Einstein's twin paradox. Twin paradox

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Well, today we will consider perhaps the most famous of the paradoxes of relativity, which is called the "twin paradox".
I say right away that there is actually no paradox, but it stems from a misunderstanding of what is happening. And if everything is understood 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 is obtained and what logical errors lead to it. And then we will move on to the subject part, in which we will look at the mechanics of what happens with a paradox.

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

Remember the joke about Zhora Batareikin, when a colonel was sent to follow Zhora, and a lieutenant colonel was sent to follow the colonel? We need imagination to imagine ourselves in the place of a 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, scientifically speaking). So, even if the observer flies after the light at a speed of 2/3 of 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. Light flies forward at a speed of 300,000 km / s, and an observer flies after it, at a speed of 200,000 km / s. We see that the distance between the observer and the light increases ( in the original, the author has a typo - approx. Quantuz) at a speed of 100,000 km/s, but the observer himself does not see this, but sees the same 300,000 km/s. How can it be so? The only (almost! ;-) reason for such a phenomenon may be that the observer is slowed down. He moves slowly, breathes slowly and slowly measures his speed on a slow watch. As a result, he perceives a removal at a speed of 100,000 km/s as a removal at a speed of 300,000 km/s.

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

Description of the experiment

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

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

But it doesn't work like that.

As a teenager, Kostya, let's call him a space brother, gets into a rocket and goes to a star located several tens of light years from Earth.
The flight is made at near-light speed and therefore the way there and back takes sixty years.

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

Relativity Prediction

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

However, since the space brother was always on the move, his time passed more slowly, so, upon his return, he will appear to have aged only 30 years. One twin will be older than the other!

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

Let's look at the logic of prediction again. Suppose an earthly brother is inseparably watching the cosmic one.

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

For continuous observation, it is necessary that the space brother, for example, send faxes to Earth every month (by radio, at the speed of light) with his image, and the earthly brother would hang them on the calendar, taking into account the transmission delay. It would turn out that at first the brother hangs up his photograph on earth, and hangs the photograph of his brother of the same time 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 journey, in the first quarter of the journey, in the last quarter of the journey, at the end of the journey. And because of this, the backlog will constantly accumulate. Only during the space brother's turn, the moment 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 backlog will still be. Consequently, at the time of the return of the cosmic brother, the lag will persist, which means that it will already remain forever.

As you can see, there are no logical errors here. However, the conclusion looks very surprising. But there's nothing to be done: we live in an amazing world. This conclusion was repeatedly confirmed, both for elementary particles that lived longer if they were in motion, and for the most ordinary, only very accurate (atomic) clocks that went on a space flight and then it was found that they lag behind laboratory ones by a fraction seconds.

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

Seeming contradiction

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

But another question arises. After all, movement is relative! Therefore, we can assume that the space brother did not fly anywhere, but remained motionless all the time. But instead of him, an earthly brother flew on a journey, along with the planet Earth itself and everything else. And if so, it means that the cosmic brother should grow old more, 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 twin 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 figure out why we didn’t take this into account, because of which a contradiction arose?

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

I said in the very first issue that not all relativity that seems to exist actually exists. For example, it might seem that if a space brother accelerates away from the Earth, then this is tantamount to the fact that he remains in place, and the Earth itself accelerates away from him. But it's not. Nature does not agree with this. For some reason, nature creates overload for the one who accelerates: he is pressed to the chair. And for someone who does not accelerate, it does 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 away from the Earth and return to it only speeding up, turning around and slowing down. In all these cases, the space brother experienced overloads.

What is the conclusion? The logical conclusion is simple: we have no right to declare that the brothers are equal. Therefore, the arguments about the dilation of time are correct only from the point of view of one of them. What? Of course, earthly. Why? Because we did not think about overloads and imagined everything as if they did not exist. For example, we cannot assert that the speed of light remains constant under conditions of g-forces. Therefore, we cannot assert that time dilation occurs under congestion conditions. Everything that we have asserted - we have asserted for the case of the absence of overloads.

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

What is an inertial frame of reference (ISO)

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

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

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

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

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

Part last: Kostya's story

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

To guess before this, you need to pay attention to the following point: during the journey of the space brother there are sections on which he moves evenly and rectilinearly. Suppose, at the start, the brother accelerates with great force so that it reaches cruising speed in 1 day. After that, it flies evenly for many years. Then, in the middle of the way, it also rapidly turns around in one day and flies back evenly again. At the end of the journey, he very sharply, in one day, slows down.

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 seats, and the ship is made of alien steel.

What will happen?

In the very first moment of the flight, as we know, the ages of the brothers are equal. During the first half of the flight, it occurs inertially, which means that the time dilation rule applies to it. That is, the space brother will see that the earth 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'll use a 10 year old picture with a "+5" added to it.

A similar result is obtained from the analysis of the end of the path. 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 also proceeded inertially, which means that the appearance of the world from Kostya's point of view corresponds to our conclusions about time dilation. Consequently, 10 years before the end of the flight, when the space brother is 30 years old, he will conclude that the earthly one is already 65, because before the end of the flight, when the ratio is 40/70, he will age twice as slowly.

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

I have placed a summary of observations of the space brother below.

As you can see, the space brother has a discrepancy. Throughout the first half of the journey, he observes that the earthly brother is aging slowly and barely breaks away from the initial age of 10 years. Throughout the second half of the flight, he watches how the earthly brother is barely pulling himself up to the age of 70 years.

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.

In fact, we will not continue to darken and wonder what is happening there. We will simply directly and honestly draw the conclusion that follows with inevitability. If an instant before the reversal the earthly brother was 17.5 years old, and after the reversal it became 52.5, then this means nothing more than the fact that 35 years have passed for the earthly brother during the reversal of the cosmic brother!

findings

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

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

We 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.

Note. Quantuz: The author also gave a link to an additional explanation of the flash animation twin paradox. You can try following the link to the web archive where this article is carefully preserved. Recommended for a deeper understanding. See you on the pages of our cozy.

What was the reaction of world famous scientists and philosophers to the strange, new world of relativity? She was different. Most physicists and astronomers, embarrassed by the violation of "common sense" and the mathematical difficulties of the general theory of relativity, kept a prudent silence. But scientists and philosophers capable of understanding the theory of relativity greeted it with joy. We have already mentioned how quickly Eddington realized the importance of Einstein's achievements. Maurice Schlick, Bertrand Russell, Rudolf Kernap, Ernst Cassirer, Alfred Whitehead, Hans Reichenbach and many other eminent philosophers were the first enthusiasts who wrote about this theory and tried to find out all its consequences. Russell's The ABCs of Relativity was first published in 1925, but it remains one of the best popular expositions of relativity to this day.

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

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

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

The most varied objections were put forward against the theory of relativity. One of the earliest and most persistent objections was made to a paradox, first mentioned by Einstein himself in 1905 in his paper on special relativity (the word "paradox" is used to denote something opposite to the conventional, but logically consistent).

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

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

As long as the cosmic route is limited by the solar system and takes place at a relatively low speed, this time difference will be negligible. But at great distances and at speeds close to the speed of light, the "time contraction" (as this phenomenon is sometimes called) will increase. It is not unbelievable that, over time, a way will be discovered by which a spacecraft, by slowly accelerating, can achieve speeds only slightly less than the speed of light. This will make it possible to visit other stars in our Galaxy, and possibly even other galaxies. So, the twin paradox is more than just a living room puzzle; someday it will become a daily routine for space travelers.

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

The answer depends on the speed of the ship.

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

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

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

You immediately noticed that there are a variety of attractive opportunities. A forty-year-old scientist and his young laboratory assistant fell in love with each other. They feel that the age difference makes their wedding impossible. Therefore, he goes on a long space journey, moving at a speed close to the speed of light. He returns at the age of 41. Meanwhile, his girlfriend on Earth had become a thirty-three-year-old woman. Probably, she could not wait for the return of her beloved for 15 years and married someone else. The scientist cannot bear this and goes on another long journey, especially since he is interested in finding out the attitude of subsequent generations to one theory he created, whether they confirm it or refute it. He returns to Earth at the age of 42. The girlfriend of his past years had died long ago, and what was worse, there was nothing left of his theory, so dear to him. Insulted, he sets out on an even longer journey to return at the age of 45 to see the world that has lived for several millennia. It is possible that, like the traveler in Wells' novel The Time Machine, he will find that humanity has degenerated. And this is where he "runs aground." Wells' "time machine" could move in both directions, and our lone scientist will have no way to go back to the segment of human history familiar to him.

If such time travel becomes possible, then quite unusual moral questions will arise. Would it be illegal, for example, for a woman to marry her own great-great-great-great-great-great-grandson?

Please note: this sort of time travel bypasses all logical traps (that scourge of science fiction), such as being able to go into the past and kill your own parents before you were born, or slip into the future and shoot yourself with a bullet in the forehead. .

Consider, for example, the situation with Miss Kat from the well-known joke rhyme:

A young lady named Kat

Moved much faster than light.

But it always got in the wrong place:

You rush quickly - you will come to yesterday.

Translation by A. I. Baz

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

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

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

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

It is always possible to choose a moving object as a fixed frame of reference without violating any laws of nature. When the Earth is taken as the reference frame, the astronaut makes a long journey, returns and finds that he has become younger than his homebody brother. And what happens if the frame of reference is connected with the spacecraft? Now we must consider that the Earth has made a long journey and returned back.

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

Dingle would like to draw the conclusion from this: either the age of the twins must be assumed to be exactly the same at the end of the journey, or the principle of relativity must be abandoned.

Without performing any calculations, it is not difficult to understand that there are others besides these two alternatives. It is true that all motion is relative, but in this case there is one very important difference between the relative motion of an astronaut and the relative motion of a couch potato. The homebody is motionless relative to the universe.

How does this difference affect the paradox?

Let's say an astronaut goes to visit planet X somewhere in the galaxy. His journey takes place at a constant speed. The homebody's clock is linked to the Earth's inertial frame of reference, and its readings match those of all other clocks on Earth because they are all stationary with respect to each other. The astronaut's watch is connected to another inertial frame of reference, to the ship. If the ship were constantly heading in the same direction, there would be no paradox due to the fact that there would be no way to compare the readings of both clocks.

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

From one point of view, one could say that the forces of inertia created by the acceleration "cause" the astronaut's clock to slow down; from another point of view, the occurrence of acceleration simply reveals a change in the frame of reference. As a result of such a change, the world line of the spacecraft, its path on the graph in four-dimensional space - time Minkowski changes so that the total "proper time" of the return trip is less than the total proper time along the homebody twin's world line. When the reference system changes, acceleration is involved, but only special theory equations are included in the calculation.

Dingle's objection still holds, since exactly the same calculations could be made under the assumption that the fixed reference frame is connected to the ship and not to the Earth. Now the Earth goes on its way, then it comes back, changing the inertial frame of reference. Why not do the same calculations and, on the basis of the same equations, show that time on Earth is behind? And these calculations would be correct, if there were not one extraordinary importance of the fact: when the Earth moved, the whole Universe would move along with it. If the Earth rotated, the Universe would also rotate. This acceleration of the universe would create a powerful gravitational field. And as already shown, gravity slows down the clock. Clocks on the Sun, for example, tick less frequently than those on Earth, and less frequently on Earth than those on the Moon. After doing all the calculations, it turns out that the gravitational field created by the acceleration of space would slow down the clocks in the spacecraft compared to the earth by exactly the same amount as they slowed down in the previous case. The gravitational field, of course, did not affect the earth clock. The Earth is motionless relative to space, therefore, no additional gravitational field appeared on it.

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

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

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

And now an interesting question can be asked.

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

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

Since the traveling twin's clock slowing down can be seen as a gravitational phenomenon, any experiment that shows time slowing down under the influence of gravity is an indirect confirmation of the twin paradox. Several such confirmations have been made in recent years with a remarkable new laboratory method based on the Mössbauer effect. The young German physicist Rudolf Mössbauer in 1958 discovered a method for making "nuclear clocks" that measure time with inconceivable accuracy. Imagine a clock “ticking five times a second, and other clocks ticking so that after a million million ticks they are only one-hundredth of a tick behind. The Mössbauer effect can immediately detect that the second clock is running slower than the first!

Experiments using the Mössbauer effect showed that time near the foundation of a building (where the gravity is greater) flows somewhat more slowly than on its roof. As Gamow remarked: “A typist working on the first floor of the Empire State Building ages more slowly than her twin sister working under the very roof.” Of course, this difference in age is imperceptibly small, but it is there and can be measured.

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

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

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

Notes:

Building in New York with 102 floors. - Note. translation.

8 The Twin Paradox

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

The answer depends on the speed of the ship.

If such time travel becomes possible, then quite unusual moral questions will arise. Would it be illegal, for example, for a woman to marry her own great-great-great-great-great-great-grandson?

Consider, for example, the situation with Miss Kat from the well-known joke rhyme:

A young lady named Kat

Moved much faster than light.

But it always got in the wrong place:

You rush quickly - you will come to yesterday.

Translation by A. I. Baz

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

Dingle would like to draw the conclusion from this: either the age of the twins must be assumed to be exactly the same at the end of the journey, or the principle of relativity must be abandoned.

How does this difference affect the paradox?

And now an interesting question can be asked.

From the author's book

8. The Twin Paradox What was the reaction of world famous scientists and philosophers to the strange, new world of relativity? She was different. Most physicists and astronomers, embarrassed by the violation of "common sense" and the mathematical difficulties of the general theory

Otyutsky Gennady Pavlovich

The article considers the existing approaches to the consideration of the twin paradox. It is shown that although the formulation of this paradox is connected with the special theory of relativity, the general theory of relativity is involved in most attempts to explain it, which is not methodologically correct. The author substantiates the proposition that the very formulation of the "twin paradox" is initially incorrect, because it describes an event that is impossible within the framework of the special theory of relativity. Address of the article: otm^.agat^a.ne^t^epa^/Z^SIU/b/3b.^t!

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

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

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Philosophical Sciences

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

Otyutsky Gennady Pavlovich, Doctor of Philosophy n., professor

Russian State Social University, Moscow

oII2ku [email protected] Tai-gi

THE GEMINI PARADOX AS A LOGICAL ERROR

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

To exacerbate the “paradoxicality”, the fact is emphasized that due to the slowdown of the clock, the returning traveler must be younger than the homebody. J. Thomson once showed that an astronaut in flight to the "nearest Centauri" star will grow old (at a speed of 0.5 of s) by 14.5 years, while 17 years will pass on Earth. However, in relation to the astronaut, the Earth was in inertial motion, so the earth clock slows down, and the homebody must become younger than the traveler. The seeming violation of the symmetry of the brothers reveals the paradoxical nature of the situation.

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

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

Other authors believe that the paradox under consideration is apparent, and it does not indicate the inconsistency of SRT, but, on the contrary, is its reliable confirmation. They give complex mathematical calculations to take into account the change in the reference frame by the traveler and strive to prove that SRT does not contradict the facts. There are three approaches to substantiating the paradox: 1) identifying logical errors in reasoning that led to an apparent contradiction; 2) detailed calculations of the amount of time dilation from the positions of each of the twins; 3) inclusion in the paradox substantiation system of theories other than SRT. Explanations of the second and third groups often intersect.

The generalizing logic of "refutations" of the SRT conclusions includes four consecutive theses: 1) A traveler, flying past any clock that is motionless in the homebody's system, observes their slow running. 2) Their accumulated readings during a long flight can lag behind the readings of the traveler's watch as much as you like. 3) Having quickly stopped, the traveler observes the lag of the clock located at the “stopping point”. 4) All clocks in the “fixed” system run synchronously, so the brother’s clock on Earth will also fall behind, which contradicts the conclusion of SRT.

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The fourth thesis is taken for granted and acts as if the final conclusion about the paradoxical nature of the situation with twins in relation to SRT. The first two theses really follow logically from the postulates of SRT. However, the authors who share this logic do not want to see that the third thesis has nothing to do with SRT, since one can “quickly stop” from a speed comparable to the speed of light only by obtaining a gigantic deceleration due to a powerful external force. However, the "refuters" pretend that nothing significant is happening: the traveler still "must observe the lag of the clock located at the stopping point." But why "must observe", because the laws of SRT cease to operate in this situation? There is no clear answer, more precisely, it is postulated without evidence.

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

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

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

Similar disturbances are associated with explanations based on comparisons of the world lines of both twins. It is directly indicated that "the world line of a traveler who flew away from the Earth and returned to it is not a straight line", i.e. the situation moves from the sphere of SRT to the sphere of general relativity. But "if the twin paradox is an internal problem of SRT, then it should be solved by SRT methods, without going beyond it."

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

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

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

light (Fig. 1). All three were born at the same time at the moment the rocket passed point C. After the meeting of twins C and B, the ages of A and C can be compared using the intermediary B, which is a copy of twin A (Fig. 2).

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

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

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

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

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

The classical formulation of the "twin paradox" in the case when it correlates with SRT is an elementary logical fallacy. Being a logical fallacy, the twin paradox in its "classical" formulation cannot be an argument either for or against SRT.

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

List of sources

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

2. Born M. Space travel and the clock paradox // Uspekhi fizicheskikh nauk. 1959. T. LXIX. pp. 105-110.

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

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

5. Twin paradox [Electronic resource]. URL: https://ru.wikipedia.org/wiki/Twin_Paradox (Accessed: 03/31/2017).

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

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

8. Ya. P. Terletsky, Paradoxes of the Theory of Relativity. M.: Nauka, 1966. 120 p.

9. Thomson J.P. Foreseeable future. M.: Foreign Literature, 1958. 176 p.

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

THE TWIN PARADOX AS A LOGIC ERROR

Otyutskii Gennadii Pavlovich, Doctor in Philosophy, Professor, Russian State Social University in Moscow [email protected] en

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

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

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 STO

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 homebody, 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 acceleration moments of an object that are significant, but the fact that it falls into a different inertial frame of reference when it finds itself on Earth.

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

Explanation of the paradox

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

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

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

controversial points

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

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

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

Verification by practice

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

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

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

The physical cause of the paradox and the exchange of signals

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

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

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

Formulation of an explanation in paradoxical situations

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

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

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

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

Conclusion

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

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

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

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