Time dilation paradox. Twin paradox or clock paradox

Imaginary paradoxes of SRT. Twin paradox

Numerous discussions about this paradox are still going on 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 SRT and its falsity. For the first time, the thesis that served as the basis for the formulation of the paradox was 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 this clock, upon arrival at A, will lag behind compared to hours that remained motionless ... ".

This thesis was later proper names the clock paradox, the Langevin paradox, and the twin paradox. The last name has taken root, and at present the wording is more common not with watches, but with twins and space flights: if one of the twins flies on a spaceship to the stars, then upon returning 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 following immediately after the first one, that clocks at the equator lag behind clocks at the Earth's pole. The meanings of both theses are the same:

“... a clock with a balancer located at the earth's equator should run somewhat slower than exactly the same clock placed at the pole, but otherwise set 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 rate of the clock is affected by the instantaneous speed, which, although it continuously changes its direction (the tangential velocity of the equator), but in total they give the expected lag of the clock.

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

Most often, the paradox is usually given a simple explanation: the two frames of reference under consideration are in fact not 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. Likewise with watches.

From here it should be concluded that in SRT the "clock paradox" cannot be correctly formulated, the special theory does not make two mutually exclusive predictions. The problem was completely solved after the creation of the general theory of relativity, which solved the problem exactly and showed that, indeed, in the cases described, moving clocks lag behind: the clock of the flying twin and the clock at the equator. The “paradox of twins” and clocks is thus an ordinary problem in the theory of relativity.

The clock lag problem at the equator

We rely on the definition of the concept of "paradox" in logic as a contradiction resulting from a logically formally correct reasoning leading to mutually contradictory conclusions (Encyclopedic Dictionary), or as two opposite statements, for each of which there are convincing arguments (Logical Dictionary). From this position, the "paradox of twins, clocks, Langevin" 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 clocks at the equator completely coincides with the thesis about the lag of moving clocks. The figure shows conditionally (top view) the clock at the T1 pole and the 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 in the ISO pole, like clock T1, and therefore run in sync with them. But now we see that clock T2 clearly has a relative speed with respect to clock T3: first clock T2 is at close range from clock T3, then they move away and approach again. Therefore, from the point of view of the stationary clock T3, the moving clock T2 lags behind:

Fig.1 The clock moving around the circle lags behind the clock located in the center of the circle. This becomes more obvious if we add stationary clocks close to the trajectory of the moving clocks.

Therefore, the clock T2 also lags behind the clock T1. Now let's move the clock T3 so close to the trajectory T2 that at some initial time they will be nearby. In this case, we get the 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 on the equator T2 began to move away from the clocks T3 and after a while returned to the starting point along a closed curve:

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

This fully corresponds to the formulation of the first thesis about the clock lag, which served as the basis of the “twin paradox”. But the clocks T1 and T3 run synchronously, therefore, the clocks T2 are behind the clocks T1 as well. Thus, both theses from Einstein's work can equally serve as the basis for the formulation of the "twin paradox".

The magnitude of the clock lag in this case is determined by the Lorentz equation, into which we must substitute the tangential velocity of the moving clock. Indeed, at each point of the trajectory, the clock T2 has velocities equal in absolute value, but different in directions:

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

As these different speeds add to the equation? Very simple. Let's put our own fixed clock at each point of the T2 clock trajectory. All of these new clocks run in sync with clocks T1 and T3 because they are all 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. For an instantaneous time interval according to this clock, the clock T2 will also lag behind by an instantaneously small time, which can be calculated using the Lorentz equation. Here and below we will use the same designations for clocks and their readings:

Obviously, the upper limit of integration is the readings of the clock T3 at the moment when the clocks T2 and T3 meet again. As you can see, the clock readings T2< 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 you can see, we have obtained a solution that completely coincides with the solution of the first thesis (with an accuracy up to the values of the fourth and higher orders). For this reason, the following discussion can be seen as referring to all sorts of "twin paradox" formulations.

Variations on the "Twin Paradox"

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

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

x, y are the coordinates of the moving clock T2 in the reference frame 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 them and the stationary clock T1 is also equal to R at any time. But it is known that the locus of points equidistant from the given one 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 - the coordinates of the fixed clock T1 in the moving frame of reference;

R is the radius of the circle described by the fixed 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, in this case, too, a clock lag should occur. Obviously, in this case, on the contrary: 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 next to 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, always facing it with the same side. Observers next to the clocks T1 and T2 in their reference frames will see an object far from them at infinity always at the same angle.

Thus, an observer moving with a clock T2 must take into account the fact that his frame of reference is non-inertial 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, in relation to the stationary (according to the conditions of the experiment) clock T1, he must admit that these clocks are in a gravitational field of lesser intensity, therefore they go faster than his own, and a gravitational correction should be added to their expected readings.

On the contrary, the observer next to the stationary clock T1 states that the moving clock T2 is in the field of inertial gravity, so they go slower and the gravitational correction should be subtracted from their expected readings.

As you can see, the opinions of both observers completely coincided in that the clock T2 moving in the original sense e will lag behind. Consequently, the special theory of relativity in its "extended" interpretation makes two strictly consistent predictions, which does not give any grounds for declaring 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 an object of the special theory of relativity. But, as you know, an incorrect premise can lead to both correct and false results.

An experiment confirming SRT

It should be noted that all of these considered imaginary paradoxes 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 give a result confirming it.

In this regard, of particular interest is an experiment that, generally recognized in real conditions, showed exactly the same result as the considered thought experiment. This directly means that mathematical model 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, carried out in 1971. Four clocks, made on the basis of cesium frequency standards, were placed on two aircraft and performed trip around the world. One watch traveled to eastbound, others circled the Earth in a westerly direction. The difference in the speed of the passage of time arose due to the additional speed of the Earth's rotation, and the influence of the gravitational field at the flight altitude compared to the Earth's level was also taken into account. As a result of the experiment, it was possible to confirm the general theory of relativity, to measure the difference in the speed of clocks on board two aircraft. The results obtained 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 well studied; it is impossible to refute the impeccable mathematics of a theory by its own mathematical means, except by imperceptibly 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. P. V. Putenikhin, So, there is no more paradox (twins)! [animated diagrams - solution of the twin paradox by means of general relativity; the solution has an error due to the use of the approximate equation potential a; time axis - horizontal, distances - 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 effect of SRT on slowing down a moving clock], URL:
https://ru.wikipedia.org/wiki/Experiment_Hafele_—_Keating (Accessed 10/12/2015)

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

The special and general theories of relativity say that each observer has his own time. That is, roughly speaking, one person moves and determines one time by his watch, another person somehow moves and determines another time by his watch. Of course, if these people move relative to each other with small speeds and accelerations, they measure almost the same time. According to our watch, which we use, we are unable to measure this difference. I do not rule out that if two people are equipped with watches that measure time with an accuracy of one second during the lifetime of the Universe, then, looking somehow differently, they may see some difference in some n sign. However, these differences are weak.

Special and general relativity predict that these differences will be significant if two companions are moving relative to each other at high speeds, accelerations, or near a black hole. For example, one of them is far from the black hole, and the other is close to the black hole or some strongly gravitating body. Or one is at rest, and the other is moving at some speed relative to it or with a large acceleration. Then the differences will be significant. How big, I don't say, and this is measured in an experiment with high-precision atomic clocks. People fly on an airplane, then they bring it back, compare what the clock on the ground showed, what the clock on the plane showed, and not only. There are many such experiments, all of them are consistent with the shape predictions of general and special relativity. In particular, if one observer is at rest, and the other moves relative to him at a constant speed, then the recalculation of the clock from one to another is given by Lorentz transformations, as an example.

In the special theory of relativity, based on this, there is the so-called twin paradox, which is described in many books. It consists in the following. Just imagine that you have two twins: Vanya and Vasya. Let's say Vanya stayed on Earth, while Vasya flew to Alpha Centauri and returned. Now it is said that relative to Vanya, Vasya moved at a constant speed. His time moved more slowly. He's back, so he should be younger. On the other hand, the paradox is formulated as follows: now, on the contrary, relative to Vasya (moving at a constant speed relative to) Vanya moves at a constant speed, despite the fact that he was on Earth, that is, when Vasya returns to Earth, in theory, Vanya the clock should show less time. Which of them is younger? Some kind of logical contradiction. Absolute nonsense this special theory of relativity, it turns out.

Fact number one: you need to understand right away that Lorentz transformations can be used if you move from one inertial frame of reference to another inertial frame of reference. And this logic is that for one, time moves more slowly due to the fact that it moves at a constant speed, only on the basis of the Lorentz transformation. And in this case, we have one of the observers almost inertial - the one that is on the Earth. Almost inertial, that is, these accelerations with which the Earth moves around the Sun, the Sun moves around the center of the Galaxy, and so on - these are all small accelerations, for this problem, this can certainly be neglected. And the second should fly to Alpha Centauri. It must accelerate, decelerate, then accelerate again, decelerate - these are all non-inertial movements. Therefore, such a naive recalculation does not immediately work.

What is the right way to explain this twin paradox? It's actually quite simple to explain. In order to compare the lifetime of two comrades, they must meet. They must first meet for the first time, be at the same point in space at the same time, compare hours: 0 hours 0 minutes on January 1, 2001. Then fly apart. One of them will move in one way, his clock will somehow tick. The other will move in a different way, and his clock will tick in his own way. Then they will meet again, return to the same point in space, but at a different time in relation to the original. At the same time they will be at the same point in relation to some additional clock. The important thing is that now they can compare clocks. One had so much, the other had so much. How is this explained?

Imagine these two points in space and time where they met at the initial moment and at the final moment, at the moment of departure to Alpha Centauri, at the moment of arrival from Alpha Centauri. One of them moved inertially, we will assume for the ideal, that is, it moved in a straight line. The second of them moved non-inertially, so it moved along some kind of curve in this space and time - it accelerated, slowed down, and so on. So one of these curves has the property of extremality. It is clear that among all possible curves in space and time, the line is extreme, that is, it has an extreme length. Naively, it seems that it should have the smallest length, because in the plane, among all curves, the straight line has the smallest length between two points. In Minkowski's space and time, the metric is arranged in such a way, the method of measuring lengths is arranged in such a way, the straight line has the longest length, however strange it may sound. The straight line is the longest. Therefore, the one that moved inertially, stayed on Earth, will measure a longer period of time than the one that flew to Alpha Centauri and returned, so it will be older.

Usually such paradoxes are invented in order to disprove a particular theory. They are invented by the scientists themselves who are engaged in this field of science.

Initially when it appears new theory, it is clear that no one perceives it at all, especially if it contradicts some well-established data at that time. And people simply resist, it certainly is, they come up with all sorts of counterarguments and so on. It all goes through a difficult process. Man fights to be recognized. This is always associated with long periods of time and a lot of hassle. There are such paradoxes.

In addition to the twin paradox, there is, for example, such a paradox with a rod and a shed, the so-called Lorentz contraction of lengths, that if you stand and look at a rod that flies past you at a very high speed, then it looks shorter than it actually is in the frame of reference in which it is at rest. There is a paradox associated with this. Imagine a hangar or a through shed, it has two holes, it is of some length, no matter what. Imagine that this rod is flying at him, going to fly through him. The barn in its rest system has one length, say 6 meters. The rod in its rest system has a length of 10 meters. Imagine that their approach speed is such that in the frame of reference of the barn the rod was reduced to 6 meters. You can calculate what this speed is, but now it doesn’t matter, it is close enough to the speed of light. The rod was reduced to 6 meters. This means that in the reference frame of the shed, the rod will at some point fit entirely into the shed.

A person who is standing in a barn - a rod is flying past him - at some point will see this rod lying entirely in the barn. On the other hand, motion at a constant speed is relative. Accordingly, it can be considered as if the rod is at rest, and a barn is flying at it. This means that in the frame of reference of the bar the barn has contracted, and it has contracted by the same number of times as the bar in the frame of reference of the barn. This means that in the frame of reference of the rod, the barn was reduced to 3.6 meters. Now, in the frame of reference of the rod, there is no way for the rod to fit into the shed. In one frame of reference it fits, in another frame of reference it does not fit. Some nonsense.

It is clear that such a theory cannot be correct - it seems at first glance. However, the explanation is simple. When you see a rod and say, "It's a given length," that means you're receiving a signal from this and that end of the rod at the same time. That is, when I say that the rod fits into the barn, moving at some speed, this means that the event of the coincidence of this end of the rod with this end of the barn is simultaneously with the event of the coincidence of this end of the rod with this end of the barn. These two events are simultaneous in the frame of the barn. But you have probably heard that in the theory of relativity simultaneity is relative. So it turns out that these two events are not simultaneous in the frame of reference of the rod. It's just that at first the right end of the rod coincides with the right end of the shed, then the left end of the rod coincides with the left end of the shed after a certain period of time. This period of time is exactly equal to the time for which these 10 meters minus 3.6 meters will fly through the end of the rod at this given speed.

Most often, the theory of relativity is refuted for the reason that such paradoxes are very easily invented for it. There are many such paradoxes. There is such a book by Taylor and Wheeler "Physics of Space-Time", it is written in a fairly accessible language for schoolchildren, where the vast majority of these paradoxes are analyzed and explained using fairly simple arguments and formulas, as this or that paradox is explained within the framework of the theory of relativity.

Can you think of a way to explain each this fact, which looks simpler than the way relativity provides. However, an important property of the special theory of relativity is that it explains not every single fact, but the entire set of facts taken together. Now, if you come up with an explanation for a single fact, isolated from this entire set, let it explain this fact better than the special theory of relativity, in your opinion, but you still need to check that it explains all the other facts too. And as a rule, all these explanations, which sound more simple, do not explain everything else. And we must remember that at the moment when this or that theory is invented, this is really some kind of psychological, scientific feat. Because there are one, two or three facts at this moment. And so a person, based on this one or three observations, formulates his theory.

At that moment it seems that it contradicts everything that was known before, if the theory is cardinal. Such paradoxes are invented to refute it, and so on. But, as a rule, these paradoxes are explained, some new additional experimental data appear, they are checked whether they correspond to this theory. Also some predictions follow from the theory. It is based on some facts, it claims something, something can be deduced from this statement, obtained, and then it can be said that if this theory is true, then it must be so-and-so. Let's go and see if it's true or not. So that. So the theory is good. And so on ad infinitum. In general, it takes an infinite number of experiments to confirm a theory, but at the moment, in the area in which special and general relativity are applicable, there are no facts that disprove these theories.

The main purpose of the thought experiment called "Twin Paradox" was to refute the logic and validity of the special theory of relativity (SRT). It is worth mentioning right away that there is actually no question of any paradox, and the word itself appears in this topic because the essence of the thought experiment was initially misunderstood.

The main idea of SRT

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

The twin paradox in brief

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

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

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

Twin paradox from Einstein's theory of relativity

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

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

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

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

Explanation of the paradox

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

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

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

controversial points

There are assumptions that the moment of acceleration is significant enough to change the speed of the clock. But during numerous experimental checks it was proved that under the action of acceleration the movement of time neither speeds up nor slows down.

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

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

Verification by practice

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

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

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

The physical cause of the paradox and the exchange of signals

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

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

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

Formulation of an explanation in paradoxical situations

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

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

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

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

Conclusion

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

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

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

First, let's figure out who are twins and who are twins. Both are born to the same mother almost at the same time. But if twins can have different heights, weights, facial features and character, then the twins are almost 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 two spermatozoa fertilized the egg at the same time, or the already fertilized egg was divided in two, and each of its half began to develop into an independent fetus. In the first case, which is not difficult to guess, twins are born that are different from each other, in the second - monozygotic twins 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 observed that any stressful effect 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 the 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 constant feeling that they once had a twin who disappeared. Researchers consider this statement not as strange as it might seem at first glance. It has now been proven that at conception, much more twins develop - both identical and just twins - than are born into the world. Researchers estimate that 25 to 85% of pregnancies begin with two embryos but end with one.

Here are just two of those hundreds and thousands of examples known to physicians 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 assumed, but not the resorbed remains of the body of the twin brother. This was evidenced by the 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 soccer ball, was cut open, a terrible picture appeared before the eyes of the surprised doctors: bones, long hair, teeth, cartilage, fatty tissues, pieces of skin were sticking out from the inside ...

The fact that a significant part of the fertilized eggs, indeed, begin their development with two embryos, was also confirmed by ultrasound studies of the course of pregnancy in tens and hundreds of women. So, in 1973, the American physician Lewis Helman reported that out of 140 risky pregnancies he examined, 22 began with two embryonic bags - 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 fetus 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 this may be the natural way of removing a damaged fetus, thereby maintaining a healthy one.

Adherents of another hypothesis explain this phenomenon by the fact that multiple pregnancy is 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 turns into a singleton. Scientists went even further in their theoretical constructions, who state 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 their surviving brother. That is, at the first stages of pregnancy, a real embryonic cannibalism takes place 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 fixed much more often than in later periods. Previously, it was thought that these were early diagnostic errors. Now, judging by the above facts, this discrepancy in the statistical data has been fully explained.

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

A lot of curious features are shown by twins in adulthood. You can verify this in the following example.

The "Jima twins" were separated immediately after birth, grew up separately and became a sensation when they found each other. Both were named the same, both were married to women named Linda, whom they divorced. When both married a second time, their wives also had the same name - Betty. Everyone had a dog named Toy. Both worked as sheriff's representatives, as well as at McDonald's and at gas stations. They spent their holidays on the beach of St. Petersburg (Florida) and drove a Chevrolet. Both 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 life to the similarities and differences in the behavior of twins. On the basis of observations of twins brought up from early childhood in different families and in different environments, he came to the conclusion that heredity plays a much greater role. big role than previously thought, in the formation of personality traits, her intellect and psyche, in susceptibility to certain diseases. Many of the twins he examined, despite the significant difference in upbringing, showed very similar behavioral traits.

For example, Jack Youf and Oscar Storch, born in 1933 in Trinidad, were separated immediately after their birth. They met only once in their early 20s. They were 45 when they saw each other again at Bochard's in 1979. They both had mustaches, matching thin metal-rimmed glasses, and blue shirts with double pockets and epaulettes. Oskar, raised by a German mother and her family in the Catholic faith, joined the Hitler Youth during the Nazi era. Jack was raised in Trinidad by a 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 share the same habits. For example, both enjoyed reading out loud in the elevator just to see how others would react. Both read magazines back to front, had a stern disposition, wore a rubber band around their wrists, and flushed the toilet before using it. Strikingly similar behavior was demonstrated by other pairs of twins studied. Bridget Harrison and Dorothy Lowe, born in 1945 and separated when they were a week old, came to Bochard with watches and bracelets on one hand, two bracelets and seven rings on the other. It was later revealed that each of the sisters has a cat named Tiger, that Dorothy's son is named Richard Andrew, and Bridget's son is 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 varied, they were recorded or skipped 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 questions. The studies revealed the similarity of the twins in the manner of speaking, gesticulating, moving. It has also been found that identical twins even sleep the same way, and their sleep phases coincide. It is assumed that they can develop the same diseases.

This study of twins can be completed with the words of Luigi Geld, who said: “If one has a hole in the tooth, then the other will have it in the same tooth or will soon appear.”

Do you want to surprise everyone with your youth? Embark on a long space flight! Although, when you return, there will most likely be no one to be surprised ...

Let's analyze history two twin brothers.
One of them - a "traveler" goes on a space flight (where the speed of rockets is near light), the second - a "homebody" remains on Earth. And what is the question? - at the age of brothers!
Will they remain the same age after space travel, or will one of them (and who exactly) become older?

Back in 1905, Albert Einstein in Special Theory Relativity (SRT) was formulated relativistic time dilation effect, according to which clocks moving relative to an inertial frame of reference run slower than stationary clocks and show a shorter time interval between events. Moreover, this slowdown is noticeable at near-light speeds.

It was after the nomination of SRT by Einstein that the French physicist Paul Langevin formulated "twin paradox" (or otherwise "clock paradox"). The twin paradox (otherwise the “clock paradox”) is a thought experiment with which they tried to explain the contradictions that arose in SRT.

So, back to the twin brothers!

It should seem to the homebody that the clock of the moving traveler has a slow motion of time, therefore, when returning, it should lag behind the clock of the homebody.
And on the other hand, the Earth is moving relative to the traveler, so he believes that the homebody's clock should fall behind.

But, both brothers cannot be at the same time one older than the other!
This is where the paradox lies...

From the point of view of the “twin paradox” that existed at the time of the emergence, a contradiction arose in this situation.

However, the paradox, as such, does not really exist, since we must remember that SRT is a theory for inertial frames of reference! Ah, the frame of reference for at least one of the twins was not inertial!

At the stages of acceleration, deceleration or turnaround, the traveler experienced accelerations, and therefore, at these moments, the provisions of SRT are not applicable.

Here you have to use General Theory of Relativity, where it is proved by calculations that:

Let's get back, to the question of slowing down time in flight!
If light travels any path in time t.
Then the duration of the flight of the ship for the "homebody" will be T = 2vt / s

And for the “traveler” on the spaceship, his clock (based on the Lorentz transformation) will take only To=T times the square root of (1-v2/c2)
As a result, calculations (in general relativity) of the magnitude of time dilation from the position of each brother will show that brother-traveler will be younger than his brother-homebody.

For example, you can mentally calculate the flight to the star system Alpha Centauri, which is 4.3 light years away from Earth (a light year is the distance that light travels in a year). Let time be measured in years and distances in light years.

Let half way spaceship moves with an acceleration close to the acceleration of free fall, and slows down the other half with the same acceleration. Making the way back, the ship repeats the stages of acceleration and deceleration.

In this situation the flight time in the earth's reference system will be approximately 12 years, while according to the clock on the ship, 7.3 years will pass. The maximum speed of the ship will reach 0.95 of the speed of light.

In 64 years of proper time, the spacecraft with a similar acceleration can travel to the Andromeda galaxy (back and forth). On Earth, during such a flight, about 5 million years will pass.

The reasoning behind the story of the twins only leads to an apparent logical contradiction. With any formulation of the “paradox”, there is no complete symmetry between the brothers.

The relativity of the simultaneity of events plays an important role in understanding why time slows down precisely for a traveler who has changed his frame of reference.

Already conducted experiments on lengthening the lifetime of elementary particles and slowing down the clock during their movement confirm the theory of relativity.

This gives grounds to assert that the time dilation described in the story of the twins will also occur in the actual implementation of this thought experiment.