоторые из них находятся рядом с движущимся объектом, с которым эти часы имеют строго определённое относительное (инерциальное) движение.

As we can see, a solution has been obtained that completely coincides with the solution to the first thesis (up to quantities of the fourth and higher orders). For this reason, the following discussion can be considered to apply to all types of formulations of the “twin paradox”.

Variations on the theme of the "twin paradox"

The clock paradox, as noted above, means that special relativity appears to make two mutually contradictory predictions. Indeed, as we just calculated, a clock moving around a circle lags behind a clock located in the center of the circle. But clock T2, moving in a circle, has every reason to claim that they are in the center of the circle around which the stationary clock T1 moves.

The equation for the trajectory of the moving clock T2 from the point of view of the stationary clock T1:

x, y - coordinates of the moving clock T2 in the reference system of the stationary ones;

R is the radius of the circle described by the moving clock T2.

Obviously, from the point of view of the moving clock T2, the distance between it and the stationary clock T1 is also equal to R at any time. But it is known that the locus of points equally distant from a given point is a circle. Consequently, in the reference frame of the moving clock T2, the stationary clock T1 moves around them in a circle:

x 1 2 + y 1 2 = R 2

x 1 , y 1 - coordinates of the stationary clock T1 in the moving frame of reference;

R is the radius of the circle described by the stationary clock T1.

Fig.4 From the point of view of the moving clock T2, the stationary clock T1 moves around them in a circle.

And this, in turn, means that from the point of view of the special theory of relativity, the clock should lag in this case too. Obviously, in this case, it’s the other way around: T2 > T3 = T. It turns out that in fact the special theory of relativity makes two mutually exclusive predictions T2 > T3 and T2< T3? И это действительно так, если не принять во внимание, что теор ия была создана для инерциальных систем отсчета. Здесь же движущиеся часы Т2 не находятся в инерциальной системе. Само по себе это не запрет, а лишь указание на необходимость учесть это обстоятельство. И это обстоятельство разъясняет общая теор ия относительности . Применять его или нет, можно определить простым опытом. В инерциальной системе отсчета на тела не действуют никакие внешние силы. В неинерциальной системе и согласно принципу эквивалентности общей теор ии относительности на все тела действует сила инерции или тяготения. Следовательно, маятник в ней отклонится, все незакреплённые тела будут стремиться переместиться в одном направлении.

Such an experiment near a stationary clock T1 will give a negative result, weightlessness will be observed. But next to the clock T2 moving in a circle, a force will act on all bodies, tending to throw them away from the stationary clock. We, of course, believe that there are no other gravitating bodies nearby. In addition, the T2 clock moving in a circle does not rotate by itself, that is, it does not move in the same way as the Moon around the Earth, which always faces the same side. Observers near clocks T1 and T2 in their reference frames will see an object at infinity from them always at the same angle.

Thus, an observer moving with clock T2 must take into account the fact of non-inertiality of his frame of reference in accordance with the provisions of the general theory of relativity. These provisions say that a clock in a gravitational field or in an equivalent field of inertia slows down. Therefore, with regard to the stationary (according to the experimental conditions) clock T1, he must admit that this clock is in a gravitational field of lower intensity, therefore it goes faster than his own and a gravitational correction should be added to its expected readings.

On the contrary, an observer next to the stationary clock T1 states that the moving clock T2 is in the field of inertial gravity, therefore it moves slower and the gravitational correction should be subtracted from its expected readings.

As we see, the opinion of both observers completely coincided that the clock T2, moving in the original sense, will lag behind. Consequently, the special theory of relativity in its “extended” interpretation makes two strictly consistent predictions, which does not provide any grounds for proclaiming paradoxes. This is an ordinary problem with a very specific solution. A paradox in SRT arises only if its provisions are applied to an object that is not the object of the special theory of relativity. But, as you know, an incorrect premise can lead to both a correct and a false result.

Experiment confirming SRT

It should be noted that all of these imaginary paradoxes discussed correspond to thought experiments based on a mathematical model called the Special Theory of Relativity. The fact that in this model these experiments have the solutions obtained above does not necessarily mean that in real physical experiments the same results will be obtained. The mathematical model of the theory has passed many years of testing and no contradictions have been found in it. This means that all logically correct thought experiments will inevitably produce results that confirm it.

In this regard, of particular interest is an experiment that is generally accepted in real conditions to show exactly the same result as the considered thought experiment. This directly means that the mathematical model of the theory correctly reflects and describes real physical processes.

This was the first experiment to test the lag of a moving clock, known as the Hafele-Keating experiment, conducted in 1971. Four clocks made using cesium frequency standards were placed on two airplanes and traveled around the world. Some clocks traveled in an easterly direction, while others circled the Earth in a westerly direction. The difference in the speed of time arose due to the additional speed of rotation of the Earth, and the influence of the gravitational field at the flight altitude compared to the level of the Earth was also taken into account. As a result of the experiment, it was possible to confirm the general theory of relativity and measure the difference in the speed of the clocks on board two aircraft. The results were published in the journal Science in 1972.

Literature

1. Putenikhin P.V., Three mistakes of anti-SRT [before criticizing a theory, it should be studied well; it is impossible to refute the impeccable mathematics of a theory using its own mathematical means, except by quietly abandoning its postulates - but this is another theory; well-known experimental contradictions in SRT are not used - the experiments of Marinov and others - they need to be repeated many times], 2011, URL:
http://samlib.ru/p/putenihin_p_w/antisto.shtml (accessed 10/12/2015)

2. Putenikhin P.V., So, the paradox (twins) is no more! [animated diagrams - solving the twin paradox using general relativity; the solution has an error due to the use of the approximate equation potential a; time axis is horizontal, distance axis is vertical], 2014, URL:
http://samlib.ru/editors/p/putenihin_p_w/ddm4-oto.shtml (accessed 10/12/2015)

3. Hafele-Keating experiment, Wikipedia, [convincing confirmation of the SRT effect on the slowdown of a moving clock], URL:
https://ru.wikipedia.org/wiki/Hafele_-_Keating Experiment (accessed 10/12/2015)

4. Putenikhin P.V. Imaginary paradoxes of SRT. The twin paradox, [the paradox is imaginary, apparent, since its formulation is made with erroneous assumptions; correct predictions of special relativity are not contradictory], 2015, URL:
http://samlib.ru/p/putenihin_p_w/paradox-twins.shtml (accessed 10/12/2015)

Otyutsky Gennady Pavlovich

The article discusses existing approaches to considering the twin paradox. It is shown that although the formulation of this paradox is associated with the special theory of relativity, most attempts to explain it involve the general theory of relativity, which is not methodologically correct. The author substantiates the position that the very formulation of the “twin paradox” is initially incorrect, because it describes an event that is impossible within the framework of the special theory of relativity. Article address: otm^.agat^a.pe^t^epa^/Z^SIU/b/Zb.^t!

Source

Historical, philosophical, political and legal sciences, cultural studies and art history. Questions of theory and practice

Tambov: Gramota, 2017. No. 5(79) P. 129-131. ISSN 1997-292X.

Journal address: www.gramota.net/editions/3.html

© Publishing house "Gramota"

Information about the possibility of publishing articles in the journal is posted on the publisher’s website: www.gramota.net The editors ask questions related to the publication of scientific materials to be sent to: [email protected]

Philosophical Sciences

The article discusses existing approaches to considering the twin paradox. It is shown that although the formulation of this paradox is associated with the special theory of relativity, most attempts to explain it involve the general theory of relativity, which is not methodologically correct. The author substantiates the position that the very formulation of the “twin paradox” is initially incorrect, because it describes an event that is impossible within the framework of the special theory of relativity.

Key words and phrases: twin paradox; general theory of relativity; special theory of relativity; space; time; simultaneity; A. Einstein.

Otyutsky Gennady Pavlovich, Doctor of Philosophy. Sc., professor

Russian State Social University, Moscow

oIi2ku1@taI-gi

THE GEMINI PARADOX AS A LOGICAL ERROR

Thousands of publications have been devoted to the twin paradox. This paradox is interpreted as a thought experiment, the idea of ​​which is generated by the special theory of relativity (STR). From the main provisions of STR (including the idea of ​​equality of inertial reference systems - IRS), the conclusion follows that from the point of view of “stationary” observers, all processes occurring in systems moving at speeds close to the speed of light must inevitably slow down. Initial condition: one of the twin brothers - a traveler - goes on a space flight at a speed comparable to the speed of light c, and then returns to Earth. The second brother - the homebody - remains on Earth: “From the point of view of the homebody, the moving traveler’s clock has a slow passage of time, so when returning, it must lag behind the homebody’s clock. On the other hand, the Earth was moving relative to the traveler, so the couch potato’s clock must fall behind. In fact, the brothers have equal rights, therefore, after returning, their watches should show the same time.”

To aggravate the “paradoxy”, the fact is emphasized that due to the slowdown of the clock, the returning traveler must be younger than the couch potato. J. Thomson once showed that an astronaut on a flight to the star “nearest Centauri” will age (at a speed of 0.5 from s) by 14.5 years, while 17 years will pass on Earth. However, relative to the astronaut, the Earth was in inertial motion, so the Earth's clock slows down, and the homebody should become younger than the traveler. In the apparent violation of the symmetry of the brothers, the paradox of the situation is seen.

P. Langevin put the paradox into the form of a visual story of twins in 1911. He explained the paradox by taking into account the accelerated movement of the astronaut when returning to Earth. The visual formulation gained popularity and was later used in the explanations of M. von Laue (1913), W. Pauli (1918) and others. There was a surge of interest in the paradox in the 1950s. associated with the desire to predict the foreseeable future of manned space exploration. The works of G. Dingle, who in 1956-1959 were critically interpreted. tried to refute the existing explanations of the paradox. An article by M. Bourne was published in Russian, containing counterarguments to Dingle's arguments. Soviet researchers did not stand aside either.

The discussion of the twin paradox continues to this day with mutually exclusive goals - either substantiating or refuting SRT as a whole. The authors of the first group believe: this paradox is a reliable argument for proving the inconsistency of STR. Thus, I. A. Vereshchagin, classifying SRT as a false teaching, remarks about the paradox: ““Younger, but older” and “older, but younger” - as always since the time of Eubulides. Theorists, instead of making a conclusion about the falsity of the theory, issue a judgment: either one of the disputants will be younger than the other, or they will remain the same age.” On this basis, it is even argued that SRT stopped the development of physics for a hundred years. Yu. A. Borisov goes further: “Teaching the theory of relativity in schools and universities in the country is flawed, devoid of meaning and practical expediency.”

Other authors believe: the paradox under consideration is apparent, and it does not indicate the inconsistency of SRT, but, on the contrary, is its reliable confirmation. They present complex mathematical calculations to take into account the change in the traveler’s frame of reference and seek to prove that STR does not contradict the facts. Three approaches to substantiating the paradox can be distinguished: 1) identifying logical errors in reasoning that led to a visible contradiction; 2) detailed calculations of the magnitude of time dilation from the positions of each of the twins; 3) inclusion of theories other than SRT into the system of substantiating the paradox. Explanations of the second and third groups often overlap.

The generalizing logic of “refutations” of the conclusions of SRT includes four sequential theses: 1) A traveler, flying past any clock that is motionless in the couch potato’s system, observes its slow motion. 2) During a long flight, their accumulated readings can lag behind the traveler’s watch readings as much as desired. 3) Having stopped quickly, the traveler observes the lag of the clock located at the “stopping point”. 4) All clocks in the “stationary” system run synchronously, so the brother’s clock on Earth will also lag behind, which contradicts the conclusion of SRT.

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The fourth thesis is taken for granted and acts as a final conclusion about the paradoxical nature of the situation with twins in relation to SRT. The first two theses indeed logically follow from the postulates of SRT. However, authors who share this logic do not want to see that the third thesis has nothing to do with SRT, since it is possible to “quickly stop” from a speed comparable to the speed of light only after receiving a gigantic deceleration due to a powerful external force. However, the “deniers” pretend that nothing significant happens: the traveler still “must observe the lag of the clock located at the stopping point.” But why “must observe”, since the laws of STR cease to apply in this situation? There is no clear answer, or rather, it is postulated without evidence.

Similar logical leaps are also characteristic of authors who “substantiate” this paradox by demonstrating the asymmetry of twins. For them, the third thesis is decisive, since they associate clock jumps with the acceleration/deceleration situation. According to D.V. Skobeltsyn, “it is logical to consider the cause of the effect [of clock slowdown] to be the “acceleration” that B experiences at the beginning of its movement, in contrast to A, which... remains motionless all the time in the same inertial frame.” Indeed, in order to return to Earth, the traveler must exit the state of inertial motion, slow down, turn around, and then accelerate again to a speed comparable to the speed of light, and upon reaching Earth, slow down and stop again. The logic of D. V. Skobeltsyn, like many of his predecessors and followers, is based on the thesis of A. Einstein himself, who, however, formulates the paradox of clocks (but not “twins”): “If at point A there are two synchronously running clocks, and we move one of them along a closed curve at a constant speed until they return to A (which will take, say, t seconds), then these clocks, upon arrival at A, will lag behind in comparison with the clocks that remained motionless.” Having formulated the general theory of relativity (GTR), Einstein tried to apply it in 1918 to explain the clock effect in a humorous dialogue between a Critic and a Relativist. The paradox was explained by taking into account the influence of the gravitational field on the change in the rhythm of time [Ibid., p. 616-625].

However, relying on A. Einstein does not save the authors from theoretical substitution, which becomes clear if a simple analogy is given. Let’s imagine the “Rules of the Road” with a single rule: “No matter how wide the road, the driver must drive evenly and straight at a speed of 60 km per hour.” We formulate the problem: one twin is a homebody, the other is a disciplined driver. What age will each twin be when the driver returns home from a long trip?

This problem not only has no solution, but is also formulated incorrectly: if the driver is disciplined, he will not be able to return home. To do this, he must either describe a semicircle at a constant speed (non-linear movement!), or slow down, stop and start accelerating in the opposite direction (uneven movement!). In any of the options, he ceases to be a disciplined driver. The traveler from the paradox is the same undisciplined astronaut, violating the postulates of the SRT.

Explanations based on comparisons of the world lines of both twins are associated with similar violations. It is directly stated that “the world line of a traveler who has flown away from the Earth and returned to it is not straight,” i.e. the situation from the sphere of STR moves to the sphere of GRT. But “if the twin paradox is an internal problem of SRT, then it should be solved by SRT methods, without going beyond its scope.”

Many authors who “prove” the consistency of the twin paradox consider the thought experiment with twins and real experiments with muons to be equivalent. Thus, A. S. Kamenev believes that in the case of the movement of cosmic particles, the phenomenon of the “twin paradox” manifests itself “very noticeably”: “an unstable muon (mu-meson) moving at sublight speed exists in its own reference frame for approximately 10-6 seconds, then how its lifetime relative to the laboratory frame of reference turns out to be approximately two orders of magnitude longer (about 10-4 sec) - but here the speed of the particle differs from the speed of light by only hundredths of a percent.” D.V. Skobeltsyn writes about the same thing. The authors do not see or do not want to see the fundamental difference between the situation of twins and the situation of muons: the twin traveler is forced to break from subordination to the postulates of STR, changing the speed and direction of movement, and muons behave like inertial systems throughout the entire time, so their behavior can be explained with the help of a service station.

A. Einstein specifically emphasized that STR deals with inertial systems and only with them, asserting the equivalence of only all “Galilean (non-accelerated) coordinate systems, i.e. such systems in relation to which sufficiently isolated material points move rectilinearly and uniformly.” Since SRT does not consider such movements (uneven and non-linear), thanks to which the traveler could return to Earth, SRT imposes a ban on such a return. The twin paradox, therefore, is not at all paradoxical: within the framework of SRT, it simply cannot be formulated if we strictly accept as prerequisites the initial postulates on which this theory is based.

Only very rare researchers try to consider the position about twins in a formulation compatible with SRT. In this case, the behavior of the twins is considered to be similar to the already known behavior of muons. V. G. Pivovarov and O. A. Nikonov introduce the idea of ​​two “homebodies” A and B at a distance b in ISO K, as well as of a traveler C in a rocket K flying at a speed V comparable to the speed

light (Fig. 1). All three were born at the same time as the rocket flew past point C. After twins C and B meet, the ages of A and C can be compared using proxy B, who is a copy of twin A (Fig. 2).

Twin A believes that when B and C meet, Twin C's watch will show a shorter time. Twin C believes that he is at rest, therefore, due to the relativistic slowdown of the clock, less time will pass for twins A and B. A typical twin paradox is obtained.

Rice. 1. Twins A and C are born at the same time as twin B according to the clock ISO K"

Rice. 2. Twins B and C meet after twin C has flown a distance L

We refer the interested reader to the mathematical calculations given in the article. Let us dwell only on the qualitative conclusions of the authors. In ISO K, twin C flies the distance b between A and B at speed V. This will determine the own age of twins A and B at the time B and C meet. However, in ISO K, twin C’s own age is determined by the time during which he and the same flies at speed L" - the distance between A and B in system K". According to SRT, b" is shorter than the distance b. This means that the time spent by twin C, according to his own clock, on the flight between A and B is less than the age of twins A and B. The authors of the article emphasize that at the moment of the meeting of twins B and C, the own age of twins A and B differs from the own age of the twin C, and “the reason for this difference is the asymmetry of the initial conditions of the problem” [Ibid., p. 140].

Thus, the theoretical formulation of the situation with twins proposed by V. G. Pivovarov and O. A. Nikonov (compatible with the postulates of SRT) turns out to be similar to the situation with muons, confirmed by physical experiments.

The classic formulation of the “twin paradox”, in the case when it is correlated with SRT, is an elementary logical error. Being a logical error, the twin paradox in its “classical” formulation cannot be an argument either for or against SRT.

Does this mean that the twin thesis cannot be discussed? Of course you can. But if we are talking about a classical formulation, then it should be considered as a thesis-hypothesis, but not as a paradox associated with SRT, since concepts that are outside the framework of SRT are used to substantiate the thesis. The further development of the approach of V. G. Pivovarov and O. A. Nikonov and the discussion of the twin paradox in a formulation different from the understanding of P. Langevin and compatible with the postulates of SRT are worthy of attention.

List of sources

1. Borisov Yu. A. Review of criticism of the theory of relativity // International Journal of Applied and Fundamental Research. 2016. No. 3. P. 382-392.

2. Born M. Space travel and the clock paradox // Advances in physical sciences. 1959. T. LXIX. pp. 105-110.

3. Vereshchagin I. A. False teachings and parascience of the twentieth century. Part 2 // Advances in modern natural science. 2007. No. 7. P. 28-34.

4. Kamenev A. S. A. Einstein’s theory of relativity and some philosophical problems of time // Bulletin of the Moscow State Pedagogical University. Series "Philosophical Sciences". 2015. No. 2 (14). pp. 42-59.

5. The twin paradox [Electronic resource]. URL: https://ru.wikipedia.org/wiki/Twin_paradox (access date: 03/31/2017).

6. Pivovarov V. G., Nikonov O. A. Notes on the twin paradox // Bulletin of the Murmansk State Technical University. 2000. T. 3. No. 1. P. 137-144.

7. Skobeltsyn D.V. The twin paradox and the theory of relativity. M.: Nauka, 1966. 192 p.

8. Terletsky Ya. P. Paradoxes of the theory of relativity. M.: Nauka, 1966. 120 p.

9. Thomson J.P. The foreseeable future. M.: Foreign literature, 1958. 176 p.

10. Einstein A. Collection of scientific works. M.: Nauka, 1965. T. 1. Works on the theory of relativity 1905-1920. 700 s.

THE TWIN PARADOX AS A LOGIC ERROR

Otyutskii Gennadii Pavlovich, Doctor in Philosophy, Professor Russian State Social University in Moscow otiuzkyi@mail. ru

The article deals with the existing approaches to the consideration of the twin paradox. It is shown that although the formulation of this paradox is related to the special theory of relativity, the general theory of relativity is also used in most attempts to explain it, which is not methodologically correct. The author grounds a proposition that the formulation of the "twin paradox" itself is initially incorrect, because it describes the event that is impossible within the framework of the special theory of relativity.

Key words and phrases: twin paradox; general theory of relativity; special theory of relativity; space; time; simultaneity; A. Einstein.

What was the reaction of world famous scientists and philosophers to the strange, new world of relativity? She was different. Most physicists and astronomers, embarrassed by the violation of “common sense” and the mathematical difficulties of the general theory of relativity, remained prudently silent. But scientists and philosophers who were able to understand the theory of relativity greeted it with joy. We have already mentioned how quickly Eddington realized the importance of Einstein's achievements. Maurice Schlick, Bertrand Russell, Rudolf Kernap, Ernst Cassirer, Alfred Whitehead, Hans Reichenbach and many other outstanding philosophers were the first enthusiasts who wrote about this theory and tried to clarify all its consequences. Russell's ABC of Relativity was first published in 1925 and remains one of the best popular expositions of the theory of relativity.

Many scientists have found themselves unable to free themselves from the old, Newtonian way of thinking.

They were in many ways like the scientists of Galileo's distant days who could not bring themselves to admit that Aristotle might be wrong. Michelson himself, whose knowledge of mathematics was limited, never accepted the theory of relativity, although his great experiment paved the way for special theory. Later, in 1935, when I was a student at the University of Chicago, Professor William MacMillan, a well-known scientist, taught us an astronomy course. He openly said that the theory of relativity is a sad misunderstanding.

« We, the modern generation, are too impatient to wait for anything.", wrote Macmillan in 1927. " In the forty years since Michelson's attempt to discover the expected motion of the Earth relative to the ether, we have abandoned everything we had been taught before, created a postulate that was the most meaningless we could come up with, and created a non-Newtonian mechanics consistent with this postulate. The success achieved is an excellent tribute to our mental activity and our wit, but it is not certain that our common sense».

A wide variety of objections have been raised against the theory of relativity. One of the earliest and most persistent objections was made to a paradox first mentioned by Einstein himself in 1905 in his paper on the special theory of relativity (the word “paradox” is used to mean something that is contrary to what is generally accepted, but is logically consistent).

This paradox has received a lot of attention in modern scientific literature, since the development of space flights, along with the construction of fantastically accurate instruments for measuring time, may soon provide a way to test this paradox in a direct way.

This paradox is usually stated as a mental experience involving twins. They check their watches. One of the twins on a spaceship makes a long journey through space. When he returns, the twins compare their watches. According to the special theory of relativity, the traveler's watch will show a slightly shorter time. In other words, time moves slower in a spaceship than on Earth.

As long as the space route is limited to the solar system and occurs at a relatively low speed, this time difference will be negligible. But over large distances and at speeds close to the speed of light, the “time reduction” (as this phenomenon is sometimes called) will increase. It is not implausible that in time a way will be discovered by which a spacecraft, slowly accelerating, can reach a speed only slightly less than the speed of light. This will make it possible to visit other stars in our Galaxy, and perhaps even other galaxies. So, the twin paradox is more than just a living room puzzle; it will one day become a daily occurrence for space travelers.

Let us assume that an astronaut - one of the twins - travels a distance of a thousand light years and returns: this distance is small compared to the size of our Galaxy. Is there any confidence that the astronaut will not die long before the end of the journey? Would its journey, as in so many works of science fiction, require an entire colony of men and women, generations living and dying as the ship made its long interstellar journey?

The answer depends on the speed of the ship.

If travel occurs at a speed close to the speed of light, time inside the ship will flow much more slowly. According to earthly time, the journey will continue, of course, more than 2000 years. From an astronaut's point of view, in a spacecraft, if it is moving fast enough, the journey may only last a few decades!

For those readers who like numerical examples, here is the result of recent calculations by Edwin McMillan, a physicist at the University of California at Berkeley. A certain astronaut went from Earth to the spiral nebula of Andromeda.

It is a little less than two million light years away. The astronaut travels the first half of the journey with a constant acceleration of 2g, then with a constant deceleration of 2g until reaching the nebula. (This is a convenient way of creating a constant gravitational field inside the ship for the entire duration of a long journey without the aid of rotation.) The return journey is accomplished in the same way. According to the astronaut's own watch, the duration of the journey will be 29 years. According to the earth's clock, almost 3 million years will pass!

You immediately noticed that a variety of attractive opportunities were arising. A forty-year-old scientist and his young laboratory assistant fell in love with each other. They feel that the age difference makes their wedding impossible. Therefore, he sets off on a long space journey, moving at a speed close to the speed of light. He returns at the age of 41. Meanwhile, his girlfriend on Earth became a thirty-three-year-old woman. She probably couldn’t wait 15 years for her beloved to return and married someone else. The scientist cannot bear this and sets off on another long journey, especially since he is interested in finding out the attitude of subsequent generations to one theory he created, whether they will confirm or refute it. He returns to Earth at the age of 42. The girlfriend of his past years died long ago, and, even worse, nothing remained of his theory, so dear to him. Insulted, he sets out on an even longer journey so that, returning at the age of 45, he sees a world that has already lived for several millennia. It is possible that, like the traveler in Wells's The Time Machine, he will discover that humanity has degenerated. And here he “runs aground.” Wells's "time machine" could move in both directions, and our lone scientist would have no way to return back to his usual segment of human history.

If such time travel becomes possible, then completely unusual moral questions will arise. Would there be anything illegal about, for example, a woman marrying her own great-great-great-great-great-great-great-grandson?

Please note: this kind of time travel bypasses all the logical pitfalls (that scourge of science fiction), such as the possibility of going back in time and killing your own parents before you were born, or dashing into the future and shooting yourself with a bullet in the forehead .

Consider, for example, the situation with Miss Kate from the famous joke rhyme:

A young lady named Kat

It moved much faster than light.

But I always ended up in the wrong place:

If you rush quickly, you will come back to yesterday.

Translation by A. I. Bazya

If she had returned yesterday, she would have met her double. Otherwise it wouldn't really be yesterday. But yesterday there could not be two Miss Kats, because, going on a trip through time, Miss Kat did not remember anything about her meeting with her double that took place yesterday. So, here you have a logical contradiction. This type of time travel is logically impossible unless one assumes the existence of a world identical to ours, but moving along a different path in time (one day earlier). Even so, the situation becomes very complicated.

Note also that Einstein's form of time travel does not attribute any true immortality or even longevity to the traveler. From the point of view of a traveler, old age always approaches him at a normal speed. And only the “own time” of the Earth seems to this traveler rushing at breakneck speed.

Henri Bergson, the famous French philosopher, was the most prominent of the thinkers who crossed swords with Einstein over the twin paradox. He wrote a lot about this paradox, making fun of what seemed to him logically absurd. Unfortunately, everything he wrote proved only that one can be a great philosopher without significant knowledge of mathematics. In the last few years, protests have resurfaced. Herbert Dingle, an English physicist, “most loudly” refuses to believe in the paradox. For many years now he has been writing witty articles about this paradox and accusing specialists in the theory of relativity of being either stupid or cunning. The superficial analysis that we will carry out, of course, will not fully explain the ongoing debate, the participants of which are quickly delving into complex equations, but it will help to understand the general reasons that led to the almost unanimous recognition by specialists that the twin paradox will be carried out exactly as I wrote about it Einstein.

Dingle's objection, the strongest ever raised against the twin paradox, is this. According to the general theory of relativity, there is no absolute motion, no “chosen” frame of reference.

It is always possible to select a moving object as a fixed frame of reference without violating any laws of nature. When the Earth is taken as the reference system, the astronaut makes a long journey, returns and discovers that he has become younger than his stay-at-home brother. What happens if the reference frame is connected to a spacecraft? Now we must assume that the Earth made a long journey and returned back.

In this case, the homebody will be the one of the twins who was in the spaceship. When the Earth returns, will the brother who was on it become younger? If this happens, then in the current situation the paradoxical challenge to common sense will give way to an obvious logical contradiction. It is clear that each of the twins cannot be younger than the other.

Dingle would like to conclude from this: either it is necessary to assume that at the end of the journey the twins will be exactly the same age, or the principle of relativity must be abandoned.

Without performing any calculations, it is easy to understand that in addition to these two alternatives, there are others. It is true that all motion is relative, but in this case there is one very important difference between the relative motion of an astronaut and the relative motion of a couch potato. The couch potato is motionless relative to the Universe.

How does this difference affect the paradox?

Let's say that an astronaut goes to visit Planet X somewhere in the Galaxy. Its journey takes place at a constant speed. The couch potato's clock is connected to the Earth's inertial frame of reference, and its readings coincide with the readings of all other clocks on Earth because they are all stationary in relation to each other. The astronaut's watch is connected to another inertial reference system, to the ship. If the ship always kept one direction, then no paradox would arise due to the fact that there would be no way to compare the readings of both clocks.

But at planet X the ship stops and turns back. In this case, the inertial reference system changes: instead of a reference system moving from the Earth, a system moving towards the Earth appears. With such a change, enormous inertial forces arise, since the ship experiences acceleration when turning. And if the acceleration during a turn is very large, then the astronaut (and not his twin brother on Earth) will die. These inertial forces arise, of course, because the astronaut is accelerating relative to the Universe. They do not occur on Earth because the Earth does not experience such acceleration.

From one point of view, one could say that the inertial forces created by the acceleration "cause" the astronaut's watch to slow down; from another point of view, the occurrence of acceleration simply reveals a change in the frame of reference. As a result of such a change, the world line of the spacecraft, its path on the graph in four-dimensional Minkowski space-time, changes so that the total “proper time” of the journey with a return turns out to be less than the total proper time along the world line of the stay-at-home twin. When changing the reference frame, acceleration is involved, but only the equations of a special theory are included in the calculation.

Dingle's objection still stands, since exactly the same calculations could be done under the assumption that the fixed frame of reference is associated with the ship, and not with the Earth. Now the Earth sets off on its journey, then it returns back, changing the inertial frame of reference. Why not do the same calculations and, based on the same equations, show that time on Earth is behind? And these calculations would be fair if it weren’t for one extremely important fact: when the Earth moved, the entire Universe would move along with it. When the Earth rotated, the Universe would also rotate. This acceleration of the Universe would create a powerful gravitational field. And as has already been shown, gravity slows down the clock. A clock on the Sun, for example, ticks less often than the same clock on Earth, and on Earth less often than on the Moon. After all the calculations are done, it turns out that the gravitational field created by the acceleration of space would slow down the clock in the spaceship compared to the clock on earth by exactly the same amount as they slowed down in the previous case. The gravitational field, of course, did not affect the earth's clock. The Earth is motionless relative to space, therefore, no additional gravitational field arose on it.

It is instructive to consider a case in which exactly the same difference in time occurs, although there are no accelerations. Spaceship A flies past the Earth at a constant speed, heading towards planet X. As the spaceship passes the Earth, its clock is set to zero. Spaceship A continues toward planet X and passes spaceship B, which is moving at a constant speed in the opposite direction. At the moment of closest approach, ship A radios to ship B the time (measured by its clock) that has passed since it passed the Earth. On ship B they remember this information and continue to move towards Earth at a constant speed. As they pass by the Earth, they report back to the Earth the time it took A to travel from Earth to Planet X, as well as the time it took B (measured by his watch) to travel from Planet X to the Earth. The sum of these two time intervals will be less than the time (measured by the earth's clock) that elapsed from the moment A passed the Earth until the moment B passed.

This time difference can be calculated using special theory equations. There were no accelerations here. Of course, in this case there is no twin paradox, since there is no astronaut who flew away and returned back. One might assume that the traveling twin went on ship A, then transferred to ship B and returned back; but this cannot be done without moving from one inertial frame of reference to another. To make such a transfer, he would have to be subjected to amazingly powerful inertial forces. These forces would be caused by the fact that his frame of reference has changed. If we wanted, we could say that inertial forces slowed down the twin's clock. However, if we consider the entire episode from the point of view of the traveling twin, connecting it with a fixed frame of reference, then the shifting space creating a gravitational field will enter into the reasoning. (The main source of confusion when considering the twin paradox is that the situation can be described from different points of view.) Regardless of the point of view taken, the equations of relativity always give the same difference in time. This difference can be obtained using only one special theory. And in general, to discuss the twin paradox, we invoked the general theory only in order to refute Dingle’s objections.

It is often impossible to determine which possibility is “correct.” Does the traveling twin fly back and forth, or does the couch potato do it along with the cosmos? There is a fact: the relative motion of twins. There are, however, two different ways to talk about this. From one point of view, a change in the astronaut's inertial frame of reference, which creates inertial forces, leads to an age difference. From another point of view, the effect of gravitational forces outweighs the effect associated with the Earth's change in the inertial system. From any point of view, the homebody and the cosmos are motionless in relation to each other. So the position is completely different from different points of view, although the relativity of motion is strictly preserved. The paradoxical age difference is explained regardless of which twin is considered to be at rest. There is no need to discard the theory of relativity.

Now an interesting question may be asked.

What if there is nothing in space except two spaceships, A and B? Let ship A, using its rocket engine, accelerate, make a long journey and return back. Will the pre-synchronized clocks on both ships behave the same?

The answer will depend on whether you follow Eddington's or Dennis Sciama's view of inertia. From Eddington's point of view, yes. Ship A is accelerating relative to the space-time metric of space; ship B is not. Their behavior is asymmetrical and will result in the usual age difference. From Skjam's point of view, no. It makes sense to talk about acceleration only in relation to other material bodies. In this case, the only objects are two spaceships. The position is completely symmetrical. And indeed, in this case it is impossible to talk about an inertial frame of reference because there is no inertia (except for the extremely weak inertia created by the presence of two ships). It's hard to predict what would happen in space without inertia if the ship turned on its rocket engines! As Sciama put it with English caution: “Life would be completely different in such a Universe!”

Since the slowing of the traveling twin's clock can be thought of as a gravitational phenomenon, any experience that shows time slowing due to gravity represents indirect confirmation of the twin paradox. In recent years, several such confirmations have been obtained using a remarkable new laboratory method based on the Mössbauer effect. In 1958, the young German physicist Rudolf Mössbauer discovered a method for making a “nuclear clock” that measures time with incomprehensible accuracy. Imagine a clock ticking five times a second, and another clock ticking so that after a million million ticks it will only be slow by one hundredth of a tick. The Mössbauer effect can immediately detect that the second clock is running slower than the first!

Experiments using the Mössbauer effect have shown that time flows somewhat slower near the foundation of a building (where the gravity is greater) than on its roof. As Gamow notes: “A typist working on the ground floor of the Empire State Building ages more slowly than her twin sister working under the roof itself.” Of course, this age difference is elusively small, but it exists and can be measured.

English physicists, using the Mössbauer effect, discovered that a nuclear clock placed on the edge of a rapidly rotating disk with a diameter of only 15 cm slows down somewhat. A rotating clock can be considered as a twin, continuously changing its inertial frame of reference (or as a twin, which is affected by the gravitational field, if we consider the disk to be at rest and the cosmos to be rotating). This experiment is a direct test of the twin paradox. The most direct experiment will be carried out when a nuclear clock is placed on an artificial satellite, which will rotate at high speed around the Earth.

The satellite will then be returned and the clock readings will be compared with the clocks that remained on Earth. Of course, the time is quickly approaching when an astronaut will be able to make the most accurate check by taking a nuclear clock with him on a distant space journey. None of the physicists, except Professor Dingle, doubts that the readings of the astronaut's watch after his return to Earth will differ slightly from the readings of the nuclear clocks remaining on Earth.

However, we must always be prepared for surprises. Remember the Michelson-Morley experiment!

Notes:

A building in New York with 102 floors. - Note translation.