Twin paradox

The twin paradox , also clock paradox, is a thought experiment that describes an apparent contradiction in the special theory of relativity . In the thought experiment, one twin flies at almost the speed of light to a distant star, while the other twin remains on Earth. Then the traveling twin returns at the same speed. After returning to Earth, it turns out that the twin left there is older than the one who traveled. This is a consequence of time dilation .

The amount of age difference between the twins on return is determined by the speed of the journey and the distance traveled. The travel speed close to the speed of light, together with the astronomical distances to the distant star, causes an age difference of sometimes many years. The time dilation also occurs at small speeds and distances, but the time difference is then correspondingly smaller. In everyday life, the difference is negligibly small with the speeds and distances that occur there.

The twin paradox as a consequence of time dilation

According to the relativistic time dilation , a clock that is moved between two synchronized clocks from A to B will lag behind these clocks. Albert Einstein pointed out in 1905 that this is also the case when the clock returns to the starting point. This means that a clock that moves away from and returns to an arbitrary point will lag behind an immobile clock left at the starting point. In 1911 Einstein extended this consideration to living organisms:

"If we z. For example, if you put a living organism in a box and let it perform the same back and forth movement as the clock did before, it could be achieved that this organism, after a flight of any length, returned to its original location with any desired little change, while correspondingly Organisms created that have remained dormant in their original locations have long since given way to new generations. For the moving organism, the long time of the journey was only a moment, if the movement occurred approximately at the speed of light! This is an unavoidable consequence of the principles on which we are based and which experience imposes on us. "

- Albert Einstein: The Relativity Theory

The time dilation itself is symmetrical according to the principle of relativity . This means that everyone must be able to view the other's watch as moving and thus be able to view its rate as slowed down. This raises the question of why the clock that remains in the same place does not slow down from the point of view of the returning clock when it meets. That would result in a contradiction, because when they meet, the hand positions of both clocks cannot follow the other. This problem was answered correctly by Paul Langevin in 1911 . In 1911/13, Max von Laue succeeded in making Langevin's Declaration clearer and more vivid with the help of Minkowski diagrams .

The paradox is based on intuitive but impermissible assumptions about the nature of time , such as simultaneity . In particular, the change in direction at the turning point of the journey is ignored. Due to this reversal, the two reference systems in which the twins rest are not equivalent, but the reference system of the traveling twin is not an inertial system due to the reversal of direction , which changes his assessment of the simultaneity of the relevant events. In order for the traveling twin to be in an inertial system during the outward and return journey, it would have to change its reference system at the moment of the reversal of direction. For the other twin staying on earth, however, the reversal of direction of the traveling twin does not change anything, so that the consideration of the pure time dilation provides the correct end result. For the age difference of the twins, in addition to the relative speed, the distance covered between the changes in direction is important. Since the duration of the acceleration during the reversal can be kept as short as desired in comparison to the travel time with uniform movement, this duration of the acceleration (in contrast to the fact that the acceleration actually took place) is of negligible importance for the size of the age difference. The case in which, at the turning point, only the time is transmitted by radio signal to another observer traveling in the opposite direction, delivers the same result as in the case with a negligibly short duration of the acceleration.

In the theory of relativity, space and time are combined into what is known as spacetime . Every traveler describes a curve in it, called the world line , the length of which is proportional to the time that passes for him (the traveler's own time ). In flat space, a straight stretch in space is the shortest connection between two points that a traveler can cover. An unaccelerated observer moves in a straight line through flat space-time. That is, both the traveling twin and the lagging twin move in a straight line through spacetime away from the initial spacetime point. The length of the lines is the time that each twin measures. In order to meet again at a spacetime point in the future, one of the twins must change its direction of movement so that it then moves on a different straight line through spacetime that intersects with the world line of the other twin. In Euclidean space, the total length of the line with the reversal of direction is always longer than the straight line between the two points. Due to the Minkowski metric of the flat space-time, in which the location and time coordinates contribute with the opposite sign, the total length of the line with the reversal of direction in the flat space-time is always less than the length of the straight line, the traveled twin is therefore younger. If both twins were to change direction, the distances covered on the world lines would have to be compared for the age difference.

Experimental evidence

The detection of time dilation is routine in particle accelerators . A time difference in the type of twin paradox (i.e. with a return flight) was also detected in storage rings , where muons came back several times on a circular path to the starting point. By comparing atomic clocks , this effect could also be demonstrated in commercial aircraft in best agreement with the prediction of the theory of relativity. In this Hafele-Keating experiment , a test of the time dilation following from the theory of relativity, the rotation of the earth and the effects of the general theory of relativity also play a role.

Resolution of the twin paradox

To resolve the twin paradox in detail, the following two questions must be answered:

How is it that each twin sees the other age more slowly?
Why does the twin left behind on earth turn out to be the older after the trip?

The mutually slower aging of the twins

To answer the first question, consider how the twin on Earth even determines that the flying one ages more slowly. To do this, he compares the display on a clock that the flying twin is carrying with two stationary clocks that are located at the beginning and end of a certain test route that the flying twin passes. For this, these two clocks must of course have been set to the same time from the point of view of the resting twin. The flying twin reads the same clock readings as the resting one during the passages, but he will object that, in his opinion, the clock at the end of the test track is faster than the one at the beginning. The same effect occurs when the flying twin judges the aging of the earthly with two clocks.

The reason is the fact that, according to the theory of relativity, there is no absolute simultaneity. The simultaneity of events at different locations and thus also the displayed time difference between two clocks there is assessed differently by observers who move at different speeds. A closer look at the situation shows that the mutual assessment of a slowing down of time does not lead to a contradiction. The comparatively clear Minkowski diagrams are helpful here, as they can be used to understand this fact graphically and without formulas.

The mutual slowing down is in accordance with the principle of relativity , which says that all observers who move against each other at a constant speed are completely equal. One speaks of inertial systems in which these observers are located.

The different ages of the twins

Variant with acceleration phases

To answer the second question, consider the deceleration and acceleration phases that are required for the flying twin to return. According to the flying twin, time on earth passes faster during these two phases. The twin remaining there ages so much that despite the slower aging during the phases with constant speed, in the end result it is the older, so that there is no contradiction from the perspective of the flying twin. The result after the return does not contradict the principle of relativity, since the two twins cannot be regarded as equivalent in terms of the overall journey due to the acceleration that only the flying one experiences.

The cause of this aging is again the relativity of simultaneity. During the acceleration, the flying twin changes constantly to new inertial systems. In each of these inertial systems, however, a different value arises for the point in time that simultaneously prevails on earth, in such a way that the flying twin concludes that the earthly one is aging. The further the twins are apart, the greater this effect. ( with as relative speed and "original" distance in the non-accelerated system). ${\ displaystyle \ Delta \ tau = \ Delta v \ cdot x '/ c ^ {2}}$ ${\ displaystyle \ Delta v}$ ${\ displaystyle x '= x \ gamma}$

Distance-time diagram for v = 0.6 c. The twin on earth moves on the time axis from A ₁ to A ₄ . The traveling twin takes the path via B. Lines of simultaneity from the traveling twin's point of view are shown in red for the outward journey and blue for the return journey. The points on the travel routes each mark one year of personal time.

The relationships are shown in the distance-time diagram of a journey from A to B and back again at 60% of the speed of light . The trajectory of the remaining twin runs along the time axis from A ₁ to A ₄ , the flying one takes the path via B. Each horizontal line in the diagram corresponds to events that occur simultaneously on Earth from the twin's point of view. The flying twin, on the other hand, assesses all events on the red lines as simultaneous on the outward flight and those on the blue lines on the return flight. Immediately before its arrival at target B, the dormant twin is, in the opinion of the flying twin, therefore at A ₂ and therefore appears less aged. During the reversal phase, which was assumed here to be so short that it cannot be seen in the diagram, the lines of simultaneity for the flying twin swing, and his brother on earth ages to point A ₃ . During the return trip to A ₄ of the twin on earth again seems slow to age. Since the inclinations of the lines of simultaneity shown depend only on the travel speed before and after the reversal phase, the strength of the acceleration is not relevant for the post-aging time. ${\ displaystyle c}$

The earthly twin does not feel anything of its apparent aging. As described, this is an effect within the framework of the special theory of relativity, which results from a representation of the processes in different coordinate systems between which the traveling twin changes.

Even the traveling twin cannot directly observe the post-aging leap of the earthly twin, but can only identify it based on the incoming light or radio signals in connection with the knowledge of the distance, the relative speed and the speed of propagation of light . (An on-board computer could take over this task and display the time on earth that is considered to be the same; the display would advance by leaps and bounds while turning.) The only thing that the traveling twin can directly observe is that from the turning point B to the incoming light or radio signals have changed in their frequency and time interval (compare the following section "Exchange of light signals"). Signals from points A ₂ and A ₃ then arrive at him only after his turning point - in the same time cycle that applies to the incoming signals during the entire return flight phase.

Variant without acceleration phases

By introducing a third person, a variant of the twin paradox can be formulated that does not require any acceleration phases. This variant (“three brothers approach”) was introduced by Lord Halsbury and others. This can be achieved if rocket A passes the earth at a negligibly small distance and both of them synchronize their clocks with radio signals during this time. Then rocket A passes a star at a constant speed and hits a second rocket B at a negligibly short distance, which at the same time passes the star at the same speed but is directed towards earth, with A transmitting its time to B with radio signals. If now rocket B arrives at the earth, the rocket clock follows the earth clock here too. The mathematical treatment of this scenario and its end result are identical to the one previously described. This variant with three people demonstrates that it is not the duration of the acceleration that resolves the twin paradox (because this can be made as small as you want in comparison to the inertial flight time), but the fact that the events take place in different inertial systems during the outward and return journey which the definition of simultaneity is necessarily different. So there must always be three inertial systems:

${\ displaystyle S}$ in which the earth clock rests.
${\ displaystyle S '}$ , in which the rocket clock rests on the way there.
${\ displaystyle S ''}$ , in which the rocket clock rests on the way back.

In the example with accelerations, the same rocket clock changes from to . In the example without accelerations, the rocket clocks remain in their systems, and it is the information of the time at the point where A and B meet that changes from to . ${\ displaystyle S '}$ ${\ displaystyle S ''}$ ${\ displaystyle S '}$ ${\ displaystyle S ''}$

Numerical example

For a return trip at 60% of the speed of light to a destination at a distance of 3 light years, the following conditions apply (see graphic above): From the point of view of the twin on earth, 5 years are required for the trip there and back. The factor for time dilation and length contraction is 0.8. This means that the flying twin only ages by 5 × 0.8 = 4 years on the way there. This explains this lower time requirement with the fact that the distance has been shortened by the length contraction at its travel speed to 3 × 0.8 = 2.4 light years. Since, according to his estimation, time also passes more slowly on Earth, it seems that only 4 × 0.8 = 3.2 years have passed on Earth immediately before his arrival at the distant star. From the point of view of the earth dweller, however, the event of the arrival on the distant star does not occur until 1.8 years later. At this point the paradox arises due to intuitive flawed assumptions about simultaneity. Due to the change of the inertial systems during the reversal phase, the perception of the traveling twin with regard to the "simultaneous" event on earth to the arrival on the star shifts by 3.6 years. Together with the 3.2 years on the way back, a total of 10 years have passed from the perspective of the flying twin on earth, while he himself has only aged by 8 years.

Exchange of light signals

Paths of light signals sent out annually. Signals sent by the earthly twin on the left and those of the traveler on the right. The traveler receives or sends the ones shown in red before turning back and the blue ones afterwards. Due to the Doppler effect, the signals are initially received at half and later at double the frequency.

So far it has been shown what the observers consider to be real events, taking into account the speed of propagation of light known to them. The following describes what both twins see immediately when they send a light signal to their brother once a year. The paths of light signals in the above path-time diagram are straight lines with an increase of 45 ° in the direction of transmission. The diagrams on the right are based on this example.

First, the twins move away from each other so that the light rays are red-shifted due to the Doppler effect . These light rays are shown in red in the diagram. Halfway through the journey, the twins are moving towards each other so that the light rays are shifted blue, therefore these light rays are shown in blue in the picture. Due to the relativity principle, both observers measure the same time intervals of 2 years each between the red signals and half a year between the blue signals, which is immediately clear in the picture.

The assumption that both twins would be the same age after their return, so that both twins would have received the same number of signals from the other, now leads to a contradiction. Because while the traveling twin immediately receives the temporally compressed signals at the turning point and thus after half the travel time, the earthly twin receives the stretched signals even longer. Due to the principle of relativity, the observer who receives blue-shifted signals for a longer period of time receives more signals than the other. The traveling twin thus receives more signals than the twin on earth, so that both agree that the traveling twin has aged more slowly.

In the numerical example in the adjacent picture, the traveling twin sees the earthly one age by 2 years in 4 years and by 8 years in a further 4 years, i.e. a total of 10 years, due to a combination of relativistic effects and transit time effects. According to the travelers, the earthly twin will initially age by 4 years in 8 years and then by 4 years in 2 years, so a total of 8 years.

Accelerated movements

Instead of assuming an infinitely high acceleration in an infinitely short time when reversing, it is also possible with the means of the special theory of relativity to describe more realistic movements of the twins with continuously finite acceleration. The case with constant intrinsic acceleration is often discussed (i.e. the acceleration that the traveling twin can measure with an acceleration sensor), which is also referred to as relativistic hyperbolic movement .

A twin should start with constant positive self-acceleration, fly on with constant positive speed, turn around with constant negative self-acceleration, keep flying with constant negative speed, and come to a standstill with constant positive self-acceleration again at the starting point. This is shown in the picture as follows: phase 1 (a = 0.6, τ = 2); Phase 2 (a = 0, τ = 2); Phase 3-4 (a = −0.6, 2τ = 4); Phase 5 (a = 0, τ = 2); Phase 6 (a = 0.6, τ = 2). Here is a the self-acceleration (in units in which the speed of light equal to 1), and τ the proper time of the accelerated twin. The thinner red lines are the simultaneity lines of the momentary inertial systems, which are continuously changed by the red twin. The thicker red points mark the points in time at which the red twin changes its own acceleration. The twins meet at T = 17.3 and τ = 12, so the red twin is younger.

Web links

Commons : twin paradox - collection of images, videos and audio files

Wikibooks: The »twin paradox« - a closer look - learning and teaching materials

Franz Embacher: Special theory of relativity: twin paradox and geodesics of space-time.
Klaus Kassner: Crash Course in Special Theory of Relativity: The Twin Paradox.
Georg Bernhardt: To the twin paradox in the special theory of relativity. ( PDF ; 340 kB).
Geometry of relativity: twin paradox. At: gsi.de.

Single receipts

↑ Albert Einstein: On the electrodynamics of moving bodies . (PDF) In: Annals of Physics . 322, No. 10, 1905, pp. 891-921.
↑ Albert Einstein: The Relativity Theory . In: Naturforschende Gesellschaft, Zurich, quarterly journal . 56, 1911, pp. 1-14.
^ Paul Langevin: L'Évolution de l'espace et du temps. In: Scientia. 10, 1911, pp. 31-54.
↑ Max von Laue : Two objections to the relativity theory and its refutation . In: Physikalische Zeitschrift . 13, pp. 118-120.
^ Max von Laue: The principle of relativity , 2nd edition. Edition, Vieweg, Braunschweig 1913.
^ Arthur I. Miller: Albert Einstein's special theory of relativity. Emergence (1905) and early interpretation (1905-1911) . Addison-Wesley, Reading 1981, ISBN 0-201-04679-2 , pp. 257-264.
↑ In 1918 Albert Einstein described the paradox also with the help of the general theory of relativity. See
A. Einstein: Dialogue on Objections to Relativity. In: The natural sciences. Issue 48, pp. 697-702 (1918) ( online PDF ).
↑ Talal A. Debs, Michael LG Redhead: The twin “paradox” and the conventionality of simultaneity . In: American Journal of Physics . 64, No. 4, 1996, pp. 384-392. doi : 10.1119 / 1.18252 .
^ Hermann Bondi: Relativity and Common Sense. Doubleday 1964 (Reprinted in Dover 2008, ISBN 0-486-24021-5 ), p. 80.
↑ T. Müller, A. King, D. Adis: A trip to the end of the universe and the twin “paradox” . In: American Journal of Physics . 76, No. 4, 2006, pp. 360-373. arxiv : physics / 0612126 . bibcode : 2008AmJPh..76..360M . doi : 10.1119 / 1.2830528 .
^ E. Minguzzi: Differential aging from acceleration: An explicit formula . In: American Journal of Physics . 73, 2005, pp. 876-880. arxiv : physics / 0411233 . doi : 10.1119 / 1.1924490 .

This version was added to the list of articles worth reading on June 2, 2005 .

[1] Albert Einstein: On the electrodynamics of moving bodies . (PDF) In: Annals of Physics . 322, No. 10, 1905, pp. 891-921.

[2] Albert Einstein: The Relativity Theory . In: Naturforschende Gesellschaft, Zurich, quarterly journal . 56, 1911, pp. 1-14.

[3] Paul Langevin: L'Évolution de l'espace et du temps. In: Scientia. 10, 1911, pp. 31-54.

[4] Max von Laue : Two objections to the relativity theory and its refutation . In: Physikalische Zeitschrift . 13, pp. 118-120.

[5] Max von Laue: The principle of relativity , 2nd edition. Edition, Vieweg, Braunschweig 1913.

[6] Arthur I. Miller: Albert Einstein's special theory of relativity. Emergence (1905) and early interpretation (1905-1911) . Addison-Wesley, Reading 1981, ISBN 0-201-04679-2 , pp. 257-264.

[7] In 1918 Albert Einstein described the paradox also with the help of the general theory of relativity. See
A. Einstein: Dialogue on Objections to Relativity. In: The natural sciences. Issue 48, pp. 697-702 (1918) ( online PDF ).

[8] Talal A. Debs, Michael LG Redhead: The twin “paradox” and the conventionality of simultaneity . In: American Journal of Physics . 64, No. 4, 1996, pp. 384-392. doi : 10.1119 / 1.18252 .

[9] Hermann Bondi: Relativity and Common Sense. Doubleday 1964 (Reprinted in Dover 2008, ISBN 0-486-24021-5 ), p. 80.

[10] T. Müller, A. King, D. Adis: A trip to the end of the universe and the twin “paradox” . In: American Journal of Physics . 76, No. 4, 2006, pp. 360-373. arxiv : physics / 0612126 . bibcode : 2008AmJPh..76..360M . doi : 10.1119 / 1.2830528 .

[11] E. Minguzzi: Differential aging from acceleration: An explicit formula . In: American Journal of Physics . 73, 2005, pp. 876-880. arxiv : physics / 0411233 . doi : 10.1119 / 1.1924490 .