What was the reaction of world famous scientists and philosophers to the strange, new world relativity? She was different. Most physicists and astronomers, confused by the violation of "common sense" and mathematical difficulties general theory relativity, maintained a prudent silence. But scientists and philosophers who were able to understand 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 outstanding philosophers were the first enthusiasts who wrote about this theory and tried to clarify all its consequences. Russell's ABC of Relativity was first published in 1925 and remains one of the best popular expositions of the theory of relativity.
Many scientists have found themselves unable to free themselves from the old, Newtonian way of thinking.
They were in many ways like the scientists of Galileo's distant days who could not bring themselves to admit that Aristotle might be wrong. Michelson himself, whose knowledge of mathematics was limited, never accepted the theory of relativity, although his great experiment paved the way for special theory. Later, in 1935, when I was a student at the University of Chicago, Professor William MacMillan, a well-known scientist, taught us an astronomy course. He openly said that the theory of relativity is a sad misunderstanding.
« We, the modern generation, are too impatient to wait for anything.", wrote Macmillan in 1927. " In the forty years since Michelson's attempt to discover the expected motion of the Earth relative to the ether, we have abandoned everything we had been taught before, created a postulate that was the most meaningless we could come up with, and created a non-Newtonian mechanics consistent with this postulate. Success achieved- an excellent tribute to our mental activity and our wit, but it is not certain that our common sense».
A wide variety of objections have been raised against the theory of relativity. One of the earliest and most persistent objections was to a paradox first mentioned by Einstein himself in 1905 in his paper on the special theory of relativity (the word “paradox” is used to mean something that is contrary to what is generally accepted, but is logically consistent).
This paradox has received a lot of attention in modern scientific literature, since the development of space flights, 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 stated as a mental experience involving twins. They check their watches. One of the twins on a spaceship makes a long journey through space. When he returns, the twins compare their watches. According to the special theory of relativity, the traveler's watch will show a slightly shorter time. In other words, time moves slower in a spaceship than on Earth.
As long as the space route is limited solar system and occurs at a relatively low speed, this time difference will be negligible. But over large distances and at speeds close to the speed of light, the “time reduction” (as this phenomenon is sometimes called) will increase. It is not implausible that in time a way will be discovered by which a spacecraft, slowly accelerating, can reach a speed only slightly less than the speed of light. This will make it possible to visit other stars in our Galaxy, and perhaps even other galaxies. So, the twin paradox is more than just a living room puzzle; it will one day become a daily occurrence 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 confidence that the astronaut will not die long before the end of the journey? Would its journey, as in so many works of science fiction, require an entire colony of men and women, generations living and dying as the ship made its long interstellar journey?
The answer depends on the speed of the ship.
If travel occurs 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, more than 2000 years. From an astronaut's point of view, in a spacecraft, if it is moving fast enough, the journey may only last a few decades!
For those readers who like numerical examples, here is the result of recent calculations by Edwin McMillan, a physicist at the University of California at Berkeley. A certain astronaut went from Earth to the spiral nebula of 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 reaching the nebula. (This is a convenient way of creating a constant gravitational field inside the ship for the entire duration of a long journey without the aid of rotation.) The return journey is accomplished in the same way. According to the astronaut's own watch, the duration of the journey will be 29 years. According to the earth's clock, almost 3 million years will pass!
You immediately noticed that a variety of attractive opportunities were arising. 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 sets off 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 became a thirty-three-year-old woman. She probably couldn’t wait 15 years for her beloved to return and married someone else. The scientist cannot bear this and sets off on another long journey, especially since he is interested in finding out the attitude of subsequent generations to one theory he created, whether they will confirm or refute it. He returns to Earth at the age of 42. The girlfriend of his past years died long ago, and, even worse, nothing remained of his theory, so dear to him. Insulted, he sets out on an even longer journey so that, returning at the age of 45, he sees a world that has already lived for several thousand years. It is possible that, like the traveler in Wells's The Time Machine, he will discover that humanity has degenerated. And here he “runs aground.” Wells's "time machine" could move in both directions, and our lone scientist would have no way to return back to his usual segment of human history.
If such time travel becomes possible, then completely unusual moral questions will arise. Would there be anything illegal about, for example, a woman marrying her own great-great-great-great-great-great-great-grandson?
Please note: this kind of time travel bypasses all the logical pitfalls (that scourge of science fiction), such as the possibility of going back in time and killing your own parents before you were born, or dashing into the future and shooting yourself with a bullet in the forehead .
Consider, for example, the situation with Miss Kate from the famous joke rhyme:
A young lady named Kat
It moved much faster than light.
But I always ended up in the wrong place:
If you rush quickly, you will come back to yesterday.
Translation by A. I. Bazya
If she had returned yesterday, she would have met her double. Otherwise it wouldn't really be yesterday. But yesterday there could not be two Miss Kats, because, going on a trip through time, Miss Kat did not remember anything about her meeting with her double that took place yesterday. So, here you have a logical contradiction. This type of time travel is logically impossible unless one assumes the existence of a world identical to ours, but moving along a different path in time (one day earlier). Even so, the situation becomes very complicated.
Note also that Einstein's form of time travel does not attribute any true immortality or even longevity to the traveler. From the point of view of a traveler, old age always approaches him at a normal speed. And only the “own 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 significant knowledge of mathematics. In the last few years, protests have resurfaced. Herbert Dingle, an English physicist, “most loudly” refuses to believe in the paradox. For many years now he has been writing witty articles about this paradox and accusing specialists in the theory of relativity of being either stupid or cunning. The superficial analysis that we will carry out, of course, will not fully explain the ongoing debate, the participants of which are quickly delving into complex equations, but it will help to understand the general reasons that led to the almost unanimous recognition by specialists that the twin paradox will be carried out exactly as I 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, no “chosen” frame of reference.
It is always possible to select a moving object as a fixed frame of reference without violating any laws of nature. When the Earth is taken as the reference system, the astronaut makes a long journey, returns and discovers that he has become younger than his stay-at-home brother. What happens if the reference frame is connected to a spacecraft? Now we must assume that the Earth 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 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 conclude from this: either it is necessary to assume that at the end of the journey the twins will be exactly the same age, or the principle of relativity must be abandoned.
Without performing any calculations, it is easy to understand that in addition to these two alternatives, there are others. 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 couch potato is motionless relative to the Universe.
How does this difference affect the paradox?
Let's say that an astronaut goes to visit Planet X somewhere in the Galaxy. Its journey takes place at a constant speed. The couch potato's clock is connected to the Earth's inertial frame of reference, and its readings coincide with the readings of all other clocks on Earth because they are all stationary in relation to each other. The astronaut's watch is connected to another inertial reference system, to the ship. If the ship always kept one direction, then no paradox would arise 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 reference system changes: instead of a reference system moving from the Earth, a system moving towards the Earth appears. With such a change, enormous inertial forces arise, since the ship experiences acceleration when turning. And if the acceleration during a turn is very large, then the astronaut (and not his twin brother on Earth) will die. These inertial forces arise, of course, because the astronaut is accelerating relative to the Universe. They do not occur on Earth because the Earth does not experience such acceleration.
From one point of view, one could say that the inertial forces created by the acceleration "cause" the astronaut's watch 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 Minkowski space-time, changes so that the total “proper time” of the journey with a return turns out to be less than the total proper time along the world line of the stay-at-home twin. When changing the reference frame, acceleration is involved, but only the equations of a special theory are included in the calculation.
Dingle's objection still stands, since exactly the same calculations could be done under the assumption that the fixed frame of reference is associated with the ship, and not with the Earth. Now the Earth sets off on its journey, then it returns back, changing the inertial frame of reference. Why not do the same calculations and, based on the same equations, show that time on Earth is behind? And these calculations would be fair if it weren’t for one extremely important fact: when the Earth moved, the entire Universe would move along with it. When the Earth rotated, the Universe would also rotate. This acceleration of the Universe would create a powerful gravitational field. And as has already been shown, gravity slows down the clock. A clock on the Sun, for example, ticks less often than the same clock on Earth, and on Earth less often than on the Moon. After all the calculations are done, it turns out that the gravitational field created by the acceleration of space would slow down the clock in the spaceship compared to the clock on earth by exactly the same amount as they slowed down in the previous case. The gravitational field, of course, did not affect the earth's clock. The Earth is motionless relative to space, therefore, no additional gravitational field arose on it.
It is instructive to consider a case in which exactly the same difference in time occurs, although there are no accelerations. Spaceship A flies past the Earth at a constant speed, heading towards planet X. As the spaceship passes the Earth, its clock is set to zero. Spaceship A continues its motion towards planet X and passes spaceship B, which is moving at a constant speed in the opposite direction. At the moment of closest approach, ship A radios to ship B the time (measured by its clock) that has passed since it passed the Earth. On ship B they remember this information and continue to move towards Earth at a constant speed. As they pass by the Earth, they report back to the Earth the time it took A to travel from Earth to Planet X, as well as the time it took B (measured by his watch) to travel from Planet X to the Earth. The sum of these two time intervals will be less than the time (measured by the earth's clock) that elapsed from the moment A passed the Earth until the moment B passed.
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. One might assume that the traveling twin went on ship A, then transferred to ship B and returned back; but this cannot be done without moving from one inertial frame of reference to another. To make such a transfer, he would have to be subjected to amazingly powerful inertial forces. These forces would be caused by the fact that his frame of reference has changed. If we wanted, we could say that inertial forces slowed down the twin's clock. However, if we consider the entire episode from the point of view of the traveling twin, connecting it with a fixed frame of reference, then the shifting space creating a gravitational field will enter into the reasoning. (The main source of confusion when considering the twin paradox is that the situation can be described from different points of view.) Regardless of the point of view taken, 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 possibility is “correct.” Does the traveling twin fly back and forth, or does the couch potato do it along with the cosmos? There is a fact: the relative motion of twins. There are, however, two different ways to talk about this. From one point of view, a change in the astronaut's inertial frame of reference, which creates inertial forces, leads to an age difference. From another point of view, the effect of gravitational forces outweighs the effect associated with the Earth's change in the inertial system. From any point of view, the homebody and the cosmos are motionless in relation to each other. So the position is completely different from different points of view, although the relativity of motion is strictly preserved. The paradoxical age difference is explained regardless of which twin is considered to be at rest. There is no need to discard the theory of relativity.
Now an interesting question may be asked.
What if there is nothing in space except 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 follow Eddington's or Dennis Sciama's view of inertia. From Eddington's point of view, yes. Ship A is accelerating relative to the space-time metric of space; ship B is not. Their behavior is asymmetrical and will lead to usual difference aged. From Skjam's point of view, no. It makes sense to talk about acceleration only in relation to other material bodies. In this case, the only objects are two spaceships. The position is completely symmetrical. And indeed, in this case it is impossible to talk about 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 turned on its rocket engines! As Sciama put it with English caution: “Life would be completely different in such a Universe!”
Since the slowing of the traveling twin's clock can be thought of as a gravitational phenomenon, any experience that shows time slowing due to gravity represents indirect confirmation of the twin paradox. IN last years Several such confirmations have been obtained using a remarkable new laboratory method based on the Mössbauer effect. In 1958, the young German physicist Rudolf Mössbauer discovered a method for making a “nuclear clock” that measures time with incomprehensible accuracy. Imagine a clock ticking five times a second, and another clock ticking so that after a million million ticks it will only be slow by one hundredth of a tick. The Mössbauer effect can immediately detect that the second clock is running slower than the first!
Experiments using the Mössbauer effect have shown that time flows somewhat slower near the foundation of a building (where the gravity is greater) than on its roof. As Gamow notes: “A typist working on the ground floor of the Empire State Building ages more slowly than her twin sister working under the roof itself.” Of course, this age difference is elusively small, but it exists and can be measured.
English physicists, using the Mössbauer effect, discovered 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 considered as a twin, continuously changing its inertial frame of reference (or as a twin, which is affected by the gravitational field, if we consider the disk to be at rest and the cosmos to be rotating). This experiment 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.
The satellite will then be returned and the clock readings will be compared with the clocks that remained on Earth. Of course, the time is quickly approaching when an 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 watch after his return to Earth will differ slightly from the readings of the nuclear clocks remaining on Earth.
However, we must always be prepared for surprises. Remember the Michelson-Morley experiment!
Notes:
A building in New York with 102 floors. - Note translation.
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 of relativity, remained prudently silent. But scientists and philosophers who were able to understand 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 outstanding philosophers were the first enthusiasts who wrote about this theory and tried to clarify all its consequences. Russell's ABC of Relativity was first published in 1925 and remains one of the best popular expositions of the theory of relativity.
Many scientists have found themselves unable to free themselves from the old, Newtonian way of thinking.
They were in many ways like the scientists of Galileo's distant days who could not bring themselves to admit that Aristotle might be wrong. Michelson himself, whose knowledge of mathematics was limited, never accepted the theory of relativity, although his great experiment paved the way for special theory. Later, in 1935, when I was a student at the University of Chicago, Professor William MacMillan, a well-known scientist, taught us an astronomy course. He openly said that the theory of relativity is a sad misunderstanding.
« We, the modern generation, are too impatient to wait for anything.", wrote Macmillan in 1927. " In the forty years since Michelson's attempt to discover the expected motion of the Earth relative to the ether, we have abandoned everything we had been taught before, created a postulate that was the most meaningless we could come up with, and created a 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».
A wide variety of objections have been raised against the theory of relativity. One of the earliest and most persistent objections was to a paradox first mentioned by Einstein himself in 1905 in his paper on the special theory of relativity (the word “paradox” is used to mean something that is contrary to what is generally accepted, but is logically consistent).
This paradox has received a lot of attention in modern scientific literature, since the development of space flights, 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 stated as a mental experience involving twins. They check their watches. One of the twins on a spaceship makes a long journey through space. When he returns, the twins compare their watches. According to the special theory of relativity, the traveler's watch will show a slightly shorter time. In other words, time moves slower in a spaceship than on Earth.
As long as the space route is limited to the solar system and occurs at a relatively low speed, this time difference will be negligible. But over large distances and at speeds close to the speed of light, the “time reduction” (as this phenomenon is sometimes called) will increase. It is not implausible that in time a way will be discovered by which a spacecraft, slowly accelerating, can reach a speed only slightly less than the speed of light. This will make it possible to visit other stars in our Galaxy, and perhaps even other galaxies. So, the twin paradox is more than just a living room puzzle; it will one day become a daily occurrence 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 confidence that the astronaut will not die long before the end of the journey? Would its journey, as in so many works of science fiction, require an entire colony of men and women, generations living and dying as the ship made its long interstellar journey?
The answer depends on the speed of the ship.
If travel occurs 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, more than 2000 years. From an astronaut's point of view, in a spacecraft, if it is moving fast enough, the journey may only last a few decades!
For those readers who like numerical examples, here is the result of recent calculations by Edwin McMillan, a physicist at the University of California at Berkeley. A certain astronaut went from Earth to the spiral nebula of 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 reaching the nebula. (This is a convenient way of creating a constant gravitational field inside the ship for the entire duration of a long journey without the aid of rotation.) The return journey is accomplished in the same way. According to the astronaut's own watch, the duration of the journey will be 29 years. According to the earth's clock, almost 3 million years will pass!
You immediately noticed that a variety of attractive opportunities were arising. 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 sets off 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 became a thirty-three-year-old woman. She probably couldn’t wait 15 years for her beloved to return and married someone else. The scientist cannot bear this and sets off on another long journey, especially since he is interested in finding out the attitude of subsequent generations to one theory he created, whether they will confirm or refute it. He returns to Earth at the age of 42. The girlfriend of his past years died long ago, and, even worse, nothing remained of his theory, so dear to him. Insulted, he sets out on an even longer journey so that, returning at the age of 45, he sees a world that has already lived for several thousand years. It is possible that, like the traveler in Wells's The Time Machine, he will discover that humanity has degenerated. And here he “runs aground.” Wells's "time machine" could move in both directions, and our lone scientist would have no way to return back to his usual segment of human history.
If such time travel becomes possible, then completely unusual moral questions will arise. Would there be anything illegal about, for example, a woman marrying her own great-great-great-great-great-great-great-grandson?
Please note: this kind of time travel bypasses all the logical pitfalls (that scourge of science fiction), such as the possibility of going back in time and killing your own parents before you were born, or dashing into the future and shooting yourself with a bullet in the forehead .
Consider, for example, the situation with Miss Kate from the famous joke rhyme:
A young lady named Kat
It moved much faster than light.
But I always ended up in the wrong place:
If you rush quickly, you will come back to yesterday.
Translation by A. I. Bazya
If she had returned yesterday, she would have met her double. Otherwise it wouldn't really be yesterday. But yesterday there could not be two Miss Kats, because, going on a trip through time, Miss Kat did not remember anything about her meeting with her double that took place yesterday. So, here you have a logical contradiction. This type of time travel is logically impossible unless one assumes the existence of a world identical to ours, but moving along a different path in time (one day earlier). Even so, the situation becomes very complicated.
Note also that Einstein's form of time travel does not attribute any true immortality or even longevity to the traveler. From the point of view of a traveler, old age always approaches him at a normal speed. And only the “own 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 significant knowledge of mathematics. In the last few years, protests have resurfaced. Herbert Dingle, an English physicist, “most loudly” refuses to believe in the paradox. For many years now he has been writing witty articles about this paradox and accusing specialists in the theory of relativity of being either stupid or cunning. The superficial analysis that we will carry out, of course, will not fully explain the ongoing debate, the participants of which are quickly delving into complex equations, but it will help to understand the general reasons that led to the almost unanimous recognition by specialists that the twin paradox will be carried out exactly as I 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, no “chosen” frame of reference.
It is always possible to select a moving object as a fixed frame of reference without violating any laws of nature. When the Earth is taken as the reference system, the astronaut makes a long journey, returns and discovers that he has become younger than his stay-at-home brother. What happens if the reference frame is connected to a spacecraft? Now we must assume that the Earth 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 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 conclude from this: either it is necessary to assume that at the end of the journey the twins will be exactly the same age, or the principle of relativity must be abandoned.
Without performing any calculations, it is easy to understand that in addition to these two alternatives, there are others. 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 couch potato is motionless relative to the Universe.
How does this difference affect the paradox?
Let's say that an astronaut goes to visit Planet X somewhere in the Galaxy. Its journey takes place at a constant speed. The couch potato's clock is connected to the Earth's inertial frame of reference, and its readings coincide with the readings of all other clocks on Earth because they are all stationary in relation to each other. The astronaut's watch is connected to another inertial reference system, to the ship. If the ship always kept one direction, then no paradox would arise 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 reference system changes: instead of a reference system moving from the Earth, a system moving towards the Earth appears. With such a change, enormous inertial forces arise, since the ship experiences acceleration when turning. And if the acceleration during a turn is very large, then the astronaut (and not his twin brother on Earth) will die. These inertial forces arise, of course, because the astronaut is accelerating relative to the Universe. They do not occur on Earth because the Earth does not experience such acceleration.
From one point of view, one could say that the inertial forces created by the acceleration "cause" the astronaut's watch 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 Minkowski space-time, changes so that the total “proper time” of the journey with a return turns out to be less than the total proper time along the world line of the stay-at-home twin. When changing the reference frame, acceleration is involved, but only the equations of a special theory are included in the calculation.
Dingle's objection still stands, since exactly the same calculations could be done under the assumption that the fixed frame of reference is associated with the ship, and not with the Earth. Now the Earth sets off on its journey, then it returns back, changing the inertial frame of reference. Why not do the same calculations and, based on the same equations, show that time on Earth is behind? And these calculations would be fair if it weren’t for one extremely important fact: when the Earth moved, the entire Universe would move along with it. When the Earth rotated, the Universe would also rotate. This acceleration of the Universe would create a powerful gravitational field. And as has already been shown, gravity slows down the clock. A clock on the Sun, for example, ticks less often than the same clock on Earth, and on Earth less often than on the Moon. After all the calculations are done, it turns out that the gravitational field created by the acceleration of space would slow down the clock in the spaceship compared to the clock on earth by exactly the same amount as they slowed down in the previous case. The gravitational field, of course, did not affect the earth's clock. The Earth is motionless relative to space, therefore, no additional gravitational field arose on it.
It is instructive to consider a case in which exactly the same difference in time occurs, although there are no accelerations. Spaceship A flies past the Earth at a constant speed, heading towards planet X. As the spaceship passes the Earth, its clock is set to zero. Spaceship A continues its motion towards planet X and passes spaceship B, which is moving at a constant speed in the opposite direction. At the moment of closest approach, ship A radios to ship B the time (measured by its clock) that has passed since it passed the Earth. On ship B they remember this information and continue to move towards Earth at a constant speed. As they pass by the Earth, they report back to the Earth the time it took A to travel from Earth to Planet X, as well as the time it took B (measured by his watch) to travel from Planet X to the Earth. The sum of these two time intervals will be less than the time (measured by the earth's clock) that elapsed from the moment A passed the Earth until the moment B passed.
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. One might assume that the traveling twin went on ship A, then transferred to ship B and returned back; but this cannot be done without moving from one inertial frame of reference to another. To make such a transfer, he would have to be subjected to amazingly powerful inertial forces. These forces would be caused by the fact that his frame of reference has changed. If we wanted, we could say that inertial forces slowed down the twin's clock. However, if we consider the entire episode from the point of view of the traveling twin, connecting it with a fixed frame of reference, then the shifting space creating a gravitational field will enter into the reasoning. (The main source of confusion when considering the twin paradox is that the situation can be described from different points of view.) Regardless of the point of view taken, 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 possibility is “correct.” Does the traveling twin fly back and forth, or does the couch potato do it along with the cosmos? There is a fact: the relative motion of twins. There are, however, two different ways to talk about this. From one point of view, a change in the astronaut's inertial frame of reference, which creates inertial forces, leads to an age difference. From another point of view, the effect of gravitational forces outweighs the effect associated with the Earth's change in the inertial system. From any point of view, the homebody and the cosmos are motionless in relation to each other. So the position is completely different from different points of view, although the relativity of motion is strictly preserved. The paradoxical age difference is explained regardless of which twin is considered to be at rest. There is no need to discard the theory of relativity.
Now an interesting question may be asked.
What if there is nothing in space except 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 follow Eddington's or Dennis Sciama's view of inertia. From Eddington's point of view, yes. Ship A is accelerating relative to the space-time metric of space; ship B is not. Their behavior is asymmetrical and will result in the usual age difference. From Skjam's point of view, no. It makes sense to talk about acceleration only in relation to other material bodies. In this case, the only objects are two spaceships. The position is completely symmetrical. And indeed, in this case it is impossible to talk about 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 turned on its rocket engines! As Sciama put it with English caution: “Life would be completely different in such a Universe!”
Since the slowing of the traveling twin's clock can be thought of as a gravitational phenomenon, any experience that shows time slowing due to gravity represents indirect confirmation of the twin paradox. In recent years, several such confirmations have been obtained using a remarkable new laboratory method based on the Mössbauer effect. In 1958, the young German physicist Rudolf Mössbauer discovered a method for making a “nuclear clock” that measures time with incomprehensible accuracy. Imagine a clock ticking five times a second, and another clock ticking so that after a million million ticks it will only be slow by one hundredth of a tick. The Mössbauer effect can immediately detect that the second clock is running slower than the first!
Experiments using the Mössbauer effect have shown that time flows somewhat slower near the foundation of a building (where the gravity is greater) than on its roof. As Gamow notes: “A typist working on the ground floor of the Empire State Building ages more slowly than her twin sister working under the roof itself.” Of course, this age difference is elusively small, but it exists and can be measured.
English physicists, using the Mössbauer effect, discovered 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 considered as a twin, continuously changing its inertial frame of reference (or as a twin, which is affected by the gravitational field, if we consider the disk to be at rest and the cosmos to be rotating). This experiment 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.
The satellite will then be returned and the clock readings will be compared with the clocks that remained on Earth. Of course, the time is quickly approaching when an 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 watch after his return to Earth will differ slightly from the readings of the nuclear clocks remaining on Earth.
From the author's book8. 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, confused by the violation of "common sense" and the mathematical difficulties of the general theory
The next famous thought experiment, the so-called twin paradox, is based on this amazing phenomenon of time dilation. Let's imagine that one of two twins goes on a long journey in a spaceship and is carried away from the Earth by an extremely high speed. Five years later he turns around and heads back. Thus the total travel time is 10 years. At home, it is discovered that the twin remaining on Earth has aged, say, 50 years. How many years younger the traveler will be than the one remaining at home depends on the speed of the flight. 50 years have actually passed on Earth, which means that the traveler twin had been on the road for 50 years, but for him the journey took only 10 years.
This thought experiment may seem absurd, but countless similar experiments have been conducted, all of which confirm the predictions of the theory of relativity. Example: ultra-precise atomic clocks fly around the Earth several times on a passenger plane. After landing, it turns out that less time has actually passed on the atomic clock on the plane than on other atomic clocks left on the ground for comparison. Since the speed of a passenger plane is much less than the speed of light, time dilation is very small - but the accuracy of atomic clocks is enough to register it. The most modern atomic clocks are so accurate that an error of one second is achieved only after 100 million years.
Another example that much better illustrates the effect of time dilation is the 15-fold increase in the lifespan of certain elementary particles - muons. Muons can be thought of as heavy electrons. They are 207 times heavier than electrons, carry a negative charge and arise in the upper layers of the earth's atmosphere under the influence of cosmic rays. Muons fly towards Earth at 99.8% the speed of light. But since their lifespan is only 2 microseconds, even at such a high speed they would have to disintegrate after 600 meters, before reaching the surface.
For us, in a resting frame of reference (Earth), muons are extremely fast-moving “decay clocks”, the lifetime of which increases by 15 times. Thanks to this, they exist for 30 microseconds and reach the surface of the Earth.
For the muons themselves, time does not stretch, but they reach the Earth. How can this be? The answer lies in another amazing phenomenon, “relativistic distance contraction,” which is also called Lorentzian. Shortening distances means that fast moving objects become shorter in the direction of travel.
In the muon reference frame at rest, the situation looks completely different: the mountain and with it the Earth approach the muons at a speed equal to 99.8% of light speed. A mountain with a height of 9000 meters, due to the reduction in distances, seems 15 times lower, and this is only 600 meters. Therefore, even with such a short lifespan - 2 microseconds - muons fall on Earth.
As we see, the main thing is from which point to consider a physical phenomenon. In the resting frame of reference “Earth”, time stretches and flows more slowly. On the contrary, in the “muons” reference frame at rest, space contracts in the direction of motion, in other words, it contracts. The distance to the earth's surface decreases from 9000 to 600 meters.
So, the constancy of the speed of light leads to two phenomena that are completely incredible from the point of view of common sense: time dilation and reduction of distances. But if we assume the speed of light to be constant and look at the formula “speed equals distance divided by time,” we can draw the following conclusion: two observers in two different inertial frames of reference, who obtained the same speed of light c as a result of measurements, will certainly receive different meanings distance and time.
Of course, it is difficult for us to accept that there is neither absolute time nor absolute space, only relative time and relative distances. However, this is due to the fact that no person has ever moved at a speed at which relativistic effects would become noticeable.
Another strange phenomenon is the so-called relativistic increase in mass. When we deal with speeds close to the speed of light, the mass of a body increases, just as time slows down or distance decreases. If the speed is 10% of light speed or more, the "relativistic effects" become so obvious that they can no longer be ignored. When the speed is equal to 99.8% of light, the mass of the body is 15 times greater than its rest mass, and when it is equal to 99.99% of light, the mass exceeds its rest mass by 700 times. If the speed is 99.9999% of the speed of light, the mass increases 700 times. So, as speed increases, the body becomes heavier, and the heavier it is, the more energy is required to accelerate it even more. As a result, the speed of light represents an upper limit that cannot be exceeded, no matter how much energy is supplied.
Of course, the queen of physical formulas, and perhaps the most famous formula in general, was also derived by Albert Einstein. It reads: E = m * c 2.
Einstein himself considered this equation to be the most important conclusion of the theory of relativity.
But what is the meaning of this formula? On the left is E, energy, on the right is mass multiplied by the squared speed of light c. It follows that energy and mass are essentially the same thing - and this is true.
As a matter of fact, one can guess this already from the relativistic increase in masses. If a body moves quickly, its mass increases. To accelerate the body, naturally, additional energy is needed.
However, the supply of energy leads not only to an increase in speed: the mass also increases at the same time. Of course, it’s difficult for us to imagine this, but this fact is 100% confirmed by experiments.
This has important applications such as generating energy through nuclear fission: a heavy uranium nucleus is split into two parts, such as krypton and barium. But the sum of the masses is somewhat less than the mass of uranium before decay. The mass difference “delta (Δ)m”, also called a mass defect, completely transforms into energy during decay. This is how electricity is generated at nuclear power plants.
The twin paradox is shrouded in the romance of interstellar travel and a fog of misinterpretations. It became widely known thanks to Paul Langevin's formulation (1911), which in a popular paraphrase reads as follows:
One twin brother remains on Earth, and the second goes on space travel at near-light speed. From the point of view of a homebody, a traveler moving relative to him has a slower passage of time. That's why upon return he will be younger. However, from the astronaut's point of view, the Earth was moving, so the stay-at-home brother should be younger.The word "paradox" has several meanings. For example, many conclusions of the theory of relativity are paradoxical, since they contradict conventional ideas. There is, of course, nothing wrong with such paradoxicality. Any new theory "unusual"and requires a change in old ideas. However, when describing the story with twins, "paradox" is synonymous with " logical contradiction"Having reasoned about the same event (the meeting of brothers) in two different ways, we get different results. Of course, in a consistent theory this should not happen.
An extensive literature is devoted to the twin paradox. The generally accepted explanation is as follows. So that the brothers can directly to compare their ages, one of them (the traveler) needs to return, and to do this, experience the stages of accelerated motion, moving to a non-inertial frame of reference. Therefore, there is no complete symmetry between the brothers. Naturally, such a removal of the paradox does not explain why the astronaut should become younger. In addition, the following objection immediately arises: “if the whole point is acceleration, then the stages of acceleration and deceleration can be made as short as desired (for each observer!) compared to arbitrarily long and symmetrical stages of uniform motion."
To this they answer that the calculation, within the framework of the general theory of relativity, gives the same answer for each brother. Of course, gravity has nothing to do with this calculation, and the differential geometry used in this case serves as a mathematical apparatus for describing non-inertial reference systems. Such calculations are absolutely correct, but the physical reasons for what happened to the brothers often turn out to be hidden.
We will begin our analysis with the remark that it is not necessary for the traveling brother to return. It is enough for him to slow down, moving into the reference system associated with the Earth. Being far away, but remaining motionless relative to each other, the brothers can easily synchronize their time and find out how their clocks (physical and biological) diverged. If you wish, you can, of course, consider a new launch of the spacecraft and its return to Earth. However, no new effects will occur, and all times will simply need to be multiplied by two. By and large, there is not even a need for an accelerated launch from Earth. One can consider the simultaneous birth of brothers in two different inertial frames of reference as they flew past each other. Leaving aside the physiological details of such a birth, we emphasize that when brothers are in different systems, but at the same spatial point, they can easily agree on the initial moment of time (the fact of their birth).
We examined this formulated story in detail in the “Time” section. As a result of the relativity of simultaneity, parts of a moving reference system located along the direction of its movement are “in the past,” and parts opposite the movement are in the future. And the further they are from the brothers’ birth point, the stronger the effect:
An astronaut flying past any “stationary” clock sees that it is moving slower than his own. However, on all such watches, those he meets on the way, he observes the future tense: 
V . Likewise, spaceport employees who pass by an astronaut see him as younger. At the same time, the “nephews of the same age” flying past the homebody brother (on the last ships of the squadron) look older than the earthling. These effects are absolute for observers of different systems located at the same spatial point, therefore will not change when stopped. To understand the twin paradox, in fact, there is no need to even consider non-inertial frames of reference! If the astronaut stops, he will “fall into the future” of the earth’s reference frame and will be younger there. In the same way, if an earthling accelerates, he will end up in the future of the astronaut system and will be younger there.
The "paradox" of the twins can be analyzed without expensive investments in the construction of spaceports. Suppose that two brothers, from the moment of separation, begin to broadcast their video images to each other. The traveler sees his brother sitting in an armchair by the fireplace, on which there is a clock. He, in turn, sees on the monitor the cockpit of a spaceship with an electronic clock above the helm, behind which sits his courageous brother-traveler. The spacecraft must reach the nearest star, distant from the Earth, and return back. Here are extracts from the spacecraft's logbook.
Travel diary. Having made a quick acceleration, I reach near-light speed. The overloads are colossal, but thanks to the latest advances in biocybernetics, I can endure them relatively easily. According to my watch, the start time of the journey coincides with the time of my stay-at-home brother. However, the frequency of the received signal from the rapidly receding Earth has noticeably decreased. My brother's movements look slow. This is understandable; the Doppler effect has not yet been canceled. The stars along the course huddled together, while behind, around our native Earth, they noticeably decreased in number and turned red. Here, too, everything is clear - aberration plus a change in frequency. The distances between the automatic beacons placed along my route have decreased, and, therefore, the flight time to the star according to my watch will be , and not, as my brother and I saw from Earth. Therefore, the travel time should be shorter 
than my brother's watch. We'll see. Speaking of my brother, the second hand on his mantel clock barely moves, and the time it shows is significantly behind mine. This result is the sum of the Doppler effect and the delay in video transmission due to the finite speed of light.
Having reached the destination of the journey, I sharply brake and take memorable photographs against the backdrop of a star. After braking, the hand on my brother’s mantel clock immediately began its natural run, although, of course, the total time that has passed since the beginning of the flight has not changed and is far behind mine. There is nothing else to do at the lonely star, so I sharply accelerate in the opposite direction. Having come to my senses after acceleration, I see that my brother’s watch has noticeably sped up, and its second hand is spinning like mad.
There is very little left to reach the Earth. During the return trip, my brother's watch managed to catch up and, moreover, overtook my chronometer. Tomorrow the braking and our long-awaited meeting. However, there is no longer any doubt that now I am the youngest brother in the family.
Let's look at the physics of the impressions described by the traveler. Let the brothers transmit exact time signals to each other every second (according to their watches). Let us assume that the accelerated movements of the spacecraft are very short (from the point of view of both brothers) compared to the time of the entire journey. While the spaceship is moving away from the Earth, each brother, due to the Doppler effect, sees a decrease in the frequency (increase in the period) of the received signals. After braking near the star, the traveler stops “running away” from earthly signals, and their period straightaway becomes equal to its second. Having turned around and accelerated, the traveler begins to “jump” at the signals coming towards him and their frequency increases (the period decreases).
According to his clock, the travel time in one direction is equal to , and the same in the opposite direction. Quantity taken "earth seconds" during the journey is equal to their frequency multiplied by time:
Therefore, when moving away from the Earth, the astronaut received significantly fewer seconds (first term), and when approaching, correspondingly, more (second term). The total number of seconds received from the Earth is greater than those transmitted to it, in exact accordance with the time dilation formula.
The arithmetic of an earthling is somewhat different. As his brother moves away, he also registers an increase in the periods of precise time transmitted from the spaceship. However, unlike his brother, the earthling observes such a slowdown longer. The flight time to the star is according to Earth clocks. An earthling will see the event of a traveler braking near a star later Extra time, required for light to travel the distance from the star. Therefore, only after the start of the journey he will see on the monitor accelerated work hours of approaching brother:
Given that the times are equal 
and , we have:
Thus, the effect of time dilation of a brother who changed his frame of reference is absolute, i.e. is the same for both brothers.
The most paradoxical thing about the twin paradox is that it is sometimes easier to explain than to formulate. This paradox is often perceived superficially, so we present the following “deep” reasoning:
Okay, let’s say the twins are not equal and the astronaut changed the frame of reference. There are no particular objections to its description based on the Doppler effect. However, this still does not remove the paradox in the following formulation. Astronaut flying by all hours, motionless in the earth's frame of reference, sees that they are moving slower than his clock. He is a “former earthling” and knows that all these watches are the same. Therefore, he must conclude that his brother’s time also flows more slowly. Time intervals, unlike ruler lengths, accumulate, and therefore, when stopped, the clock readings cannot be equal. Moreover, if the stop is very fast compared to the time of uniform motion, it cannot in any way lead to the lagging clock of the earthly brother jumping ahead of the clock of the spaceship. Therefore, time on Earth should (from the point of view of the astronaut) fall behind, and the earthly brother will be younger. However, this contradicts similar reasoning from the point of view of an earthling, in relation to which all processes in moving objects slow down. And if so, then when the traveler returns (when the clocks can be compared directly), something incomprehensible will happen...
In that incorrect reasoning forgets that, in addition to time dilation, there is another effect - the relativity of simultaneity. In classical mechanics, for all observers, regardless of their movement, there is a single present. In the theory of relativity the situation is different. Such a “single present” exists only for observers who are motionless relative to each other. However, to observers moving past such a system, it represents a continuous unification of past, present and future. Observers far ahead in motion see the distant future of a stationary frame of reference, while those moving behind see the past.
All the clocks that the astronauts fly past run slower than their own. However, this does not mean that they should show less “accumulated” time! Having a slower speed, such clocks are located in the future of the earth’s reference frame, and when the astronaut gets to them, they “do not have time” to lag behind enough to compensate for this future.
To conclude the story of the twin paradox, let's tell a fairy tale.
Relativistic world - lectures on the theory of relativity, gravity and cosmology
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Well, today we will look at perhaps the most famous of the paradoxes of relativity, which is called the “twin paradox”.
I’ll say right away that there really is no paradox, but it stems from a misunderstanding of what is happening. And if you understand everything correctly, and I assure you, this is not at all difficult, then there will be no paradox.
We will start with the logical part, where we will see how the paradox is created and what logical errors lead to it. And then we’ll move on to the subject part, in which we’ll look at the mechanics of what happens during a paradox.
First, let me remind you of our basic discussion about time dilation.
Remember the joke about Zhora Batareikin, when a colonel was sent to keep an eye on Zhora, and a lieutenant colonel to watch the colonel? We will need imagination to imagine ourselves in the place of the lieutenant colonel, that is, to watch the observer.
So, postulate of relativity states that the speed of light is the same from the point of view of all observers (in all reference systems, scientifically speaking). So, even if an observer flies after the light at a speed of 2/3 the speed of light, he will still see that the light is running away from him at the same speed.
Let's look at this situation from the outside. The light flies forward at a speed of 300,000 km/s, and the observer flies after it at a speed of 200,000 km/s. We see that the distance between the observer and the light increases ( There was a typo in the original by the author - approx. Quantuz) at a speed of 100,000 km/s, but the observer himself does not see this, but sees the same 300,000 km/s. How can this be so? The only (almost! ;-) reason for this phenomenon can be that the observer is slow. He moves slowly, breathes slowly and measures his speed slowly with a slow watch. As a result, he perceives a removal at a speed of 100,000 km/s as a removal at a speed of 300,000 km/s.
Remember another joke about two drug addicts who saw a fireball flash across the sky several times, and then it turned out that they stood on the balcony for three days, and the fireball was the sun? So this observer should be in the state of such a slow drug addict. Of course, this will only be visible to us, and he himself will not notice anything special, because all the processes around him will slow down.
Description of the experiment
To dramatize this conclusion, an unknown author from the past, perhaps Einstein himself, came up with the following thought experiment. Two twin brothers live on earth - Kostya and Yasha.
If brothers lived together on earth, they would synchronously go through the following stages of growing up and aging (I apologize for some convention):
But that's not how things happen.
While still a teenager, Kostya, let's call him a space brother, gets into a rocket and goes to a star located several tens of light years from Earth.
The flight takes place at near-light speed and therefore the round trip takes sixty years.
Kostya, whom we will call our earthly brother, is not flying anywhere, but is patiently waiting for his relative at home.
Relativity Prediction
When the space brother returns, the earthly brother turns out to be sixty years older.
However, since the space brother was constantly on the move, his time passed more slowly, therefore, upon his return, he would be only 30 years older. One twin will be older than the other!
It seems to many that this prediction is wrong and these people call this prediction itself the twin paradox. But that's not true. The prediction is absolutely true and the world works exactly like that!
Let's look at the logic of the prediction again. Let’s say that an earthly brother continuously observes the cosmic.
By the way, I have already repeatedly said that many people make a mistake here, incorrectly interpreting the concept of “observes.” They think that observation must necessarily take place with the help of light, for example, through a telescope. Then, they think, since light travels at a finite speed, everything that is observed will be seen as it was before, at the moment the light was emitted. Because of this, these people think, time dilation occurs, which is thus an apparent phenomenon.
Another version of the same misconception is to attribute all phenomena to the Doppler effect: since the cosmic brother moves away from the earthly one, each new “image frame” comes to Earth later and later, and the frames themselves, thus, follow less frequently than necessary, and entail time dilation.
Both explanations are incorrect. The theory of relativity is not so stupid as to ignore these effects. Look for yourself at our statement regarding the speed of light. We wrote there “he will still see that,” but we did not mean exactly “he will see with his eyes.” We meant “will receive as a result, taking into account all known phenomena.” Please note that the entire logic of reasoning is nowhere based on the fact that observation occurs with the help of light. And if this is exactly what you imagined all along, then re-read everything again, imagining how it should be!
For continuous observation, it is necessary that the space brother, for example, send faxes to Earth every month (by radio, at the speed of light) with his image, and the earthly brother would post them on the calendar, taking into account the transmission delay. It would turn out that first the earthly brother hangs up his photograph, and hangs up the photograph of his brother from the same time later, when it reaches him.
According to the theory, he will always see that time flows more slowly for his space brother. It will flow more slowly at the beginning of the journey, in the first quarter of the journey, in the last quarter of the journey, at the end of the journey. And because of this, the backlog will constantly accumulate. Only during the turn of the space brother, at the moment when he stops to fly back, his time will pass at the same speed as on Earth. But this will not change the final result, since the total lag will still be there. Consequently, at the time of the return of the space brother, the lag will remain and that means it will remain forever.
As you can see, there are no logical errors here. However, the conclusion looks very surprising. But there's nothing you can do about it: we live in amazing world. This conclusion has been confirmed many times, both for elementary particles, which lived longer if they were in motion, and for the most ordinary, only very accurate (atomic) clocks, which were sent into space flight and then discovered that they were fractions behind the laboratory ones seconds.
Not only the fact of the lag was confirmed, but also its numerical value, which can be calculated using formulas from one of the previous issues.
Apparent contradiction
So, there will be a lag. The space brother will be younger than the earthly one, you can be sure.
But another question arises. After all, movement is relative! Therefore, we can assume that the space brother did not fly anywhere, but remained motionless all the time. But instead of him, his earthly brother flew on the journey, along with the planet Earth itself and everything else. And if so, it means that the space brother should grow older, and the earthly brother should remain younger.
The result is a contradiction: both considerations, which should be equivalent according to the theory of relativity, lead to opposite conclusions.
This contradiction is called the twin paradox.
Inertial and non-inertial reference systems
How can we resolve this contradiction? As you know, there can be no contradictions :-)
Therefore, we must figure out what we didn’t take into account that caused the contradiction?
The very conclusion that time should slow down is impeccable, because it is too simple. Therefore, the error in reasoning must be present later, where we assumed that the brothers were equal. This means that in fact the brothers are not equal!
I already said in the very first issue that not every relativity that seems to exist in reality. For example, it may seem that if a cosmic brother accelerates away from the Earth, then this is equivalent to the fact that he remains in place, and the Earth itself accelerates, away from him. But that's not true. Nature does not agree with this. For some reason, nature creates overloads for the one who accelerates: he is pressed into the chair. And for those who do not accelerate, it does not create overloads.
Why nature does this is not important at the moment. At this moment, it is important to learn to imagine nature as correctly as possible.
So, brothers can be unequal, provided that one of them accelerates or brakes. But we have exactly this situation: you can fly away from Earth and return to it only accelerating, turning around and braking. In all these cases, the space brother experienced overloads.
What is the conclusion? The logical conclusion is simple: we have no right to declare that brothers have equal rights. Consequently, reasoning about time dilation is correct only from the point of view of one of them. Which one? Of course, earthly. Why? Because we didn’t think about overloads and imagined everything as if they didn’t exist. For example, we cannot say that under overload conditions the speed of light remains constant. Therefore, we cannot claim that time slows down under overload conditions. Everything we stated was for the case of no overloads.
When scientists got to this point, they realized that they needed a special name to describe the "normal" world, the world without overload. This description was called a description in terms of inertial reference system(abbreviated as ISO). The new description, which had not yet been created, was naturally called a description from the point of view non-inertial reference frame.
What is an inertial reference system (IRS)
It's clear that first, what we can say about ISO is a description of the world that seems “normal” to us. That is, this is the description with which we started.
In inertial frames of reference, the so-called law of inertia operates - each body, being left to itself, either remains at rest or moves uniformly and rectilinearly. Because of this, the systems were so called.
If we sit in a spaceship, car or train that moves absolutely uniformly and rectilinearly from the point of view of ISO, then inside such a vehicle we will not be able to notice the movement. This means that such a surveillance system will also be ISO.
Consequently, the second thing we can say about the ISO is that any system moving uniformly and rectilinearly relative to the ISO will also be an ISO.
What can we say about non-ISOs? For now, we can only say about them that a system moving relative to an ISO with acceleration will be a non-IFR.
Part last: Kostya's story
Now let's try to figure out what the world will look like from the point of view of our space brother? Let him also receive faxes from his earthly brother and post them on the calendar, taking into account the flight time of the fax from Earth to the ship. What will he get?
To figure this out, you need to pay attention to the following point: during the journey of the space brother, there are sections in which he moves uniformly and in a straight line. Let's say that at the start, the brother accelerates with enormous force so that he reaches cruising speed in 1 day. After that, it flies evenly for many years. Then, in the middle of the journey, it also quickly turns around in one day and flies back again evenly. At the end of the journey, he brakes very sharply, in one day.
Of course, if we calculate what speeds we need and with what acceleration we need to accelerate and turn, we get that our space brother should simply be smeared across the walls. And the walls of the spaceship themselves, if they are made of modern materials- will not be able to withstand such overloads. But that’s not what’s important to us now. Let's say Kostya has super-duper anti-g seats, and the ship is made of alien steel.
What will happen?
At the very first moment of the flight, as we know, the ages of the brothers are equal. During the first half of the flight, it occurs inertially, which means that the rule of time dilation applies to it. That is, the cosmic brother will see that the earthly one is aging twice as slow. Consequently, after 10 years of flight, Kostya will age by 10 years, and Yasha will age by only 5.
Unfortunately, I didn't draw the 15 year old twin, so I'll use the 10 year old picture with a "+5" added.
A similar result is obtained from end-of-path analysis. At the very last moment, the brothers’ ages are 40 (Yasha) and 70 (Kostya), we know this for sure. In addition, we know that the second half of the flight also proceeded inertially, which means that the appearance of the world from Kostya’s point of view corresponds to our conclusions about time dilation. Consequently, 10 years before the end of the flight, when the space brother is 30 years old, he will conclude that the earthly one is already 65, because before the end of the flight, when the ratio is 40/70, he will age twice as slow.
Again, I don't have a 65 year old design and will use a 70 year old one marked "-5".
I have posted a summary of my space brother observations below.
As you can see, the space brother has an inconsistency. Throughout the first half of the journey, he observes that his earthly brother is aging slowly and is barely breaking away from his initial age of 10 years. Throughout the second half of the flight, he watches as his earthly brother barely reaches the age of 70 years.
Somewhere between these sections, in the very middle of the flight, something must happen that “stitches” the aging process of the earthly brother together.
Actually, we won’t continue to obfuscate and guess what’s going on there. We will simply directly and honestly draw the conclusion that follows inevitably. If a moment before the reversal, the earthly brother was 17.5 years old, and after the reversal it became 52.5, then this means nothing more than the fact that during the reversal of the cosmic brother, 35 years passed for the earthly brother!
conclusions
So we saw that there is a so-called twin paradox, which consists of an apparent contradiction in which of the two twins time slows down. The very fact of time dilation is not a paradox.
We saw that there are inertial and non-inertial frames of reference, and the laws of nature that we obtained earlier applied only to inertial frames. It is in inertial systems that time dilation is observed on moving spacecraft.
We found that in non-inertial reference systems, for example, from the point of view of unfolding spaceships, time behaves even more strangely - it fast-forwards.
Note Quantuz: The author also provided a link to further explanation of the twin paradox with flash animation. You can try following the link to the web archive where this article is carefully saved. Recommended for deeper understanding. See you on the pages of our cozy little one.