The twin effect. Twin Paradox (Thought Experiment): Explanation Twin Effect

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!

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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.

Keywords 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 SRT. 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 introduce the "Rules" traffic” with the only 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 straight and evenly." 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 the 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 STR, 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're talking about about the 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. Deserves attention further development approach of V. G. Pivovarov and O. A. Nikonov and discussion of the twin paradox in a formulation different from the understanding of P. Langevin and compatible with the postulates of SRT.

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 // Successes 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 University 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. Remarks 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 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.

We apologize for not reposting exciting articles on maintenance for a long time. Let's continue. Start here:

Well, today we will look at perhaps the most famous of the paradoxes of relativity, which is called the “twin paradox”.
I’ll say right away that there really is no paradox, but it stems from an incorrect understanding of what is happening. And if you understand everything correctly, and I assure you, this is not at all difficult, then there will be no paradox.

We will start with the logical part, where we will see how the paradox is created and what logical errors lead to it. And then we’ll move on to the subject part, in which we’ll look at the mechanics of what happens during a paradox.

First, let me remind you of our basic discussion about time dilation.

Remember the joke about Zhora Batareikin, when a colonel was sent to keep an eye on Zhora, and a lieutenant colonel to watch the colonel? We will need imagination to imagine ourselves in the place of the lieutenant colonel, that is, to watch the observer.

So, postulate of relativity states that the speed of light is the same from the point of view of all observers (in all reference systems, scientifically speaking). So, even if an observer flies after the light at a speed of 2/3 the speed of light, he will still see that the light is running away from him at the same speed.

Let's look at this situation from the outside. The light flies forward at a speed of 300,000 km/s, and the observer flies after it at a speed of 200,000 km/s. We see that the distance between the observer and the light increases ( There was a typo in the original by the author - approx. Quantuz) at a speed of 100,000 km/s, but the observer himself does not see this, but sees the same 300,000 km/s. How can this be so? The only (almost! ;-) reason for this phenomenon can be that the observer is slow. He moves slowly, breathes slowly and measures his speed slowly with a slow watch. As a result, he perceives a removal at a speed of 100,000 km/s as a removal at a speed of 300,000 km/s.

Remember another joke about two drug addicts who saw a fireball flash across the sky several times, and then it turned out that they stood on the balcony for three days, and the fireball was the sun? So this observer should be in the state of such a slow drug addict. Of course, this will only be visible to us, and he himself will not notice anything special, because all the processes around him will slow down.

Description of the experiment

To dramatize this conclusion, an unknown author from the past, perhaps Einstein himself, came up with the following thought experiment. Two twin brothers live on earth - Kostya and Yasha.

If brothers lived together on earth, they would synchronously go through the following stages of growing up and aging (I apologize for some convention):

But that's not how things happen.

While still a teenager, Kostya, let's call him a space brother, gets into a rocket and goes to a star located several tens of light years from Earth.
The flight takes place at near-light speed and therefore the round trip takes sixty years.

Kostya, whom we will call our earthly brother, is not flying anywhere, but is patiently waiting for his relative at home.

Relativity Prediction

When the space brother returns, the earthly one turns out to be sixty years older.

However, since the space brother was constantly on the move, his time passed more slowly, therefore, upon his return, he would be only 30 years older. One twin will be older than the other!

It seems to many that this prediction is wrong and these people call this prediction itself the twin paradox. But that's not true. The prediction is absolutely true and the world works exactly like that!

Let's look at the logic of the prediction again. Let’s say that an earthly brother continuously observes the cosmic.

By the way, I have already repeatedly said that many people make a mistake here, incorrectly interpreting the concept of “observes.” They think that observation must necessarily take place with the help of light, for example, through a telescope. Then, they think, since light travels at a finite speed, everything that is observed will be seen as it was before, at the moment the light was emitted. Because of this, these people think, time dilation occurs, which is thus an apparent phenomenon.
Another version of the same misconception is to attribute all phenomena to the Doppler effect: since the cosmic brother moves away from the earthly one, each new “image frame” comes to Earth later and later, and the frames themselves, thus, follow less frequently than necessary, and entail time dilation.
Both explanations are incorrect. The theory of relativity is not so stupid as to ignore these effects. Look for yourself at our statement regarding the speed of light. We wrote there “he will still see that,” but we did not mean exactly “he will see with his eyes.” We meant “will receive as a result, taking into account all known phenomena.” Please note that the entire logic of reasoning is nowhere based on the fact that observation occurs with the help of light. And if this is exactly what you imagined all along, then re-read everything again, imagining how it should be!

For continuous observation, it is necessary that the space brother, for example, send faxes to Earth every month (by radio, at the speed of light) with his image, and the earthly brother would post them on the calendar, taking into account the transmission delay. It would turn out that first the earthly brother hangs up his photograph, and hangs up the photograph of his brother from the same time later, when it reaches him.

According to the theory, he will always see that time flows more slowly for his space brother. It will flow more slowly at the beginning of the journey, in the first quarter of the journey, in the last quarter of the journey, at the end of the journey. And because of this, the backlog will constantly accumulate. Only during the turn of the space brother, at the moment when he stops to fly back, his time will pass at the same speed as on Earth. But this will not change the final result, since the total lag will still be there. Consequently, at the time of the return of the space brother, the lag will remain and that means it will remain forever.

As you can see, there are no logical errors here. However, the conclusion looks very surprising. But there's nothing you can do about it: we live in amazing world. This conclusion has been confirmed many times, both for elementary particles, which lived longer if they were in motion, and for the most ordinary, only very accurate (atomic) clocks, which were sent into space flight and then it was discovered that they were behind the laboratory ones by a fraction of a second.

Not only the fact of the lag was confirmed, but also its numerical value, which can be calculated using formulas from one of the previous issues.

Apparent contradiction

So, there will be a lag. The space brother will be younger than the earthly one, you can be sure.

But another question arises. After all, movement is relative! Therefore, we can assume that the space brother did not fly anywhere, but remained motionless all the time. But instead of him, his earthly brother flew on the journey, along with the planet Earth itself and everything else. And if so, it means that the space brother should grow older, and the earthly brother should remain younger.

This results in a contradiction: both considerations, which should be equivalent according to the theory of relativity, lead to opposite conclusions.

This contradiction is called the twin paradox.

Inertial and non-inertial reference systems

How can we resolve this contradiction? As you know, there can be no contradictions :-)

Therefore, we must figure out what we didn’t take into account that caused the contradiction?

The very conclusion that time should slow down is impeccable, because it is too simple. Therefore, the error in reasoning must be present later, where we assumed that the brothers were equal. This means that in fact the brothers are not equal!

I already said in the very first issue that not every relativity that seems to exist in reality. For example, it may seem that if a cosmic brother accelerates away from the Earth, then this is equivalent to the fact that he remains in place, and the Earth itself accelerates, away from him. But that's not true. Nature does not agree with this. For some reason, nature creates overloads for the one who accelerates: he is pressed into the chair. And for those who do not accelerate, it does not create overloads.

Why does nature do this? this moment doesn't matter. At this moment, it is important to learn to imagine nature as correctly as possible.

So, brothers can be unequal, provided that one of them accelerates or brakes. But we have exactly this situation: you can fly away from Earth and return to it only accelerating, turning around and braking. In all these cases, the space brother experienced overloads.

What is the conclusion? The logical conclusion is simple: we have no right to declare that brothers have equal rights. Consequently, reasoning about time dilation is correct only from the point of view of one of them. Which one? Of course, earthly. Why? Because we didn’t think about overloads and imagined everything as if they didn’t exist. For example, we cannot say that under overload conditions the speed of light remains constant. Therefore, we cannot claim that time slows down under overload conditions. Everything we stated was for the case of no overloads.

When scientists got to this point, they realized that they needed a special name to describe the "normal" world, the world without overload. This description was called a description from the point of view inertial reference system(abbreviated as ISO). The new description, which had not yet been created, was naturally called a description from the point of view non-inertial reference frame.

What is an inertial reference system (IRS)

It's clear that first, what we can say about ISO is a description of the world that seems “normal” to us. That is, this is the description with which we started.

In inertial frames of reference, the so-called law of inertia operates - each body, being left to itself, either remains at rest or moves uniformly and rectilinearly. Because of this, the systems were so called.

If we sit in a spaceship, car or train that moves absolutely uniformly and rectilinearly from the point of view of ISO, then inside such a vehicle we will not be able to notice the movement. This means that such a surveillance system will also be ISO.

Consequently, the second thing we can say about the ISO is that any system moving uniformly and rectilinearly relative to the ISO will also be an ISO.

What can we say about non-ISOs? For now, we can only say about them that a system moving relative to an ISO with acceleration will be a non-IFR.

Part last: Kostya's story

Now let's try to figure out what the world will look like from the point of view of our space brother? Let him also receive faxes from his earthly brother and post them on the calendar, taking into account the flight time of the fax from Earth to the ship. What will he get?

To figure this out, you need to pay attention to the following point: during the journey of the space brother, there are sections in which he moves uniformly and in a straight line. Let's say that at the start, the brother accelerates with enormous force so that he reaches cruising speed in 1 day. After that, it flies evenly for many years. Then, in the middle of the journey, it also quickly turns around in one day and flies back again evenly. At the end of the journey, he brakes very sharply, in one day.

Of course, if we calculate what speeds we need and with what acceleration we need to accelerate and turn, we get that our space brother should simply be smeared across the walls. And the walls themselves spaceship, if they are made of modern materials, they will not be able to withstand such overloads. But that’s not what’s important to us now. Let's say Kostya has super-duper anti-g seats, and the ship is made of alien steel.

What will happen?

At the very first moment of the flight, as we know, the ages of the brothers are equal. During the first half of the flight, it occurs inertially, which means that the rule of time dilation applies to it. That is, the cosmic brother will see that the earthly one is aging twice as slow. Consequently, after 10 years of flight, Kostya will age by 10 years, and Yasha will age by only 5.

Unfortunately, I didn't draw the 15 year old twin, so I'll use the 10 year old picture with a "+5" added.

A similar result is obtained from end-of-path analysis. At the very last moment, the brothers’ ages are 40 (Yasha) and 70 (Kostya), we know this for sure. In addition, we know that the second half of the flight also proceeded inertially, which means that the appearance of the world from Kostya’s point of view corresponds to our conclusions about time dilation. Consequently, 10 years before the end of the flight, when the space brother is 30 years old, he will conclude that the earthly one is already 65, because before the end of the flight, when the ratio is 40/70, he will age twice as slow.

Again, I don't have a 65 year old design and will use a 70 year old one marked "-5".

I have posted a summary of my space brother observations below.

As you can see, the space brother has an inconsistency. Throughout the first half of the journey, he observes that his earthly brother is aging slowly and is barely breaking away from his initial age of 10 years. Throughout the second half of the flight, he watches as his earthly brother barely reaches the age of 70 years.

Somewhere between these sections, in the very middle of the flight, something must happen that “stitches” the aging process of the earthly brother together.

Actually, we won’t continue to obfuscate and guess what’s going on there. We will simply directly and honestly draw the conclusion that follows inevitably. If a moment before the reversal, the earthly brother was 17.5 years old, and after the reversal it became 52.5, then this means nothing more than the fact that during the reversal of the cosmic brother, 35 years passed for the earthly brother!

conclusions

So we saw that there is a so-called twin paradox, which consists of an apparent contradiction in which of the two twins time slows down. The very fact of time dilation is not a paradox.

We saw that there are inertial and non-inertial frames of reference, and the laws of nature that we obtained earlier applied only to inertial frames. It is in inertial systems that time dilation is observed on moving spacecraft.

We found that in non-inertial reference systems, for example, from the point of view of unfolding spaceships, time behaves even more strangely - it fast-forwards.

Note Quantuz: The author also provided a link to further explanation of the twin paradox with flash animation. You can try following the link to the web archive where this article is carefully saved. Recommended for deeper understanding. See you on the pages of our cozy little one.

The next famous thought experiment, the so-called twin paradox, is based on this amazing phenomenon of time dilation. Let's imagine that one of two twins goes on a long journey in a spaceship and is carried away from the Earth at extremely high speed. Five years later he turns around and heads back. Thus the total travel time is 10 years. At home, it is discovered that the twin remaining on Earth has aged, say, 50 years. How many years younger the traveler will be than the one remaining at home depends on the speed of the flight. 50 years have actually passed on Earth, which means that the traveler twin had been on the road for 50 years, but for him the journey took only 10 years.

This thought experiment may seem absurd, but countless similar experiments have been conducted, all of which confirm the predictions of the theory of relativity. Example: ultra-precise atomic clocks fly around the Earth several times on a passenger plane. After landing, it turns out that less time has actually passed on the atomic clock on the plane than on other atomic clocks left on the ground for comparison. Since the speed of a passenger plane is much less than the speed of light, time dilation is very small - but the accuracy of atomic clocks is enough to register it. The most modern atomic clocks are so accurate that an error of one second is achieved only after 100 million years.

Another example that much better illustrates the effect of time dilation is the 15-fold increase in the lifespan of certain elementary particles - muons. Muons can be thought of as heavy electrons. They are 207 times heavier than electrons, carry a negative charge and arise in the upper layers of the earth's atmosphere under the influence of cosmic rays. Muons fly towards Earth at 99.8% the speed of light. But since their lifespan is only 2 microseconds, even at such a high speed they would have to disintegrate after 600 meters, before reaching the surface.

For us, in a resting frame of reference (Earth), muons are extremely fast-moving “decay clocks”, the lifetime of which increases by 15 times. Thanks to this, they exist for 30 microseconds and reach the surface of the Earth.

For the muons themselves, time does not stretch, but they reach the Earth. How can this be? The answer lies in another amazing phenomenon, “relativistic distance contraction,” which is also called Lorentzian. Shortening distances means that fast moving objects become shorter in the direction of travel.

In the muon reference frame at rest, the situation looks completely different: the mountain and with it the Earth approach the muons at a speed equal to 99.8% of light speed. A mountain with a height of 9000 meters, due to the reduction in distances, seems 15 times lower, and this is only 600 meters. Therefore, even with such a short lifespan - 2 microseconds - muons fall on Earth.

As we see, the main thing is from which point to consider a physical phenomenon. In the resting frame of reference “Earth”, time stretches and flows more slowly. On the contrary, in the “muons” reference frame at rest, space contracts in the direction of motion, in other words, it contracts. Distance to earth's surface decreases from 9000 to 600 meters.

So, the constancy of the speed of light leads to two phenomena that are completely incredible from the point of view of common sense: time dilation and reduction of distances. But if we consider the speed of light to be constant and look at the formula “speed equals distance divided by time,” we can draw the following conclusion: two observers in two different inertial frames of reference, who obtained the same speed of light c as a result of measurements, will certainly receive different meanings distance and time.

Of course, it is difficult for us to accept that there is neither absolute time nor absolute space, only relative time and relative distances. However, this is due to the fact that no person has ever moved at a speed at which relativistic effects would become noticeable.

Another strange phenomenon is the so-called relativistic increase in mass. When we deal with speeds close to the speed of light, the mass of a body increases, just as time slows down or distance decreases. If the speed is 10% of light speed or more, the "relativistic effects" become so obvious that they can no longer be ignored. When the speed is equal to 99.8% of light, the mass of the body is 15 times greater than its rest mass, and when it is equal to 99.99% of light, the mass exceeds its rest mass by 700 times. If the speed is 99.9999% of the speed of light, the mass increases 700 times. So, as speed increases, the body becomes heavier, and the heavier it is, the more energy is required to accelerate it even more. As a result, the speed of light represents an upper limit that cannot be exceeded, no matter how much energy is supplied.

Of course, the queen of physical formulas, and perhaps the most famous formula in general, was also derived by Albert Einstein. It reads: E = m * c 2.

Einstein himself considered this equation to be the most important conclusion of the theory of relativity.

But what is the meaning of this formula? On the left is E, energy, on the right is mass multiplied by the squared speed of light c. It follows that energy and mass are essentially the same thing - and this is true.

As a matter of fact, one can guess this already from the relativistic increase in masses. If a body moves quickly, its mass increases. To accelerate the body, naturally, additional energy is needed.

However, the supply of energy leads not only to an increase in speed: the mass also increases at the same time. Of course, it’s difficult for us to imagine this, but this fact is 100% confirmed by experiments.

This has important applications such as generating energy through nuclear fission: a heavy uranium nucleus splits into two parts, such as krypton and barium. But the sum of the masses is somewhat less than the mass of uranium before decay. The mass difference “delta (Δ)m”, also called a mass defect, completely transforms into energy during decay. This is how electricity is generated at nuclear power plants.

The twin paradox is shrouded in the romance of interstellar travel and a fog of misinterpretations. It became widely known thanks to Paul Langevin's formulation (1911), which in a popular paraphrase reads as follows:

One twin brother remains on Earth, and the second goes on space travel at near-light speed. From the point of view of a homebody, a traveler moving relative to him has a slower passage of time. That's why upon return he will be younger. However, from the astronaut's point of view, the Earth was moving, so the stay-at-home brother should be younger.

The word "paradox" has several meanings. For example, many conclusions of the theory of relativity are paradoxical, since they contradict conventional ideas. There is, of course, nothing wrong with such paradoxicality. Any new theory "unusual"and requires a change in old ideas. However, when describing the story with twins, "paradox" is synonymous with " logical contradiction". Having reasoned about the same event (meeting of brothers) by two different ways, we get different results. Of course, in a consistent theory this should not happen.

An extensive literature is devoted to the twin paradox. The generally accepted explanation is as follows. So that the brothers can directly to compare their ages, one of them (the traveler) needs to return, and to do this, experience the stages of accelerated motion, moving to a non-inertial frame of reference. Therefore, there is no complete symmetry between the brothers. Naturally, such a removal of the paradox does not explain why the astronaut should become younger. In addition, the following objection immediately arises: “if the whole point is acceleration, then the acceleration and deceleration stages can be made as short as desired (for each observer!) compared to arbitrarily long and symmetrical stages uniform motion".

To this they answer that the calculation, within the framework of the general theory of relativity, gives the same answer for each brother. Of course, gravity has nothing to do with this calculation, and the differential geometry used in this case serves as a mathematical apparatus for describing non-inertial reference systems. Such calculations are absolutely correct, but the physical reasons for what happened to the brothers often turn out to be hidden.

We will begin our analysis with the remark that it is not necessary for the traveling brother to return. It is enough for him to slow down, moving into the reference system associated with the Earth. Being far away, but remaining motionless relative to each other, the brothers can easily synchronize their time and find out how their clocks (physical and biological) diverged. If you wish, you can, of course, consider a new launch of the spacecraft and its return to Earth. However, no new effects will occur, and all times will simply need to be multiplied by two. By and large, there is not even a need for an accelerated launch from Earth. One can consider the simultaneous birth of brothers in two different inertial frames of reference as they flew past each other. Leaving aside the physiological details of such a birth, we emphasize that when brothers are in different systems, but at the same spatial point, they can easily agree on the initial moment of time (the fact of their birth).

We examined this formulated story in detail in the “Time” section. As a result of the relativity of simultaneity, parts of a moving reference system located along the direction of its movement are “in the past,” and parts opposite the movement are in the future. And the further they are from the brothers’ birth point, the stronger the effect:

An astronaut flying past any “stationary” clock sees that it is moving slower than his own. However, on all such watches, those he meets on the way, he observes the future tense: V . Likewise, spaceport employees who pass by an astronaut see him as younger. At the same time, the “nephews of the same age” flying past the homebody brother (on the last ships of the squadron) look older than the earthling. These effects are absolute for observers of different systems located at the same spatial point, therefore will not change when stopped. To understand the twin paradox, in fact, there is no need to even consider non-inertial frames of reference! If the astronaut stops, he will “fall into the future” of the earth’s reference frame and will be younger there. In the same way, if an earthling accelerates, he will end up in the future of the astronaut system and will be younger there.

The "paradox" of the twins can be analyzed without expensive investments in the construction of spaceports. Suppose that two brothers, from the moment of separation, begin to broadcast their video images to each other. The traveler sees his brother sitting in an armchair by the fireplace, on which there is a clock. He, in turn, sees on the monitor the cockpit of a spaceship with an electronic clock above the helm, behind which sits his courageous brother-traveler. The spacecraft must reach the nearest star, distant from the Earth, and return back. Here are extracts from the spacecraft's logbook.

Travel diary. Having made a quick acceleration, I reach near-light speed. The overloads are colossal, but thanks latest achievements Biocybernetics can tolerate them relatively easily. According to my watch, the start time of the journey coincides with the time of my stay-at-home brother. However, the frequency of the received signal from the rapidly receding Earth has noticeably decreased. My brother's movements look slow. This is understandable; the Doppler effect has not yet been canceled. The stars along the course huddled together, while behind, around native land, they noticeably decreased, and they turned red. Here, too, everything is clear - aberration plus a change in frequency. The distances between the automatic beacons placed along my route have decreased, and, therefore, the flight time to the star according to my watch will be , and not, as my brother and I saw from Earth. Therefore, the travel time should be shorter than my brother's watch. We'll see. Speaking of my brother, the second hand on his mantel clock barely moves, and the time it shows is significantly behind mine. This result is the sum of the Doppler effect and the delay in video transmission due to the finite speed of light.

Having reached the destination of the journey, I sharply brake and take memorable photographs against the backdrop of a star. After braking, the hand on my brother’s mantel clock immediately began its natural run, although, of course, the total time that has passed since the beginning of the flight has not changed and is far behind mine. There is nothing else to do near the lonely star, so I sharply accelerate in the opposite direction. Having come to my senses after acceleration, I see that my brother’s watch has noticeably sped up, and its second hand is spinning like mad.

There is very little left to reach the Earth. During the return trip, my brother's watch managed to catch up and, moreover, overtook my chronometer. Tomorrow the braking and our long-awaited meeting. However, there is no longer any doubt that now I am the youngest brother in the family.

Let's look at the physics of the impressions described by the traveler. Let the brothers transmit to each other every second (according to their watches) signals of the exact time. Let us assume that the accelerated movements of the spacecraft are very short (from the point of view of both brothers) compared to the time of the entire journey. While the spaceship is moving away from the Earth, each brother, due to the Doppler effect, sees a decrease in the frequency (increase in the period) of the received signals. After braking near the star, the traveler stops “running away” from earthly signals, and their period straightaway becomes equal to its second. Having turned around and accelerated, the traveler begins to “jump” at the signals coming towards him and their frequency increases (the period decreases).

According to his clock, the travel time in one direction is equal to , and the same in the opposite direction. Quantity taken "earth seconds" during the journey is equal to their frequency multiplied by time:

Therefore, when moving away from the Earth, the astronaut received significantly fewer seconds (first term), and when approaching, correspondingly, more (second term). The total number of seconds received from the Earth is greater than those transmitted to it, in exact accordance with the time dilation formula.

The arithmetic of an earthling is somewhat different. While his brother is moving away, he also registers an increase in the periods of precise time transmitted from the spaceship. However, unlike his brother, the earthling observes such a slowdown longer. The flight time to the star is according to Earth clocks. An earthling will see the event of a traveler braking near a star after the additional time required for the light to travel the distance from the star. Therefore, only after the start of the journey he will see on the monitor the accelerated operation of his approaching brother’s clock:

Given that the times are equal and , we have:

Thus, the effect of time dilation of a brother who changed his frame of reference is absolute, i.e. is the same for both brothers.

The most paradoxical thing about the twin paradox is that it is sometimes easier to explain than to formulate. This paradox is often perceived superficially, so we present the following “deep” reasoning:

Okay, let’s say the twins are not equal and the astronaut changed the frame of reference. There are no particular objections to its description based on the Doppler effect. However, this still does not remove the paradox in the following formulation. Astronaut flying by all hours, motionless in the earth's frame of reference, sees that they are moving slower than his clock. He is a “former earthling” and knows that all these watches are the same. Therefore, he must conclude that his brother’s time also flows more slowly. Time intervals, unlike ruler lengths, accumulate, and therefore, when stopped, the clock readings cannot be equal. Moreover, if the stop is very fast compared to the time of uniform motion, it cannot in any way lead to the lagging clock of the earthly brother jumping ahead of the clock of the spaceship. Therefore, time on Earth should (from the point of view of the astronaut) fall behind, and the earthly brother will be younger. However, this contradicts similar reasoning from the point of view of an earthling, in relation to which all processes in moving objects slow down. And if so, then when the traveler returns (when the clocks can be compared directly), it is unclear what will happen...

In that incorrect reasoning forgets that, in addition to time dilation, there is another effect - the relativity of simultaneity. In classical mechanics, for all observers, regardless of their movement, there is a single present. In the theory of relativity the situation is different. Such a “single present” exists only for observers who are motionless relative to each other. However, to observers moving past such a system, it represents a continuous unification of past, present and future. Observers far ahead in motion see the distant future of a stationary frame of reference, while those moving behind see the past.

All the clocks that the astronauts fly past run slower than their own. However, this does not mean that they should show less “accumulated” time! Having a slower speed, such clocks are located in the future of the earth’s reference frame, and when the astronaut gets to them, they “do not have time” to lag behind enough to compensate for this future.

To conclude the story about the twin paradox, let's tell a fairy tale.

Relativistic world - lectures on the theory of relativity, gravity and cosmology

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 a lot of in popular. 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 trip space trip, A 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 it is assumed that there is another " actor", which is not in the formulation of the paradox, nor 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 the SRT and, moreover, I do not call for abandoning it altogether, but I propose its further development, I propose a new "Special theory of relativity(STO* 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)):

Where V- absolute value of speed

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). 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 twin problem as 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:

1	2	3	4	5	6	7
0 2 4 6 8 10 12 14 16 18 20	0 1 2 3 4 5 6 7 8 9 10	0 1,2 2,4 3,6 4,8 6,0 4,8 3,6 2,4 1,2 0	0 2,2 4,4 6,6 8,8 11,0 10,8 10,6 10,4 10,2 10,0	-20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0	-20,0 -16,8 -13,6 -10,4 -7,2 -4,0 -3,2 -2,4 -1,6 -0,8 0	0 3,2 6,4 9,6 12,8 16,0 16,8 17,6 18,4 19,2 20,0

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