Imaginary paradoxes of SRT. The twins paradox

Imaginary paradoxes of SRT. The twins paradox

Numerous discussions on 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 about both the infallibility of SRT and its falsity. For the first time, the thesis, which served as the basis for the formulation of the paradox, was set forth by Einstein in his fundamental work on the special (particular) theory of relativity "To the electrodynamics of moving bodies" in 1905:

“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 (...), then these clocks upon arrival at A will lag behind in comparison with for hours that remained motionless ... ".

Later this thesis received its own names "clock paradox", "Langevin's paradox" and "twins paradox". The last name stuck, and nowadays, the wording is more common not with watches, but with twins and space flights: if one of the twins flies off in a spaceship to the stars, then on his return he is younger than his brother who remained on Earth.

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

"... a clock with a balance wheel, located at the Earth's equator, should run somewhat slower than exactly the same clock, placed at the pole, but otherwise placed in the same conditions."

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

The paradox, an apparent 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 off into space, should expect that the brother remaining on Earth will be younger than him. The same is with the clock: from the point of view of the clock at the equator, the clock at the pole should be considered moving. Thus, a contradiction arises: so which of the twins will be younger? Which clock will show the time with a lag?

Most often, the paradox is usually given a simple explanation: the two frames of reference in question are in fact not equal. The twin, which flew into space, in its flight was not always in the inertial frame of reference, at these moments it cannot use the Lorentz equations. So it is with the clock.

Hence, the conclusion should be drawn: the “clock paradox” cannot be correctly formulated in SRT, 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, the moving clock lags behind: the clock of the departing twin and the clock at the equator. The "paradox of twins" and the clock is thus an ordinary task of the theory of relativity.

The problem of the clock lag at the equator

We rely on the definition of the concept of "paradox" in logic as a contradiction, obtained as a result of logically formally correct reasoning leading to mutually contradictory conclusions (Enciplopedic 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 the clock at the equator completely coincides with the thesis about the lag of a moving clock. The figure shows conditionally (top view) a clock at the T1 pole and a clock at the T2 equator. We see that the distance between the clocks is invariable, that is, there seems to be no necessary relative speed between them, which can be substituted into the Lorentz equations. However, let's add a third clock T3. They are located in the ISO of the pole, like the clock T1, and therefore run synchronously with them. But now we see that the T2 clock clearly has a relative speed in relation to the T3 clock: first, the T2 clock is at a close distance from the T3 clock, then it recedes and approaches again. Therefore, from the point of view of the stationary clock T3, the moving clock T2 lags behind:

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

Therefore, the T2 clock also lags behind the T1 clock. Now let's move the clock T3 so close to the trajectory of T2 that at some initial moment of time it will be close. 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 at the equator of T2 began to move away from the clock T3 and after a while returned to the starting point along a closed curve:

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

This is fully consistent with the formulation of the first thesis about the clock lag, which served as the basis for the "twin paradox". But clocks T1 and T3 run synchronously, therefore, clock T2 also lagged behind clock T1. Thus, both theses from Einstein's work can equally serve as a basis for formulating the "paradox of twins".

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

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

How can these different speeds be brought into the equation? Very simple. Let's put our own stationary clock at each point of the trajectory of the T2 clock. All these new clocks run synchronously with clocks T1 and T3, since they are all in the same stationary IFR. The clock T2, each time passing by the corresponding clock, experiences a lag caused by the relative speed just past this clock. For an instantaneous interval of time according to this clock, the clock T2 will also lag behind for an instantly small time, which can be calculated using the Lorentz equation. Hereinafter, we will use the same designations for the clock and its readings:

Obviously, the upper limit of integration is the readings of the T3 clock at the moment when the T2 and T3 clocks meet again. As you can see, the readings of the 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, a solution was obtained that completely coincides with the solution of the first thesis (up to values \u200b\u200bof the fourth and higher orders). For this reason, the following discussion can be considered as referring to all kinds of formulations of the "twin paradox".

Variations on the "Twin Paradox"

The clock paradox, as noted above, means that special relativity appears to be making two mutually contradictory predictions. Indeed, as we just calculated, the clock moving along 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 motionless 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 - coordinates of the moving clock T2 in the stationary frame of reference;

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

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

x 1 2 + y 1 2 \u003d R 2

x 1, y 1 - coordinates of the motionless clock T1 in the moving one;

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

Fig. 4 From the point of view of the moving clock T2, the stationary clock T1 moves around it 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\u003e T3 \u003d T. It turns out that in fact the special theory of relativity makes two mutually exclusive predictions T2\u003e 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, striving to throw them away from the motionless clock. We, of course, believe that there are no other gravitating bodies nearby. In addition, the clock T2 moving in a circle does not rotate by itself, that is, it does not move like the Moon around the Earth, always facing it with the same side. Observers next to clocks T1 and T2 in their reference systems will see an object that is distant from them to infinity always at the same angle.

Thus, the observer moving with the 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 the clock in the gravitational field or in the 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 less 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, therefore it goes slower and the gravitational correction should be subtracted from its expected readings.

As you can see, the opinion of both observers completely coincided that the clock T2 moving in the original sense of e would lag behind. Consequently, the special theory of relativity in the "extended" interpretation makes two strictly consistent predictions, which does not give any grounds for proclaiming paradoxes. This is an ordinary task with a very specific solution. The paradox in SRT arises only if we use its position to an object that is not an object of the special theory of relativity. But, as you know, a wrong premise can lead to both correct and false results.

Experiment confirming SRT

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

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

This was the first experiment to test the lag of a moving clock, known as the Hafele-Keating experiment, in 1971. Four clocks, made on the basis of cesium frequency standards, were placed on two aircraft and traveled around the world. Some watches traveled eastward, others circled the Earth westward. The difference in the speed of the passage of time arose due to the additional speed of rotation of the Earth, while taking into account the influence of the gravitational field at the flight altitude in comparison with the level of the Earth. 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 were published in the journal Science in 1972.

Literature

1. Putenikhin PV, Three errors of anti-SRT [before criticizing the theory, it should be well studied; it is impossible to refute the impeccable mathematics of the theory and its own mathematical means, except for imperceptibly rejecting its postulates - but this is already another theory; the well-known experimental contradictions in SRT are not used - the experiments of Marinov and others - they must be repeated many times], 2011, URL:
http://samlib.ru/p/putenihin_p_w/antisto.shtml (date of treatment 10/12/2015)

2. Putenikhin PV, 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, distance - vertical], 2014, URL:
http://samlib.ru/editors/p/putenihin_p_w/ddm4-oto.shtml (date of treatment 10/12/2015)

3. Hafele-Keating's experiment, Wikpiedia, [convincing confirmation of the SRT effect on the slowing down of a moving clock], URL:
https://ru.wikipedia.org/wiki/Hafele_—_Kitinga_Experiment (date of treatment 10/12/2015)

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

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

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

Back in 1905, Albert Einstein in the Special Theory of Relativity (STR) formulated relativistic time dilation effect, according to which a clock moving relative to an inertial reference frame runs slower than a stationary clock and shows a shorter time interval between events. Moreover, this deceleration is noticeable at near-light speeds.

It was after Einstein's promotion of SRT by the French physicist Paul Langevin that "Paradox of twins" (or otherwise "paradox of hours")... The twins paradox (otherwise, the "clock paradox") is a thought experiment with the help of which they tried to explain the contradictions that have arisen in SRT.

So, back to the twin brothers!

It should seem to a couch potato that the watch of a moving traveler has a slow time course, so when returning it should lag behind the couch potato's clock.
On the other hand, the Earth is moving relative to the traveler, so he believes that the homebody's clock should lag behind.

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

From the point of view of the “paradox of twins” that existed at the time, 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 of at least one of the twins was not inertial!

At the stages of acceleration, deceleration or turning, the traveler experienced acceleration, and therefore to him at these moments provisions of SRT are inapplicable.

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

Let's go back, to the question of time dilation in flight!
If light travels any path in time t.
Then the duration of the flight of the ship for the "couch potato" will be T \u003d 2vt / s

And for a "traveler" on a spaceship, according to his clock (based on the Lorentz transformation), the total To \u003d T will be multiplied by 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 the traveling brother will be younger than his stay-at-home brother.

For example, you can mentally calculate a flight to the Alpha Centauri star system, which is 4.3 light years 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 the spacecraft move half the way with an acceleration close to the acceleration of gravity, and the other half with the same acceleration slows down. Making the return journey, the ship repeats the stages of acceleration and deceleration.

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

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

The reasoning conducted in the story with 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 simultaneity of events plays an important role in understanding why time slows down in the case of a traveler who changed his frame of reference.

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

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

Column editor

Hello dear readers!

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For this dish you need:

chicken in the form of cut and seasoned pieces (for example, thighs or legs), these are sold, they are already sprinkled with all kinds of garbage and even sometimes salted
one onion
microwave oven
utensils for the microwave

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Within 30 minutes you can do whatever you want, and then you can eat deliciously and even more than once!

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Introduction

Well, today we will consider, perhaps, the most famous of the paradoxes of relativity, which is called the paradox of twins.

I say right away that there is actually no paradox, but it stems from a misunderstanding of what is happening. And if you understand everything correctly, and this, I assure you, 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 turns out and what logical errors lead to it. And then let's move on to the subject part, in which we will look at the mechanics of what happens in a paradox.

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

Remember the anecdote about Zhora Batareikin, when the colonel was sent to follow Zhora, and the colonel was sent to follow the colonel? We need imagination to imagine ourselves in the place of the lieutenant colonel, that is, to observe 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 frames of reference, in scientific terms). So, even if the observer flies in pursuit of 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 an observer flies in pursuit of it at a speed of 200,000 km / s. We see that the distance between the observer and the light decreases 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 may be that the observer is slowed down. He moves slowly, breathes slowly and slowly measures the speed on a slow clock. As a result, he perceives removal at a speed of 100,000 km / s as removal at a speed of 300,000 km / s ...

Remember another anecdote about two drug addicts who saw a fireball fly 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-moving addict. Of course, this will only be visible to us, and he himself will not notice anything special, because all processes around him will slow down.

Experiment Description

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.


Kostya	Yasha

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

10	20	30	40	50	60	70

teenager	a difficult age	young rake	young worker	honored worker	pensioner	decrepit old man

But that's not how it goes.

As a teenager, Kostya, let's call him a space brother, gets into a rocket and goes to a star located several dozen light-years from Earth.

The flight takes place at near-light speed, and therefore the journey back and forth takes sixty years.

Kostya, whom we will call his earthly brother, does not fly anywhere, but patiently waits for his relative at home.

Predicting relativity

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

However, since the space brother was in motion all the time, his time passed more slowly, therefore, upon his return, he will be aged only 30 years. One twin will be older than the other!


Kostya	Yasha

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

Let's take another look at the prediction logic. Let us suppose that the earthly brother is inseparably watching the cosmic.

By the way, I have already said more than once that many make a mistake here by misinterpreting the concept of observing. They think that observation must necessarily take place with the help of light, for example, through a telescope. Then, they think, since light travels with a finite speed, everything that is observed will be seen as it was before, at the moment of emission of light. Because of this, these people think, time dilation occurs, which is thus an apparent phenomenon.

Another version of the same delusion is to write off all phenomena to the Doppler effect: since the cosmic brother moves away from the earthly, each new frame of the image comes to the Earth more and more later, and the frames themselves, therefore, follow less often than necessary, and entail a slowdown in time.

Both explanations are wrong. The theory of relativity isn't stupid enough to ignore these effects. See for yourself. We wrote there all the same will see that, but we did not mean to see with our eyes. We had in mind the result, taking into account all known phenomena. Note that the whole logic of reasoning is nowhere based on the fact that observation occurs with the help of light. And if you have been imagining exactly this all the time, then re-read everything again, imagining how it should be!

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

In theory, he will see all the time that the space brother's time flows more slowly. It will flow more slowly at the beginning of the path, in the first quarter of the path, in the last quarter of the path, at the end of the path. 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 go at the same speed as on Earth. But this will not change the final result, since the total lag will still be. Consequently, at the moment of the return of the space brother, the lag will remain and it means that it will remain forever.

Space brother
10	20	30	40

Earth brother
10	30	50	70

As you can see, there are no logical errors here. However, the conclusion looks very surprising. But nothing can be done about it: we live in a wonderful world. This conclusion has been repeatedly confirmed, both for elementary particles that lived more time if they were in motion, and for the most ordinary, only very accurate (atomic) clocks that were sent into space flight and then it was found that they lagged behind laboratory ones by a fraction seconds.

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

Seeming 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, the 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, the earthly brother flew on the journey, along with the planet Earth itself and everything else. And if so, it means that the cosmic brother should grow old, and the earthly brother should remain younger.

It turns out a contradiction: both considerations, which should be equivalent according to the theory of relativity, lead to opposite conclusions.

This contradiction is called the twins paradox.

Inertial and non-inertial frames of reference

How can we resolve this contradiction? As you know, there can be no contradictions 🙂

Therefore, we have to come up with something that we did not take into account, because of what a contradiction arose?

The very conclusion that time should slow down is flawless, because it is too simple. Therefore, the error in reasoning must be present later, where we assumed that the brothers are equal. So, in fact, the brothers are unequal!

I said in the very first issue that not all relativity that seems to exist in reality. For example, it may seem that if the space 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 this is not the case. Nature does not agree with this. For some reason, nature creates for the one who accelerates overload: he is pushed to the chair. And for those who do not accelerate, they do not create overloads.

Why nature does this is not important at the moment. At the 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 slows down. But we have just such a situation: you can fly off the Earth and return to it only accelerating, turning 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 claim that brothers are equal. Therefore, the reasoning about time dilation is correct only from the point of view of one of them. Which one? Of course, earthly. Why? Because we did not think about overloads and presented everything as if there were none. For example, we cannot say that under conditions of overload the speed of light remains constant. Therefore, we cannot assert that under conditions of overloads time slows down. All we have stated is that we have stated for the case of no overload.

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

What is an inertial reference system (IFR)

It's clear that firstwhat we can say about ISO is such a description of the world that seems normal to us. That is, this is the description we started with.

In inertial reference frames, 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 get into a spaceship, car or train, which move absolutely evenly and rectilinearly from the point of view of ISO, then we will not be able to notice the movement inside such a vehicle. This means that such an observation system will also be ISO.

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

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

The last part: the story of Bones

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

To guess this, you need to pay attention to the following moment: during the travel of the space brother, there are areas in which it moves evenly and rectilinearly. For example, at the start, the brother accelerates with tremendous force so that it reaches cruising speed in 1 day. After that, it flies evenly for many years. Then, in the middle of the journey, it also turns around rapidly in one day and flies back again evenly. At the end of the path, it slows down very abruptly, in one day.

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

What happens?

In the very first moment of flight, as we know, the brothers are equal in age. 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 slowly. Consequently, after 10 years of flight, Kostya will age by 10 years, and Yasha - only by 5.

Unfortunately, I didn’t draw a 15 year old twin, so I will use a 10 year old picture with a + 5.

A similar result is obtained from end-of-path analysis. At the very last moment, the ages of the brothers are 40 (Yasha) and 70 (Kostya), we know this for sure. In addition, we know that the second half of the flight was also inertial, which means that the appearance of the world from Kostya's point of view corresponds to our conclusions about time dilation. Therefore, 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.

Somewhere between these areas, in the very middle of the flight, something must be happening that sews the aging process of the earthly brother together.

We, in fact, will not continue to darken and wonder what is happening there. We will simply draw the conclusion directly and honestly, which inevitably follows. If a moment before the turn, the terrestrial brother was 17.5 years old, and after the turn it became 52.5, then this means nothing more than the fact that 35 years have passed since the turn of the cosmic brother!

conclusions

So we saw that there is the so-called paradox of twins, which consists in an apparent contradiction in which of the two twins has time slowing 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, obtained by us earlier, related only to inertial frames. It is in inertial systems that time dilation is observed on moving spaceships.

We got that in non-inertial frames of reference, for example, from the point of view of unfolding spaceships, time behaves even more strangely - it scrolls forward.

A glimpse into the twins paradox from four-dimensional spacetime can be seen in.

Dims.

The twins paradox

Then, in 1921, Wolfgang Pauli suggested a simple explanation based on the invariance of proper time.

For some time, the "twins paradox" attracted little attention. In 1956-1959, Herbert Dingle came out with a number of articles in which it was argued that the known explanations for the "paradox" are incorrect. Despite the fallacy of Dingle's argumentation, his work has caused numerous discussions in scientific and popular science journals. As a result, a number of books have appeared on this topic. From Russian-language sources, it is worth noting books, as well as an article.

Most researchers do not consider the "paradox of twins" to be a demonstration of the contradiction of the theory of relativity, although the history of the emergence of various explanations of the "paradox" and giving it new forms does not stop to this day.

Classification of explanations for the paradox

There are two approaches to explaining a paradox similar to the twin paradox:

1) Reveal the origin of a logical error in reasoning that led to a contradiction; 2) Carry out detailed calculations of the magnitude of the time dilation effect from the perspective of each of the brothers.

The first approach depends on the details of the paradox formulation. In the sections “ Simplest explanations"And" The physical cause of the paradox"Various versions of the" paradox "will be given and explanations will be given of why the contradiction does not actually arise.

In the second approach, the clock readings of each of the brothers are calculated both from the point of view of a couch potato (which is usually not difficult) and from the point of view of a traveler. Since the latter changed its frame of reference, there are various options for taking this fact into account. They can be conditionally divided into two large groups.

The first group includes calculations based on the special theory of relativity within the framework of inertial reference frames. In this case, the stages of accelerated movement are considered negligible compared to the total flight time. Sometimes a third inertial reference system is introduced, moving towards the traveler, with the help of which the readings of his clock are "transmitted" to his homebody brother. In chapter " Signal exchange»The simplest calculation based on the Doppler effect will be presented.

The second group includes calculations that take into account the details of accelerated motion. In turn, they are divided according to the use or non-use of Einstein's theory of gravity (GR). Calculations using general relativity are based on the introduction of an effective gravitational field, equivalent to the acceleration of the system, and taking into account the change in the rate of time in it. In the second method, non-inertial reference frames are described in flat space-time and the concept of the gravitational field is not involved. The main ideas of this group of calculations will be presented in the section " Non-inertial frames of reference».

Kinematic effects of SRT

In this case, the shorter the moment of acceleration, the greater it is, and as a consequence, the greater the difference in the speed of clocks on Earth and the spacecraft, if it is removed from the Earth at the moment of speed change. Therefore, acceleration can never be neglected.

Of course, the statement of the brothers' asymmetry in itself does not explain why the hours of the traveler should slow down, and not the stay-at-home. In addition, misunderstandings often arise:

Why does the violation of the equality of brothers for such a short time (stopping the traveler) lead to such a striking violation of symmetry?

In order to better understand the reasons for asymmetry and the consequences to which they lead, it is necessary to once again highlight the key premises that are explicitly or implicitly present in any formulation of the paradox. To do this, we will assume that along the trajectory of the traveler in the "stationary" frame of reference associated with the couch potato, synchronously running (in this frame) clock is located. Then the following line of reasoning is possible, as if "proving" the inconsistency of the SRT conclusions:

A traveler, flying past any clock stationary in the couch potato system, observes their slow motion.
A slower pace of the watch means that it is accumulated the readings will lag behind the traveler's watch readings, and during a long flight - as much as you like.
Having stopped quickly, the traveler must still observe the lag of the clock located at the “stopping point”.
All clocks in the "stationary" system run synchronously, therefore, the brother's clock on Earth will also lag behind, which contradicts the conclusion of the SRT.

So, why would a traveler actually observe the lag of his clock from the clock of the "stationary" system, despite the fact that all such clocks, from his point of view, run slower? The simplest explanation in the framework of SRT is that it is impossible to synchronize all clocks in two inertial frames of reference. Let's take a closer look at this explanation.

The physical cause of the paradox

During the flight, the traveler and the couch potato are at different points in space and cannot compare their watches directly. Therefore, as above, we will assume that along the trajectory of the traveler in the “stationary” system associated with the couch potato, there are identical synchronously running clocks that can be observed by the traveler during the flight. Thanks to the synchronization procedure, a single time is introduced in the "stationary" frame of reference, which determines the "present" of this system at the moment.

After the start, the traveler "passes" into an inertial reference system moving relatively "stationary" with speed. This moment in time is taken by the brothers as the initial one. Each of them will observe the slow speed of the clock of the other brother.

However, the single "present" of the system for the traveler ceases to exist. The frame of reference has its own "present" (a set of synchronized clocks). For the system, the further along the traveler's movement the parts of the system are, the more distant "future" (from the point of view of the "present" of the system) they are.

The traveler cannot observe this future directly. This could be done by other observers of the system, located in front of the movement and having time synchronized with the traveler.

Therefore, although all the clocks in a stationary frame of reference, past which the traveler flies by, go from his point of view more slowly, from this it does not followthat they would lag behind his watch.

At a moment in time, the further ahead along the course the "stationary" watch is, the greater is its indication from the point of view of the traveler. When it reaches this clock, it will not be behind enough time to compensate for the initial time discrepancy.

Indeed, let us set the traveler's coordinate in the Lorentz transformations equal. The law of its motion relative to the system has the form. The time elapsed after the start of the flight is less in the system than in:

In other words, the time on the traveler's clock lags behind the system clock. At the same time, the clock, past which the traveler flies by, is motionless at:. Therefore, their pace of travel looks slow for the traveler:

In this way:

despite the fact that all specific hours in the system run slower from the point of view of the observer at, different hours along its trajectory will show the time that has gone ahead.

The difference in the pace of the clock and is a relative effect, while the values \u200b\u200bof the current readings and at one spatial point are absolute. Observers located in different inertial reference systems, but "at the same" spatial point, can always compare the current readings of their clocks. The traveler, flying past the clock of the system, sees that they have gone ahead. Therefore, if the traveler decides to stop (quickly braking), nothing will change, and he will find himself in the "future" of the system. Naturally, after stopping, the pace of his clock and clock in will become the same. However, the traveler's clock will show a shorter time than the system clock at the stopping point. Due to the uniform time in the system, the traveler's clock will lag behind all clocks, including his brother's. After stopping, the traveler can return home. In this case, the entire analysis is repeated. As a result, both at the point of stopping and turning, and at the starting point upon returning, the traveler is younger than his stay-at-home brother.

If, instead of stopping the traveler, the couch potato accelerates to his speed, the latter will "fall" into the "future" of the traveler's system. As a result, the "couch potato" will be younger than the "traveler". In this way:

he who changes his frame of reference is also younger.

Signal exchange

The calculation of the time dilation from the position of each brother can be carried out by analyzing the exchange of signals between them. Although the brothers, being in different points in space, cannot directly compare the readings of their clocks, they can transmit "exact time" signals using pulses of light or video broadcasting of the clock image. It is clear that in this case they observe not the “current” time on the clock of their brother, but the “past”, since the signal takes time to propagate from the source to the receiver.

When exchanging signals, the Doppler effect must be taken into account. If the source moves away from the receiver, then the frequency of the signal decreases, and when it approaches, it increases:

where is the natural frequency of radiation, and is the frequency of the signal received by the observer. The Doppler effect has a classical component and a relativistic component directly related to time dilation. The rate included in the frequency change ratio is relative source and receiver speed.

Consider a situation in which brothers transmit exact time signals to each other every second (according to their watch). Let's first calculate from the perspective of a traveler.

Traveler's calculation

While the traveler is moving away from the Earth, he, due to the Doppler effect, registers a decrease in the frequency of the received signals. Video feed from Earth looks slower. After rapid deceleration and stopping, the traveler ceases to move away from the earth signals, and their period immediately turns out to be equal to his second. The rate of video transmission becomes "natural", although, due to the finiteness of the speed of light, the traveler still observes his brother's "past". Having turned around and accelerated, the traveler begins to "run up" on the signals coming towards him and their frequency increases. From this moment, the "brother's movements" on the video broadcast begin to look accelerated for the traveler.

The flight time according to the traveler's clock in one direction is equal, and the same in the opposite direction. amount taken "earth seconds" during the journey is equal to their frequency multiplied by time. Therefore, when moving away from the Earth, the traveler will receive significantly less "seconds":

while approaching, on the contrary, more:

The total number of "seconds" received from the Earth during the time is greater than those transmitted to it:

in exact accordance with the time dilation formula.

Calculation of a couch potato

A somewhat different arithmetic for a couch potato. While his brother is retiring, he also records an extended period of precise time as transmitted by the traveler. However, unlike his brother, the couch potato sees such a slowdown. longer... The flight time for a distance in one direction is according to the earth clock. The couch potato will see the braking and turning of the traveler after the additional time required for the light to travel the distance from the turning point. Therefore, only after a time from the start of the trip, the couch potato will register the accelerated work of the clock of the approaching brother:

The time of movement of light from the turning point is expressed in terms of the travel time of the traveler to it as follows (see figure):

Therefore, the number of "seconds" received from the traveler until the moment of his turn (according to the observations of a couch potato) is equal to:

The couch potato receives signals with an increased frequency over a period of time (see the picture above), and receives the traveler's "seconds":

The total number of received "seconds" for the time is equal to:

Thus, the ratio for the reading of the clock at the time of the meeting between the traveler () and the homebody brother () does not depend on from whose point of view it is calculated.

Geometric interpretation

, where is the hyperbolic arcsine

Consider a hypothetical flight to the Alpha Centauri star system, 4.3 light years from Earth. If time is measured in years and distances are in light years, then the speed of light is unity, and unit acceleration sv.y / yr² is close to the acceleration of gravity and is approximately equal to 9.5 m / s².

Let the spacecraft move half the way with a unit acceleration, and the other half with the same acceleration decelerates (). Then the ship turns around and repeats the stages of acceleration and deceleration. In this situation, the flight time in the earth's frame of reference will be approximately 12 years, while 7.3 years will pass by the clock on the ship. The maximum speed of the ship will reach 0.95 times the speed of light.

In 64 years of proper time, a spacecraft with a unit acceleration can potentially make a trip (returning to Earth) to the Andromeda galaxy, 2.5 million light sv. years old . On Earth, about 5 million years will pass during such a flight. Developing twice as much acceleration (to which a trained person may well get used to subject to a number of conditions and the use of a number of devices, for example, suspended animation), one can even think of an expedition to the visible edge of the Universe (about 14 billion light years), which will take astronauts about 50 years; however, having returned from such an expedition (after 28 billion years according to the earth's clock), its participants risk not to be found alive, not only the Earth and the Sun, but even our Galaxy. Based on these calculations, a reasonable radius of accessibility for interstellar return missions does not exceed several tens of light years, unless, of course, any fundamentally new physical principles of movement in space-time are discovered. However, the discovery of numerous exoplanets gives reason to believe that planetary systems are found in a fairly large proportion of stars, so astronauts will have something to explore in this radius (for example, the planetary systems ε Eridani and Gliese 581).

Traveler's calculation

To carry out the same calculation from the perspective of a traveler, it is necessary to specify a metric tensor corresponding to his non-inertial frame of reference. The traveler's speed is zero relative to this system, so the time on his watch is

Note that it is the coordinate time and in the traveler's system it differs from the time of the homebody's frame of reference.

The earth clock is free, so it moves along the geodesic defined by the equation:

where are Christoffel symbols expressed in terms of the metric tensor. For a given metric tensor of a non-inertial frame of reference, these equations make it possible to find the trajectory of the couch potato's clock in the traveler's frame of reference. Its substitution into the formula for proper time gives the time interval passed by the "stationary" clock:

where is the coordinate velocity of the earth clock.

A similar description of non-inertial frames of reference is possible either with the help of Einstein's theory of gravity, or without reference to the latter. Details of the calculation within the first method can be found, for example, in the book of Fock or Möller. The second method is discussed in Logunov's book.

The result of all these calculations shows that from the point of view of the traveler, his watch will lag behind the watch of a stationary observer. As a result, the difference in travel time from both points of view will be the same, and the traveler will be younger than a couch potato. If the duration of the stages of accelerated motion is much less than the duration of a uniform flight, then the result of more general calculations coincides with the formula obtained within the framework of inertial reference systems.

conclusions

The reasoning conducted in the story with the twins only leads to an apparent logical contradiction. With any formulation of the "paradox", there is no complete symmetry between the brothers. In addition, the relativity of the simultaneity of events plays an important role in understanding why time slows down in the case of a traveler who changed his frame of reference.

The calculation of the magnitude of the slowing down of time from the position of each brother can be performed both within the framework of elementary calculations in the SRT, and using the analysis of non-inertial frames of reference. All these calculations are consistent with each other and show that the traveler will be younger than his homebody brother.

The twins paradox is also often called the very conclusion of the theory of relativity that one of the twins will grow old stronger than the other. Although this situation is unusual, there is no internal contradiction in it. Numerous experiments on lengthening the lifetime of elementary particles and slowing down the rate of macroscopic clocks as they move confirm the theory of relativity. This suggests that the time dilation described in the story with the twins will also occur during the actual implementation of this thought experiment.

Notes

Sources

Einstein A. " To the electrodynamics of moving bodies", Ann. d. Phys., 1905 b. 17, s. 89, Russian translation into “Einstein A. Collected scientific works in four volumes. Volume 1. Works on the theory of relativity 1905-1920. " Moscow: Nauka, 1965.
Langevin P. " L'evolution de l'espace et du temps". Scientia 10: 31-54. (1911)
Laue M. (1913) " Das Relativit \\ "atsprinzip". Wissenschaft (No. 38) (2nd ed.). (1913)
Einstein A. " Dialogue on objections to the theory of relativity", Naturwiss., 6, pp. 697-702. (1918). Russian translation “A. Einstein, Collected scientific works ", vol. I, M.," Science "(1965)
Pauli V. - " Theory of relativity"Moscow: Nauka, 1991.
Dingle N. " Relativity and Space travel"Nature 177, 4513 (1956).
Dingle H. " A possible experimental test of Einstein's Second postulate"Nature 183, 4677 (1959).
Coawford F. " Experimental verification of the clock-paradox in relativity"Nature 179, 4549 (1957).
Darvin S., " The clock paradox in relativity"Nature 180, 4593 (1957).
Boyer R., " The clock paradox and general relativity", Einstein's collection," Science ", (1968).
Campbell W., " The clock paradox", Canad. Aeronaut. J.4, 9, (1958)
Frey R., Brigham V., " Paradox of the twins", Amer. J. Phys. 25, 8 (1957)
Leffert S., Donahue T., " Clock paradox and the physics of discontinuous gravitational fields", Amer. J. Phys. 26, 8 (1958)
McMillan E., " The „clock-paradox“ and Space travel", Science, 126, 3270 (1957)
Romer R., " Twin paradox in special relativity". Amer. J. Phys. 27, 3 (1957)
Schild, A. " The clock paradox in relativity theory", Amer. Math. Mouthly 66, 1, 1-8 (1959).
Singer S., " Relativity and space travel", Nature 179.4567 (1957)
Skobeltsyn D. V., " The twin paradox in relativity"," Science ", (1966).
Goldenblat I. I., " Time paradoxes in relativistic mechanics", M." Science ", (1972).
Terletskiy Ya. P. " The paradoxes of the theory of relativity", M .: Science (1965)
Ugarov V.A. - " Special theory of relativity"M .:" Science ", (1977)

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 time dilation effect. For the convenience of speech and for "greater clarity" the clock paradox also formulated as "the twins paradox". I also use this formulation. Initially, the paradox was actively discussed in the scientific literature, and especially in the popular one. Currently, the twins' 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 paradox of twins because, contrary to what was said above, it "still contains" unexplained problems and not only "not resolved", but in principle cannot be resolved within the framework of Einstein's theory of relativity, i.e. this is a paradox not so much of the "paradox of twins in the theory of relativity" as "the paradox of Einstein's theory of relativity itself."

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

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

This, we repeat, is the canonical resolution of the twins paradox. However, in all such arguments 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"). With regard to our twins, this means that, firstly, there are only two twins and there is NOTHING MORE, in particular, there is no absolute space, and secondly, twins (read - Einstein's clock) have no mass. it necessary and sufficient conditions formulations of the twin paradox. Any additional conditions lead to "another paradox of twins". Of course, one can formulate and then resolve "other paradoxes of twins", 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 takes place only in absolute space, or only under the condition that the clock has mass, etc. As you know, there is nothing like this in Einstein's theory.

Let's go over the canonical proofs again. P accelerating from time to time ... Accelerating relative to what? Only relative to the other twin (there is simply nothing else. However, in all canonical reasoning default the existence of one more "actor" is assumed, which is absent neither in the formulation of the paradox, nor in Einstein's theory - absolute space, and then P accelerates relative to this absolute space, while D rests relative to the same absolute space - there is a violation of symmetry). But kinematically acceleration is relatively the same as speed, i.e. if a twin traveler accelerates (moves away, approaches or rests) relative to his brother, then a homebody brother, in the same way, accelerates (moves away, approaches or rests) relative to his traveling brother, - symmetry is not broken in this case (!)... No inertial forces or gravitational fields in the frame of reference of the accelerated brother arise also due to the lack of mass in the twins. For the same reason, general relativity is inapplicable here. Thus, the symmetry of the twins is not broken, and the twins 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 useless to involve other, dynamic theories to solve it, as is done in canonical proofs. In conclusion, I note that the paradox of twins is not a physical paradox, but the paradox of our logic ( aporia such as Zeno's aporias), applied to the analysis of a specific pseudo-physical situation. This, in turn, means that any arguments such as the possibility or impossibility of the technical realization of such a journey, the possible connection 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, do not sin against logic , we can count the acceleration time P from zero to cruising speed, turn time, braking time when approaching the Earth, arbitrarily small, 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, v. 1, p. 8), Einstein writes: “We must pay attention to the fact that all our judgments, in which time plays a role, are always judgments about simultaneous events (Einstein's italics) "(We, in a sense, go beyond Einstein, assuming the simultaneity of events necessary condition reality events.) With regard to our twins, this means the following: regarding each of them his brother always simultaneous with him (that is, it 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 absolutely must be the same when they are at the same point in space. The latter means that their age was the same at the moment of the beginning of the journey (they are twins), when they were at the same point in space, then their age mutually changed during the journey of one of them, depending on its speed (no one canceled the theory of relativity), when they were in different points in space, and again became the same at the end of the journey, when they again found themselves in the same point in space .. Of course, they both aged, but the aging process could take place in them differently, from the point of view of one or the other, but ultimately, they aged the same. Note that this new situation for twins is still symmetrical. Now, taking into account the last remarks, the paradox of twins becomes qualitatively different - fundamentally insoluble 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 urge to abandon it altogether, but I propose its further development, I propose a new "Special theory of relativity (SRT * - new edition) ", in which, in particular, the" paradox of twins "simply does not exist as such (for those who have not yet read the article" Special "theories of relativity", I inform you that in the new special theory of relativity, time slows downonly when the movable inertial system approaching to motionless, and time is acceleratingwhen the moving frame of reference removed 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 slowing down of time in the second half (approaching the Earth), and there are no slow aging of the twin traveler, no paradoxes. Travelers of the future may not be afraid to return to the distant future of the Earth!). Two fundamentally new theories of relativity, which have no analogues, were also built - "The" special general "theory of (SOTO) "and "Quater Universe" (model of the Universe as "independent theory of relativity"). The article "Special" theories of relativity "is published on this site. I have dedicated this article to the upcoming 100th anniversary of the theory of relativity ... I invite you to speak about my ideas, as well as about the theory of relativity in connection with its 100th anniversary.

Myasnikov Vladimir Makarovich [email protected]
September 2004

Supplement (Added Oct 2007)

The "paradox" of twins in SRT *. No paradoxes!

So, the symmetry of twins is unavoidable in the problem of twins, which in Einstein's SRT leads to an insoluble 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. It is these conclusions that are obtained in the SRT * - a new edition.

Let me remind you that in SRT *, unlike Einstein's SRT, time slows down only when the moving frame of reference approaches the stationary one, and accelerates when the moving frame moves away from the fixed 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 frame of reference, which takes into account the indissoluble unity of space and time in SRT *. I define the inertial frame of reference (see Theory of relativity, new approaches, new ideas. Or Space and ether in mathematics and physics.) As a reference point and its vicinity, all points of which are determined from the reference point and whose space is homogeneous and isotropic. But the indissoluble unity of space and time necessarily requires that a reference point fixed in space be also fixed in time, in other words, a reference point in space must also be a reference point for time.

So, I consider two fixed frames of reference associated with D : stationary frame of reference at the moment of start (reference frame seeing off D) and a stationary frame of reference at the time of finish (frame of reference meeting D). A distinctive feature of these frames of reference is that in the frame of reference seeing off D time flows from the reference 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 moves away from D both in space and in time. In the frame of reference meeting D - time flows from the past to the starting point and the moment of the meeting approaches, and the path of the rocket with P to the reference point decreases, i.e. in this frame of reference P approaching D both in space and in time.

Let's go back to our twins. Let me remind you that I consider the problem of twins as a logical problem ( aporia type of Zeno's aporias) in pseudo-physical conditions of kinematics, i.e. I think that P moves all the time at a constant speed, relying on time for acceleration during acceleration, deceleration, etc. negligible (zero).

Twins P (traveler) and D (couch potato) discuss the upcoming flight on Earth P to the star Z 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 in its frame of reference equally T \u003d 2L / V... But in frame of reference seeing off D P is removed and, therefore, its flight time (the time it waits on Earth) is equal to (see (!!)), and this time is much less T, i.e. waiting time less flight time! Paradox? Of course not, since this perfectly fair conclusion "remained" in frame of reference seeing off D ... Now D meets P already in another frame of reference meeting D , and in this frame of reference P is approaching, and its waiting time is equal, in accordance with (!!!), i.e. own flight time P and own waiting time D match. 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 \u003d 6 light years. Let it go P on a rocket flies back and forth at a constant speed V = 0,6 c... Then its own flight time T = 2L / V \u003d 20 years old. Let us also calculate (see (!!) and (!!!)). We also agree that with an interval of 2 years, at control points in time, P will send a signal (at the speed of light) to Earth. The "experiment" consists in recording the time of receiving signals on Earth, analyzing them and comparing them with theory.

All measurement data of points 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. Control points in time (in years) in the rocket frame... These moments record the time intervals from the moment of launch, or the readings of the clock on the rocket, which are set to "zero" at the moment of launch. Control moments of time determine the moments of sending a signal to the Earth on the rocket. 2. The same control points in time, but in the frame of reference accompanying twin (where "zero" is also set at the time of rocket launch). They are determined by (!!) given. 3. Distances from the rocket to the Earth in light years at control times or the propagation time of the corresponding signal (in years) from the rocket to the Earth 4. in the frame of reference accompanying twin... It is defined as a control moment in the frame of reference of the accompanying twin (column 2 3 ). 5. Same milestones, but now in the frame of reference meeting twin... The peculiarity of this frame of reference is that now "zero" of time is determined at the moment of the rocket's finish, and all control points of time are in the past. We assign them a minus sign, and taking into account the invariability of the direction of time (from the past to the future), we change their sequence in the column to the opposite. The absolute values \u200b\u200bof these points in time are found from the corresponding values in the frame of reference accompanying twin (column 2 ) by multiplying by (see (!!!)). 6. The moment of reception on Earth of the corresponding signal in the frame of reference meeting twin... Defined as a milestone in the frame of reference meeting twin (column 5 ) plus the corresponding propagation time of the signal from the rocket to the Earth (column 3 ). 7. Real time moments of signal reception on Earth. The fact is that D is motionless in space (on Earth), but moves in real time, and at the moment of receiving the signal it is no longer in the frame of reference accompanying twinbut in the frame of reference moment in time signal reception... How to determine this moment in real time? The signal, by condition, propagates at the speed of light, which means that two events A \u003d (Earth at the time of receiving the signal) and B \u003d (the point in space where the rocket is at the moment of sending the signal) (I remind you that an event in space is time is called a point at a certain point in time) are simultaneoussince Δx \u003d c Δt, where Δx is the spatial distance between events, and Δt is the temporal distance, i.e. the propagation time of the signal from the rocket to the Earth (see the definition of simultaneity in the "Special" theory of relativity, formula (5)). And this, in turn, means that D, with equal right, can consider itself both in the frame of reference of event A and in the frame of reference of event B. In the latter case, the rocket approaches, and in accordance with (!!!), all time intervals (up to this control moment) in the frame of reference accompanying twin (column 2 ) should be multiplied by and then add the corresponding signal propagation time (column 3 ). The above is true for any control point in time, including the final one, i.e. travel finish moment P... This is how the column is calculated 7 ... Naturally, the real moments of signal reception do not depend on the method of their calculation, this is exactly what the actual coincidence of the columns indicates 6 and 7 .

The considered "experiment" only confirms the main conclusion that the own flight time of the twin traveler (his age) and the own waiting time of the stay-at-home twin (his age) coincide and there are no contradictions! "Contradictions" arise only in some reference systems, for example, in the frame of reference accompanying twin, but this does not affect the final result in any way, since in this frame of reference the twins, in principle, cannot meet, whereas in the frame of reference meeting twinwhere the twins actually meet, there are no more contradictions. I repeat: Travelers of the future may not be afraid, upon returning to Earth, to get to its distant future!

October 2007