Twin paradox (thought experiment): explanation. The Twin Paradox What is the Twin Paradox?

Imaginary paradoxes of SRT. Twin paradox

Putenikhin P.V.
[email protected]

There are still numerous discussions on this paradox in the literature and on the Internet. Many of its solutions (explanations) have been proposed and continue to be proposed, from which conclusions are drawn both about the infallibility of STR and its falsity. The thesis that served as the basis for the formulation of the paradox was first stated by Einstein in his fundamental work on the special (particular) theory of relativity “On the electrodynamics of moving bodies” in 1905:

“If there are two synchronously running clocks at point A and we move one of them along a closed curve at a constant speed until they return to A (...), then these clocks, upon arrival at A, will lag behind compared to for hours, remaining motionless...”

Subsequently, this thesis received proper names"Clock Paradox", "Langevin's Paradox" and "Twins' Paradox". The latter name stuck, and nowadays the formulation is more often found not with watches, but with twins and space flights: if one of the twins flies on a spaceship to the stars, then upon return he turns out to be younger than his brother who remained on Earth.

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

“... a clock with a balancer, located on the earth’s equator, should go somewhat slower than exactly the same clock placed at the pole, but otherwise placed in the same conditions.”

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

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

Most often, a simple explanation is usually given to the paradox: the two reference systems under consideration are not actually equal. The twin that flew into space was not always in the inertial frame of reference during its flight; at these moments it cannot use the Lorentz equations. It's the same with watches.

Hence the conclusion should be drawn: the “clock paradox” cannot be correctly formulated in STR; the special theory does not make two mutually exclusive predictions. The problem received a complete solution after the creation of the general theory of relativity, which solved the problem exactly and showed that, indeed, in the described cases, moving clocks lag behind: the clock of the departing twin and the clock at the equator. The “paradox of twins” and clocks is thus an ordinary problem in the theory of relativity.

Clock lag problem at the equator

We rely on the definition of the concept of “paradox” in logic as a contradiction resulting from logically formally correct reasoning, leading to mutually contradictory conclusions (Enciplopedic Dictionary), or as two opposing statements, for each of which there are convincing arguments (Dictionary of Logic). From this position, the “twin, clock, Langevin paradox” is not a paradox, since there are no two mutually exclusive predictions of the theory.

First, let us show that the thesis in Einstein's work about the clock at the equator completely coincides with the thesis about the lag of moving clocks. The figure shows conventionally (top view) a clock at the pole T1 and a clock at the equator T2. We see that the distance between the clocks is unchanged, that is, between them, it would seem, there is no necessary relative speed that can be substituted into the Lorentz equations. However, let's add a third clock T3. They are located in the ISO pole, like the T1 clock, and therefore run synchronously with them. But now we see that clock T2 clearly has a relative speed with respect to clock T3: at first, clock T2 is close to clock T3, then it moves away and approaches again. Therefore, from the point of view of the stationary clock T3, the moving clock T2 lags:

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

Therefore, clock T2 also lags behind clock T1. Let us now move the clock T3 so close to the trajectory T2 that at some initial moment of time they will be nearby. In this case, we get a classic version of the twin paradox. In the following figure we see that at first the clocks T2 and T3 were at the same point, then the clocks at the equator T2 began to move away from the clocks T3 and after some time returned to the starting point along a closed curve:

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

This is fully consistent with the formulation of the first thesis about clock lag, which served as the basis for the “twin paradox.” But clocks T1 and T3 are synchronous, therefore, clock T2 is also behind clock T1. Thus, both theses from Einstein's work can equally serve as the basis for the formulation of the “twin paradox”.

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

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

How do these different speeds fit into the equation? Very simple. Let's place our own fixed clock at each point of the trajectory of the clock T2. All of these new clocks are synchronized with clocks T1 and T3, since they are all located in the same fixed ISO. Clock T2, each time passing by the corresponding clock, experiences a lag caused by the relative speed just past these clocks. During an instantaneous time interval according to this clock, clock T2 will also lag behind by an instantaneously small time, which can be calculated using the Lorentz equation. Here and further we will use the same notation for the clock and its readings:

Obviously, the upper limit of integration is the readings of clock T3 at the moment when clocks T2 and T3 meet again. As you can see, the readings of the T2 clock< T3 = T1 = T. Лоренцев множитель мы выносим из-под знака интеграла, поскольку он является константой для всех часов. Введённое множество часов можно рассматривать как одни часы - «распределённые в пространстве часы». Это «пространство часов», в котором часы в каждой точке пространства идут синхронно и обязательно некоторые из них находятся рядом с движущимся объектом, с которым эти часы имеют строго определённое относительное (инерциальное) движение.

As we can see, a solution has been obtained that completely coincides with the solution to the first thesis (up to quantities of the fourth and higher orders). For this reason, the following discussion can be considered to apply to all types of formulations of the “twin paradox”.

Variations on the theme of the "twin paradox"

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

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

x, y - coordinates of the moving clock T2 in the reference system of the stationary ones;

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

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

x 1 2 + y 1 2 = R 2

x 1 , y 1 - coordinates of the stationary clock T1 in the moving frame of reference;

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

Fig.4 From the point of view of the moving clock T2, the stationary clock T1 moves around them in a circle.

And this, in turn, means that from the point of view of the special theory of relativity, the clock should lag in this case too. Obviously, in this case, it’s the other way around: T2 > T3 = T. It turns out that in fact the special theory of relativity makes two mutually exclusive predictions T2 > T3 and T2< T3? И это действительно так, если не принять во внимание, что теор ия была создана для инерциальных систем отсчета. Здесь же движущиеся часы Т2 не находятся в инерциальной системе. Само по себе это не запрет, а лишь указание на необходимость учесть это обстоятельство. И это обстоятельство разъясняет общая теор ия относительности . Применять его или нет, можно определить простым опытом. В инерциальной системе отсчета на тела не действуют никакие внешние силы. В неинерциальной системе и согласно принципу эквивалентности общей теор ии относительности на все тела действует сила инерции или тяготения. Следовательно, маятник в ней отклонится, все незакреплённые тела будут стремиться переместиться в одном направлении.

Such an experiment near a stationary clock T1 will give a negative result, weightlessness will be observed. But next to the clock T2 moving in a circle, a force will act on all bodies, tending to throw them away from the stationary clock. We, of course, believe that there are no other gravitating bodies nearby. In addition, the T2 clock moving in a circle does not rotate by itself, that is, it does not move in the same way as the Moon around the Earth, which always faces the same side. Observers near clocks T1 and T2 in their reference frames will see an object at infinity from them always at the same angle.

Thus, an observer moving with clock T2 must take into account the fact of non-inertiality of his frame of reference in accordance with the provisions of the general theory of relativity. These provisions say that a clock in a gravitational field or in an equivalent field of inertia slows down. Therefore, in relation to the stationary (according to the experimental conditions) clock T1, he must admit that this clock is in a gravitational field of lower intensity, therefore it goes faster than his own and a gravitational correction should be added to its expected readings.

On the contrary, an observer next to the stationary clock T1 states that the moving clock T2 is in the field of inertial gravity, therefore it moves slower and the gravitational correction should be subtracted from its expected readings.

As we see, the opinion of both observers completely coincided that the clock T2, moving in the original sense, will lag behind. Consequently, the special theory of relativity in its “extended” interpretation makes two strictly consistent predictions, which does not provide any grounds for proclaiming paradoxes. This is an ordinary problem with a very specific solution. A paradox in SRT arises only if its provisions are applied to an object that is not the object of the special theory of relativity. But, as you know, an incorrect premise can lead to both a correct and a false result.

Experiment confirming SRT

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

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

This was the first experiment to test the lag of a moving clock, known as the Hafele-Keating experiment, conducted in 1971. Four clocks made using cesium frequency standards were placed on two airplanes and traveled around the world. Some clocks traveled in an easterly direction, while others circled the Earth in a westerly direction. The difference in the speed of time arose due to the additional speed of rotation of the Earth, and the influence of the gravitational field at the flight altitude compared to the level of the Earth was also taken into account. As a result of the experiment, it was possible to confirm the general theory of relativity and measure the difference in the speed of the clocks on board two aircraft. The results were published in the journal Science in 1972.

Literature

1. Putenikhin P.V., Three mistakes of anti-SRT [before criticizing a theory, it should be studied well; it is impossible to refute the impeccable mathematics of a theory using its own mathematical means, except by quietly abandoning its postulates - but this is another theory; well-known experimental contradictions in SRT are not used - the experiments of Marinov and others - they need to be repeated many times], 2011, URL:
http://samlib.ru/p/putenihin_p_w/antisto.shtml (accessed 10/12/2015)

2. Putenikhin P.V., So, the paradox (twins) is no more! [animated diagrams - solving the twin paradox using general relativity; the solution has an error due to the use of the approximate equation potential a; time axis is horizontal, distance axis is vertical], 2014, URL:
http://samlib.ru/editors/p/putenihin_p_w/ddm4-oto.shtml (accessed 10/12/2015)

3. Hafele-Keating experiment, Wikipedia, [convincing confirmation of the SRT effect on the slowdown of a moving clock], URL:
https://ru.wikipedia.org/wiki/Hafele_-_Keating Experiment (accessed 10/12/2015)

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

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

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

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

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

So, back to the twin brothers!

It should seem to the couch potato that the clock of the moving traveler has a slow passage of time, so when he returns, it should lag behind the couch potato's clock.
On the other hand, the Earth moves relative to the traveler, so he believes that the couch potato’s clock should fall behind.

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

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

However, a paradox as such does not really exist, because we must remember that STR is a theory for inertial reference systems! Oh, the frame of reference of at least one of the twins was not inertial!

At the stages of acceleration, braking or turning, the traveler experienced acceleration, and therefore at these moments the provisions of the STO are not applicable.

Here you need to use General Theory of Relativity, where with the help of calculations it is proved that:

We'll be back, to the question of time dilation in flight!
If light travels any path in time t.
Then the flight duration of the ship for the “homebody” will be T = 2vt/s

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

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

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

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

Over 64 years of its own time, the spacecraft with similar acceleration can travel to the Andromeda galaxy (there and back). About 5 million years will pass on Earth during such a flight.

The reasoning carried out in the story with the twins leads only to an apparent logical contradiction. Whatever the formulation of the “paradox,” there is no complete symmetry between the brothers.

An important role for understanding why time slows down specifically for the traveler who changed his frame of reference is played by the relativity of the simultaneity of events.

Experiments already carried out to lengthen the lifetime of elementary particles and slow down the clock as they move confirm the theory of relativity.

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

Twin paradox

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

For some time, the “twin paradox” attracted little attention. In 1956-1959, Herbert Dingle published a series 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 devoted to this topic appeared. From 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 a contradiction in the theory of relativity, although the history of the emergence of certain explanations of the “paradox” and the giving of new forms to it does not stop to this day.

Classification of explanations of the paradox

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

1) Identify 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 " Physical reason for the paradox“Various versions of the “paradox” will be given and explanations will be given as to why the contradiction does not actually arise.

In the second approach, 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 reference system, various options for taking this fact into account are possible. They can be roughly divided into two large groups.

The first group includes calculations based on the special theory of relativity within the framework of inertial reference systems. In this case, the stages of accelerated motion are considered negligible compared to total time flight. Sometimes a third inertial reference frame is introduced, moving towards the traveler, with the help of which the readings of his watch are “transmitted” to his stay-at-home 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 according to the use or non-use of Einstein's theory of gravity (GTR). Calculations using general relativity are based on the introduction of an effective gravitational field, equivalent to the acceleration of the system, and taking into account the change in the rate of time in it. In the second method, non-inertial reference systems are described in flat space-time and the concept of a gravitational field is not used. The main ideas of this group of calculations will be presented in the section “ Non-inertial reference systems».

Kinematic effects of service station

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

Of course, the mere statement of the brothers’ asymmetry does not explain why it is the traveler’s clock that should slow down, and not the homebody’s. In addition, misunderstandings often arise:

“Why does the violation of the equality of brothers within such a short time (the traveler’s stop) lead to such a striking violation of symmetry?”

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

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

So why would a traveler actually observe his clock lagging behind the clock of a “stationary” system, despite the fact that all such clocks from his point of view are running slower? Most simple explanation within the framework of SRT is that it is impossible to synchronize all clocks in two inertial reference systems. Let's look at this explanation in more detail.

Physical reason for the paradox

During the flight, the traveler and the couch potato are in various points space and cannot compare their watches directly. Therefore, as above, we will assume that along the trajectory of the traveler’s movement in the “stationary” system associated with the couch potato, identical, synchronously running clocks are placed, which the traveler can observe during the flight. Thanks to the synchronization procedure, a single time was introduced in the “fixed” reference system, which at the moment determines the “present” of this system.

After the start, the traveler “transitions” to an inertial frame of reference, moving relatively “stationary” with a speed of . This moment in time is accepted by the brothers as the initial one. Each of them will observe the slow motion of the other brother's clock.

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

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

Therefore, although all the clocks in a fixed frame of reference, past which the traveler flies, go slower from his point of view, from this do not do it that they will lag behind his watch.

At the moment of time, the further ahead the “stationary” clock is on the course, the greater its readings from the traveler’s point of view. When he reaches these clocks, they will not have time to lag enough to compensate for the initial time discrepancy.

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

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

Thus:

despite the fact that all specific clocks in the system are running slower from the point of view of an observer in , different watches along its trajectory will show the time that has gone forward.

The difference in the clock rate and is a relative effect, while the values ​​of the current readings and at one spatial point are absolute. Observers located in different inertial reference systems, but at the “same” spatial point, can always compare the current readings of their clocks. A traveler flying past the system clock sees that it has gone ahead. Therefore, if the traveler decides to stop (by quickly braking), nothing will change, and he will end up in the “future” of the system. Naturally, after stopping, the pace of his clock and his clock will become the same. However, the traveler's clock will show less time than the system clock located at the stopping point. Due to the uniform time in the system, the traveler's clock will lag behind all clocks, including his brother's clock. After the stop, the traveler can return home. In this case, the entire analysis is repeated. As a result, both at the point of stopping and turning around, and at the starting point when returning, the traveler turns out to be younger than his stay-at-home brother.

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.” Thus:

whoever changes his frame of reference turns out to be younger.

Signal exchange

Calculation of time dilation from the position of each brother can be carried out by analyzing the exchange of signals between them. Although the brothers, being in different points in space, cannot directly compare the readings of their watches, they can transmit signals of “precise time” using light pulses or video broadcasts of the watch’s image. It is clear that in this case they observe not the “current” time on their brother’s watch, but the “past” one, since the signal requires time to propagate from the source to the receiver.

When exchanging signals, it is necessary to take into account the Doppler effect. 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 relationship is relative speed of the source and receiver.

Consider a situation in which brothers transmit exact time signals to each other every second (according to their watches). Let’s first carry out the calculation from the traveler’s position.

Traveler's calculation

While the traveler moves away from the Earth, he, due to the Doppler effect, registers a decrease in the frequency of received signals. The video feed from Earth appears slower. After quick braking and stopping, the traveler stops moving away from the earth's 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 finite speed of light, the traveler still observes his brother’s “past”. Having turned around and accelerated, the traveler begins to “run” towards the signals coming towards him and their frequency increases. “Brother’s movements” on the video broadcast from this moment begin to look accelerated for the traveler.

According to the traveler's watch, the flight time in one direction is equal, and the same in the opposite direction. Quantity taken "earth seconds" during travel is equal to their frequency multiplied by time. Therefore, when moving away from the Earth, the traveler will receive significantly fewer “seconds”:

and when approaching, on the contrary, more:

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

in exact accordance with the time dilation formula.

Homebody Calculation

The arithmetic of a homebody is slightly different. While his brother moves away, he also registers the extended period of precise time transmitted by the traveler. However, unlike his brother, the homebody observes such a slowdown longer. The flight time for a distance in one direction is according to earth clocks. The homebody will see the traveler braking and turning after the additional time required for the light to travel the distance from the turning point. Therefore, only after a while from the start of the journey will the couch potato register the accelerated operation of the approaching brother’s clock:

The travel time of light 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 until the moment of his turn (according to the observations of the couch potato) is equal to:

The couch potato receives signals with increased frequency over time (see the figure above), and receives the traveler’s “seconds”:

The total number of “seconds” received during the time is:

Thus, the ratio for the clock reading at the moment of meeting the traveler () and the stay-at-home 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 Alpha Centauri star system, 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 unity, and the unit acceleration per year/year² is close to the acceleration of gravity and is approximately equal to 9.5 m/s².

Let the spacecraft move half the way with unit acceleration, and let it slow down the second half with the same acceleration (). The ship then turns around and repeats the acceleration and deceleration stages. In this situation, the flight time in the earth's reference frame 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 its own time, a spacecraft with unit acceleration could potentially travel (returning to Earth) to the Andromeda Galaxy, 2.5 million light years away. years . About 5 million years will pass on Earth during such a flight. Developing twice the acceleration (which a trained person can easily get used to if a number of conditions are met and a number of devices are used, 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 the cosmonauts about 50 years; However, having returned from such an expedition (after 28 billion years according to Earth's clock), its participants risk not finding alive not only the Earth and the Sun, but even our Galaxy. Based on these calculations, the reasonable radius of accessibility for interstellar return expeditions does not exceed several tens of light years, unless, of course, any fundamentally new physical principles of movement in space-time are discovered. However, the discovery of numerous exoplanets gives reason to believe that planetary systems are found near a sufficiently large proportion of stars, so astronauts will have something to explore in this radius (for example, the planetary systems ε Eridani and Gliese 581).

Traveler's calculation

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

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

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

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

where is the coordinate speed of the earth's clock.

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

The result of all these calculations shows that from the point of view of the traveler, his clock will lag behind the clock of the 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 couch potato. 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 within the framework of inertial reference systems.

conclusions

The reasoning carried out in the story with the twins leads only to an apparent logical contradiction. Whatever the 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 specifically for the traveler who changed his frame of reference.

Calculation of the magnitude of time dilation from the position of each brother can be performed both within the framework of elementary calculations in SRT, and using the analysis of non-inertial reference systems. All these calculations are consistent with each other and show that the traveler will be younger than his stay-at-home 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 internal contradiction in it. Numerous experiments on extending the lifetime of elementary particles and slowing down macroscopic clocks as they move confirm the theory of relativity. This gives grounds to assert that the time dilation described in the story with the twins will also occur in the real implementation of this thought experiment.

see also

Notes

Sources

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Special and general theories of relativity say that each observer has his own time. That is, roughly speaking, one person moves and uses his watch to determine one time, another person somehow moves and uses his watch to determine another time. Of course, if these people move relative to each other with low speeds and accelerations, they measure practically the same time. With our watches that we use, we are unable to measure this difference. I do not rule out that if two people are equipped with a clock that measures time with an accuracy of one second during the life of the Universe, then, having walked differently, they may see some difference in some n sign. However, these differences are weak.

Special and general theories of relativity predict that these differences will be significant if two companions move 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 greater acceleration. Then the differences will be significant. How big I am not saying, and this is measured in an experiment with a high-precision atomic clock. People fly on an airplane, then bring it back, compare what the clock on the ground showed, what the clock on the plane showed, and more. There are many such experiments, all of them are consistent with the formal predictions of general and special relativity. In particular, if one observer is at rest, and the other is moving relative to him at a constant speed, then the recalculation of the clock rate from one to the other 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, and Vasya flew to Alpha Centauri and returned. Now it is said that relative to Vanya, Vasya moved at a constant speed. Time moved slower for him. He returned, so he must be younger. On the other hand, the paradox is formulated as follows: now, on the contrary, relative to Vasya (movement at a constant speed relative to) Vanya moves at a constant speed, despite the fact that he was on Earth, that is, when Vasya returns to Earth, in theory, Vanya the clock should show less time. Which one is younger? Some kind of logical contradiction. This special theory of relativity is complete nonsense, it turns out.

Fact number one: you immediately need to understand that Lorentz transformations can be used if you move from one inertial reference system to another inertial reference system. And this logic that time moves slower for one due to the fact that it moves at a constant speed is only based on the Lorentz transformation. And in this case, one of the observers is almost inertial - the one who is on 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, are all small accelerations; for this task, this can certainly be neglected. And the second one 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 work immediately.

How to properly explain this twin paradox? It's actually quite simple to explain. In order to compare the lifespan 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 the hours: 0 hours 0 minutes January 1, 2001. Then scatter. One of them will move in one way, somehow his clock will be ticking. 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 one. At the same time they will find themselves at the same point in relation to some additional clock. The important thing is that they can now compare watches. One had so much pressure, 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, let’s assume for the ideal, that is, it moved in a straight line. The second of them moved non-inertially, so in this space and time it moved along some kind of curve - it accelerated, slowed down, and so on. So one of these curves has the property of extremity. It is clear that among all possible curves in space and time, the straight line is extremal, that is, it has an extreme length. Naively, it seems that it should have the shortest length, because on a plane, among all curves, a straight line has the shortest length between two points. In Minkowski’s space and time, his metric is structured this way, this is how the method of measuring lengths is structured, the straight line has the longest length, no matter how strange it may sound. The straight line has the longest length. Therefore, the one that moved inertially, remained 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 refute one or another theory. They are invented by the scientists themselves who are involved 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 established data at that time. And people simply resist, of course, they come up with all sorts of counterarguments and so on. This all goes through a very difficult process. A person fights to be recognized. This always involves long periods of time and a lot of hassle. These are the paradoxes that arise.

In addition to the twin paradox, there is, for example, such a paradox with a rod and a barn, the so-called Lorentzian 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, about to fly through him. The barn in its resting system has one length, say 6 meters. The rod in its rest frame has a length of 10 meters. Imagine that their closing speed is such that in the frame of reference of the barn the rod is reduced to 6 meters. You can calculate what speed this is, but it doesn’t matter now, it’s close enough to the speed of light. The rod was reduced to 6 meters. This means that in the frame of reference of the barn, the rod will at some point fit entirely into the barn.

A person who is standing in a barn and a rod is flying past him will at some point see this rod lying entirely in the barn. On the other hand, motion at constant speed is relative. Accordingly, one can consider it as if the rod is at rest, and the barn is flying towards it. This means that in the frame of reference of the rod the barn has contracted, and it has contracted by the same number of times as the rod in the frame of reference of the barn. This means that in the reference frame of the rod the barn has shrunk to 3.6 meters. Now, in the frame of reference of the rod, there is no way the rod can fit into the shed. In one reference system it fits, in another reference system it does not fit. This is some kind of 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 is of this length,” it means that you are receiving a signal from this end of the rod and from that end of the rod at the same time. That is, when I say that the rod was placed in the barn, moving at some speed, this means that the event of the coincidence of this end of the rod with this end of the barn is simultaneously with the event of the coincidence of this end of the rod with this end of the barn. These two events are simultaneous in the frame of reference of the barn. But you’ve probably heard that in the theory of relativity, simultaneity is relative. So it turns out that in the frame of reference of the rod these two events are not simultaneous. Simply, first the right end of the rod coincides with the right end of the barn, then the left end of the rod coincides with the left end of the barn after some period of time. This period of time is exactly equal to the time during which these 10 meters minus 3.6 meters will fly past 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 a lot of these paradoxes. There is a book by Taylor and Wheeler “Physics of Space-Time”, it is written in a fairly accessible language for schoolchildren, where the vast majority of these paradoxes are analyzed and explained using fairly simple arguments and formulas, as this or that paradox is explained within the framework of the theory of relativity.

One can come up with some way to explain each given fact that looks simpler than the way provided by the theory of relativity. However important property The special theory of relativity is that it explains not each individual fact, but this entire set of facts taken together. Now, if you come up with an explanation for one 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 also explains all the other facts. And as a rule, all these explanations, which sound simpler, do not explain everything else. And we must remember that at the moment when this or that theory is invented, it is really some kind of psychological, scientific feat. Because at this moment there are one, two or three facts. 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 previously known, 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, and they are checked to see if they correspond to this theory. Some predictions also follow from the theory. It is based on some facts, it states something, from this statement you can deduce something, get it, and then say that if this theory is correct, then such and such should be the case. Let's go and check whether this is 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 apply, there is no evidence to disprove these theories.

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

The main idea of ​​SRT

The paradox (twin paradox) states that a “stationary” observer perceives the processes of moving objects as slowing down. In accordance with the same theory, inertial reference systems (systems in which the movement of free bodies occurs rectilinearly and uniformly or they are at rest) are equal relative to each other.

The Twin Paradox: Briefly

Taking into account the second postulate, an assumption of inconsistency arises. To resolve this problem clearly, it was proposed to consider the situation with two twin brothers. One (relatively 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 traveler’s watch moves more slowly, which means that when he returns, his (the traveler’s) watch will be slower. The traveler, on the contrary, sees that the Earth is moving relative to him (on which the couch potato is located with his watch), and, from his point of view, it is his brother who will have time move more slowly.

In reality, both brothers are in equal conditions, which means that when they find themselves together, the time on their watches will be the same. At the same time, according to the theory of relativity, it is the clock of the brother-traveler that should lag behind. Such a violation of obvious symmetry was considered as an inconsistency of the theory.

Twin paradox from Einstein's theory of relativity

In 1905, Albert Einstein derived a theorem that states that if a pair of clocks synchronized with each other is at point A, one can move one of them along a curvilinear closed path with a constant speed until they reach point A again (and this will take, for example, t seconds), but at the moment 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 explained by the fact that, returning to Earth, the traveler was moving at an accelerated pace.

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

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

Explanation of the paradox

The explanation for the twin paradox is quite simple: the initial assumption of equality between the two frames of reference is incorrect. The traveler was not 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 it would produce inconsistent predictions.

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

Controversial issues

There are suggestions 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 proven that under the influence of acceleration, the movement of time does not accelerate or slow down.

As a result, the segment of the trajectory along which one of the brothers accelerated demonstrates only some asymmetry that arises between the traveler and the couch potato.

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

Testing by practice

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 were carried out, the purpose of which was to prove or disprove the theory of relativity.

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

Another example: the life of muons (heavy electrons) is longer. These elementary particles several hundred times heavier than usual, 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 that of light. Given their true lifespan (2 microseconds), they would decay before they touched the surface of the planet. But during the flight they live 15 times longer (30 microseconds) and still reach their goal.

Physical reason for the paradox and signal exchange

Physics explains the twin paradox in a more accessible language. While the flight is taking place, both twin brothers are out of range of each other and cannot practically verify that their clocks move synchronously. You can determine exactly how much a traveler's watch is slowing down by analyzing the signals that they send to each other. These are conventional “precise time” signals, expressed as light pulses or a video broadcast of a watch dial.

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

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

Formulating an explanation in paradoxical situations

To explain the paradoxes of such stories with twins, two main methods can be used:

1. Careful examination of existing logical structures for contradictions and identification of logical errors in the chain of reasoning.
2. Carrying out detailed calculations in order to assess the fact of time braking from the point of view of each of the brothers.

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

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

Conclusion

All discussions related to the imaginary paradox are due to only an apparent logical error. No matter how the conditions of the problem are formulated, it is impossible to ensure that the brothers find themselves in completely symmetrical conditions. It is important to take into account that time slows down precisely on a moving clock that had to go through a change of 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 reference systems. The results of both chains of calculations can be mutually consistent and serve equally to confirm that time moves slower on a moving clock.

On this basis, we can assume that when the thought experiment is transferred into reality, the one who takes the place of a homebody will actually grow old faster than the traveler.