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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 off 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 return to his familiar segment of human history.

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 always approaches him 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 such a 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 frame is changed, 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 it were not for one extraordinary fact: when the Earth moved, the entire 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.

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

I draw your attention to the twin paradox because, contrary to what has been said above, it "still contains" unexplained problems and not only is "not resolved", but in principle cannot be resolved within the framework of Einstein's theory of relativity, i.e. this paradox is not so much "the paradox of twins in the theory of relativity" as "the paradox of Einstein's theory of relativity itself".

The essence of the twin paradox is as follows. Let P(traveler) and D(homebody) - twin brothers. P goes on a long space journey, and D stays at home. Over time P returns. Main part of the way P moves by inertia, with a constant speed (the time for acceleration, deceleration, stopping is negligibly small compared to the total travel time and we neglect it). Movement at a constant speed is relative, i.e. if P moves away (approaches, rests) relative to D, then and D also moves away (approaches, rests) relative to P- let's call it symmetry twins. Further, in accordance with SRT, the time for P, from point of view D, flows more slowly than proper time D, i.e. own travel time P less waiting time D. In this case, it is said that upon 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, with the same right consider yourself a traveler, and P homebody, and then D younger P .

The (canonical) resolution of the paradox generally accepted today is that by accelerations P cannot be neglected, i.e. its frame of reference is not inertial, forces of inertia arise from time to time in its frame of reference, and hence there is no symmetry. Moreover, 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). Thus the time P slows down as in the frame of reference D(according to SRT, when P moves by inertia), and in the frame of reference P(according to GR, when it accelerates), i.e. time dilation P becomes absolute. final conclusion : P, on return, younger D, and this is not a paradox!

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

Let's go over the canonical proofs again. P accelerates from time to time... accelerates relative to what? Only relative to the other twin(there is simply nothing else. However, in all canonical reasoning default the existence of one more "actor" is assumed, which is not present either in the formulation of the paradox or in Einstein's theory - absolute space, and then P accelerates with respect 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 the traveler twin is accelerating (moving away, approaching or resting) relative to his brother, then the homebody brother, in the same way, is accelerating (moving away, approaching or resting) relative to his traveler brother, - symmetry is not violated in this case (!). No forces of inertia or gravitational fields in the frame of reference of the accelerated brother also arise due to the absence of mass in the twins. For the same reason, the general theory of relativity does not apply here either. Thus, the symmetry of the twins is not violated, and twin paradox remains unresolved . within the framework of Einstein's theory of relativity. A purely philosophical argument can be made in defense of this conclusion: kinematic paradox must be resolved kinematically , and it is useless to involve other, dynamical theories for its solution, as is done in canonical proofs. In conclusion, the twin paradox is not a physical paradox, but a paradox of our logic ( aporia such as Zeno's aporias) applied to the analysis of a specific pseudophysical situation. This, in turn, means that any arguments such as the possibility or impossibility of the technical implementation of such a journey, the possible connection between the twins through the exchange of light signals, taking into account the Doppler effect, etc., should also not be used to resolve the paradox (in particular, without sinning against logic , we can consider the acceleration time P from zero to cruising speed, turnaround time, deceleration time when approaching the Earth, arbitrarily small, even "instantaneous").

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

The latter (together with a number of similar "claims" to Einstein's SRT, see Chapter XI of our book or an annotation to it in the article "Mathematical Principles of Modern Natural Philosophy" on this site) inevitably leads to the need to revise the special theory of relativity. I do not consider my work as a refutation of SRT and, moreover, I do not call for abandoning it at all, but I propose its further development, I propose a new "The Special Theory of Relativity"(SRT * - new edition)", in which, in particular, the "twin paradox" simply does not exist as such (for those who have not yet read the article "Special" relativity, I inform you that in the new special theory of relativity, time slows down, only when the moving inertial system approaching to the still, and time is accelerating when the moving reference frame removed from the stationary one, 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 traveling twin, no paradoxes. Travelers of the future may not be afraid, upon their return, to get into the distant future of the Earth!). Two fundamentally new theories of relativity have also been constructed, which have no analogues - ""Special general" theory of relativity(SOTO)" and "Quater Universe"(the model of the Universe as "an independent theory of relativity"). The article ""Special" theories of relativity" is published on this site. I dedicated this article to the upcoming 100th anniversary of the theory of relativity . I invite you to comment on my ideas, as well as on the theory of relativity in connection with its 100th anniversary.

Myasnikov Vladimir Makarovich [email protected]
September 2004

Supplement (Added October 2007)

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

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

Let me remind you that in SRT*, unlike Einstein's SRT, time slows down only when the moving reference frame approaches the fixed one, and accelerates when the moving reference 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 refine the concept of an inertial frame of reference, which takes into account the inseparable unity of space and time in SRT*. I define an 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 neighborhood, all points of which are determined from the reference point and whose space is homogeneous and isotropic. But the inseparable unity of space and time necessarily requires that the reference point, fixed in space, be also fixed in time, in other words, the reference point in space must also be the reference point of time.

So, I consider two fixed frames of reference associated with D: fixed frame of reference at the moment of launch (frame of reference seeing off D) and a fixed frame of reference at the moment of finish (the frame of reference meeting D). A distinctive feature of these reference systems is that in the reference frame seeing off D time flows from the starting point to the future, and the path traveled by the rocket from P grows, regardless of 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 meeting is approaching, and the path of the rocket from 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. I remind you that I consider the problem of twins as a logical problem ( aporia such as Zeno's aporias) under pseudophysical conditions of kinematics, i.e. I think that P moves all the time at a constant speed, relying on acceleration during acceleration, deceleration, etc. negligible (zero).

Two twins P(traveler) and D(homebody) discussing the upcoming flight on Earth P to the star Z located at a distance L from the Earth, and back, at a constant speed V. Estimated flight time, from start on Earth to finish on Earth, for P v its frame of reference equals T=2L/V. But in reference system seeing off D P is removed and, consequently, its flight time (time of waiting for it on Earth) is equal to (see (!!)), and this time is much less T, i.e. waiting time is less than flight time! Paradox? Of course not, since this completely fair conclusion "remained" in reference system seeing off D . Now D meets P already in another reference system 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 up. No contradictions!

I propose to consider a specific (of course, mental) "experiment", scheduled for time for each twin, and in any frame of reference. To be specific, let the star Z removed from the earth at a distance L= 6 light years. Let it go P on a rocket flies back and forth at a constant speed V = 0,6 c. Then his own flight time T = 2L/V= 20 years. We also calculate and (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 reception of signals on Earth, their analysis and comparison with theory.

All measurement data of time points are given 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 reference frame. These moments fix the time intervals from the moment of launch, or the readings of the clock on the rocket, which is set to "zero" at the moment of launch. The control points of time determine the moments of sending a signal to the Earth on the rocket. 2. The same control points in time, but in reference frame seeing off twin(where "zero" is also set at the time of rocket launch). They are determined by (!!) taking into account . 3. Distances from the rocket to the Earth in light years at the control points in time or the propagation time of the corresponding signal (in years) from the rocket to the Earth 4. in reference frame seeing off twin. Defined as a control point in time in the reference frame of the accompanying twin (column 2 3 ). 5. Same checkpoints in time, but now in reference frame meeting twin. The peculiarity of this reference system is that now the "zero" of time is determined at the moment of the rocket finish, and all control points of time are in the past. We attribute a minus sign to them, and taking into account the invariability of the direction of time (from past to future), we change their sequence in the column to the opposite one. The absolute values ​​of these moments of time are found from the corresponding values in reference frame seeing off twin(column 2 ) by multiplying by (see (!!!)). 6. The moment of reception on Earth of the corresponding signal in reference frame meeting twin. Defined as a control point in time in reference frame meeting twin(column 5 ) plus the corresponding propagation time of the signal from the rocket to the Earth (column 3 ). 7. Real times of signal reception on Earth. The fact is that D stationary in space (on Earth), but moving in real time, and at the time of receiving the signal, he is already not in reference frame seeing off twin, but in reference frame point in time signal reception. How to define this moment of real time? The signal, by condition, propagates at the speed of light, which means that two events A = (Earth at the time of receiving the signal) and B = (point in space where the rocket is located at the time 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 simultaneous, because ∆x = cΔt , where Δx is the spatial distance between events, and Δt is temporal, i.e. the propagation time of the signal from the rocket to the Earth (see the definition of simultaneity in "Special" relativity theory, 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 reference frame seeing off twin(column 2 ) should be multiplied by and then added the corresponding signal propagation time (column 3 ). The above is true for any control point in time, including the final one, i.e. end of journey 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 precisely what the actual coincidence of the columns indicates. 6 and 7 .

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

October 2007

Twin paradox

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

For some time, the "twin paradox" attracted almost no attention. In 1956-1959, Herbert Dingle published a number of papers arguing that the known explanations for the "paradox" were incorrect. Despite the fallacy of Dingle's argument, his work has generated numerous discussions in scientific and popular science journals. As a result, a number of books have appeared on this subject. Of the Russian-language sources, it is worth noting books, as well as an article.

Most researchers do not consider the “twin paradox” to be a demonstration of the contradiction of the theory of relativity, although the history of the appearance of certain explanations of the “paradox” and giving it new forms does not stop to this day.

Classification of paradox explanations

A paradox similar to the “twin paradox” can be explained using two approaches:

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

The first approach depends on the details of the formulation of the paradox. In the sections " The 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.

As part of the second approach, the calculations of the clock readings of each of the brothers are carried out both from the point of view of a homebody (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 frames of reference. In this case, the stages of accelerated motion are considered negligible compared to the total flight time. Sometimes a third inertial frame of reference is introduced, moving towards the traveler, with the help of which the readings of his watch are “transmitted” to his homebody brother. In chapter " Signal exchange"The simplest calculation based on the Doppler effect will be given.

The second group includes calculations that take into account the details of accelerated motion. In turn, they are divided on the basis of the use or non-use of Einstein's theory of gravity (GR) in them. 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 changes in the rate of time in it. In the second method, non-inertial reference systems are described in flat space-time and the concept of a 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

At the same time, the shorter the moment of acceleration, the greater it is, and as a result, the difference in the speed of the clock on Earth and the spacecraft, if it is removed from the Earth at the moment of change in speed, is greater. Therefore, acceleration can never be neglected.

Of course, the ascertainment of the asymmetry of the brothers does not in itself explain why it is the traveler's watch that should slow down, and not the homebody's. In addition, misunderstanding often arises:

“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?”

To better understand the causes of asymmetry and the consequences they lead to, 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 "fixed" frame of reference associated with the homebody, there are clocks running synchronously (in this frame). Then the following chain of reasoning is possible, as if “proving” the inconsistency of SRT conclusions:

1. The traveler, flying past any clock that is stationary in the homebody system, observes their slow running.
2. The slower pace of the clock means that they accumulated the readings will lag behind the readings of the traveler's watch, and during a long flight - arbitrarily strongly.
3. Having stopped quickly, the traveler must still observe 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.

So why would the traveler actually observe his clock lagging behind that of the “stationary” system, despite the fact that all such clocks are running slower from his point of view? The simplest explanation within 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 homebody are at different points in space and cannot directly compare their watches. Therefore, as above, we will assume that along the trajectory of the traveler in the “immobile” system associated with the homebody, there are identical, synchronously running clocks that the traveler can observe during the flight. Thanks to the synchronization procedure in the "immobile" reference system, a single time is introduced, which determines at the moment the "present" of this system.

After the start, the traveler "transfers" to an inertial reference frame , moving relatively "stationary" with a speed . This point in time is taken by the brothers as the initial one. Each of them will watch the other brother's watch slowing down.

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

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

Therefore, although all the clocks in a fixed frame of reference that the traveler flies by are slower from his point of view, from this it does not follow that they will fall behind his watch.

At time t, the farther ahead the "stationary" clock is, the greater its reading from the traveler's point of view. When he reaches those hours, they won't be far enough behind to make up for the initial time difference.

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

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

In this way:

despite the fact that all specific clocks in the system are slower from the point of view of the observer at , different clocks along its path will show the time elapsed.

The difference in the rate of the clock and - the effect is relative, while the values ​​​​of the current readings and at one spatial point - are absolute. Observers who are in different inertial frames of reference, 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 (braking quickly), nothing will change, and he will fall into the “future” of the system. Naturally, after the stop, the pace of his clock and the clock in will become the same. However, the traveler's clock will show less time than the system's clock at the stopping point. Due to the uniform time in the system, the traveler's watch will lag behind all watches, 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 on returning, the traveler is younger than his brother-homebody.

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

who changes his frame of reference, he turns out to be younger.

Signal exchange

Calculation of time dilation from the position of each brother can be done by analyzing the signal exchange between them. Although the brothers, being at different points in space, cannot directly compare the readings of their watches, they can transmit “exact time” signals using light pulses or video transmission of the clock image. It is clear that in this case they observe not the “current” time on the brother’s clock, 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 the 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 speed included in the frequency change ratio is relative source and receiver speeds.

Consider a situation in which the brothers transmit to each other every second (by their watches) the exact time signals. Let's do the calculation from the traveller's point of view first.

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. The video feed from Earth appears to be slower. After rapid braking and stopping, the traveler ceases to move away from earthly signals, and their period immediately turns out to be equal to his second. The pace of the video broadcast becomes "natural", although, due to the finiteness of the speed of light, the traveler still observes the "past" of his brother. Turning around and accelerating, the traveler begins to “run into” the signals coming towards him and their frequency increases. "Brother's movements" on the video broadcast from that moment begin to look accelerated for the traveler.

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

and when approaching, on the contrary, more:

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

in exact accordance with the time dilation formula.

Homebody calculation

Somewhat different arithmetic for a homebody. While his brother is moving away, he also registers an increased period of accurate time transmitted by the traveler. However, unlike the brother, the homebody observes such a slowdown longer. The flight time for a distance in one direction is according to earth clocks. The stay-at-home will see the traveler's braking and turning after the additional time required for the light to travel the distance from the turning point. Therefore, only after the time from the start of the journey, the homebody will register the accelerated work of the approaching brother's clock:

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

Therefore, the number of "seconds" received from the traveler, before the moment of his turn (according to the observations of the homebody) is equal to:

The stay-at-home receives signals with an increased frequency over time (see the figure above), and receives the traveler's "seconds":

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

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

Geometric interpretation

, where is the hyperbolic arcsine

Consider a hypothetical flight to the star system Alpha Centauri, distant from Earth at a distance of 4.3 light years. If time is measured in years, and distances in light years, then the speed of light is equal to one, and the unit acceleration of light year / year² is close to the acceleration of gravity and is approximately equal to 9.5 m / s².

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

In 64 years of proper time, a spacecraft with unit acceleration could potentially make a trip (returning to Earth) to the Andromeda galaxy, 2.5 million light years away. years . On Earth, during such a flight, about 5 million years will pass. By developing twice as much acceleration (to which a trained person can quite get used to under certain conditions and using a number of devices, for example, suspended animation), one can even think about an expedition to the visible edge of the Universe (about 14 billion light years), which will take astronauts about 50 years; however, returning from such an expedition (after 28 billion years according to earth clocks), its participants run the risk of not finding alive not only the Earth and the Sun, but even our Galaxy. Based on these calculations, a reasonable radius of accessibility for interstellar expeditions with a return 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 suggests that planetary systems are found near a fairly large proportion of stars, so astronauts will have something to explore in this radius (for example, the planetary systems ε Eridanus and Gliese 581).

Traveler's calculation

To carry out the same calculation from the position of the traveler, it is necessary to set the metric tensor corresponding to its non-inertial frame of reference . Relative to this system, the traveler's speed is zero, so the time on his clock is

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

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

where are the Christoffel symbols, expressed in terms of the metric tensor. For a given metric tensor of a non-inertial frame of reference, these equations allow us to find the trajectory of the homebody's clock in the traveler's frame of reference. Its substitution into the formula for proper time gives the time interval that has passed according to the “stationary” clock:

where is the coordinate velocity of the earth clock.

A similar description of non-inertial reference systems is possible either with the help of Einstein's theory of gravity, or without reference to the latter. Details of the calculation within the framework of the first method can be found, for example, in the book of Fock or Möller. The second method is considered 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 that 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 the homebody. If the duration of the stages of accelerated motion is much less than the duration of uniform flight, then the result of more general calculations coincides with the formula obtained in the framework of inertial frames of reference.

conclusions

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

The calculation of the time dilation value from the position of each brother can be performed both within the framework of elementary calculations in SRT, and with the help of 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 twin paradox is often also called the very conclusion of the theory of relativity that one of the twins will age more than the other. Although this situation is unusual, there is no inherent contradiction in it. Numerous experiments on lengthening the lifetime of elementary particles and slowing down the rate of macroscopic clocks during their movement confirm the theory of relativity. This gives grounds to assert that the time dilation described in the story of the twins will also occur in the actual implementation of this thought experiment.

see also

Notes

Sources

1. Einstein A. « On the electrodynamics of moving bodies", Ann. d. Phys., 1905 b. 17, s. 89, Russian translation in “Einstein A. Collection of scientific papers in four volumes. Volume 1. Works on the theory of relativity 1905-1920. Moscow: Nauka, 1965.
2. Langevin P. « L'evolution de l'espace et du temps". Scientia 10:31-54. (1911)
3. Laue M. (1913)" Das Relativit\"atsprinzip Wissenschaft (No. 38) (2 ed.). (1913)
4. Einstein A. « Dialogue on objections to the theory of relativity”, Naturwiss., 6, pp. 697-702. (1918). Russian translation "A. Einstein, Collection of scientific papers, vol. I, M., Science (1965)
5. Pauli V. - « Theory of relativity» M.: Nauka, 1991.
6. Dingle N. " Relativity and space travel", Nature 177, 4513 (1956).
7. Dingle H. « A possible experimental test of Einstein's Second postulate", Nature 183, 4677 (1959).
8. Coawford F. « Experimental verification of the clock-paradox in relativity", Nature 179, 4549 (1957).
9. Darvin S., " The clock paradox in relativity", Nature 180, 4593 (1957).
10. Boyer R., " The clock paradox and general relativity", Einstein's collection, "Science", (1968).
11. Campbell W., " The clock paradox", Canada. Aeronaut. J.4, 9, (1958)
12. Frey R., Brigham V., " Paradox of the twins, Amer. J Phys. 25.8 (1957)
13. Leffert S., Donahue T., " Clock paradox and the physics of discontinuous gravitational fields, Amer. J Phys. 26, 8 (1958)
14. McMillan, E. The „clock-paradox“ and Space travel, Science, 126, 3270 (1957)
15. Romer R., " Twin paradox in special relativity". amer. J Phys. 27, 3 (1957)
16. Schild, A. The clock paradox in relativity theory, Amer. Math. Mouthly 66, 1, 1-8 (1959).
17. Singer S., " Relativity and space travel', Nature 179.4567 (1957)
18. Skobeltsyn D. V., " The twin paradox in relativity”,“ Science ”, (1966).
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20. Terletsky Ya. P. " Paradoxes of the Theory of Relativity”, M.: Science (1965)
21. Ugarov V. A. - “ Special theory of relativity» M.: «Science», (1977)

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 the 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:

1. Careful consideration of existing logical constructions for contradictions and identification of logical errors in the chain of reasoning.
2. 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 motion 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 consider 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 consistent 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.