Twin paradox

The twin paradox (or clock paradox ) is a thought experiment that describes an apparent contradiction in the special theory of relativity. After flying one of the twins with almost the speed of light to a distant star and then returns at the same speed back. After returning to Earth, it turns out that there retarded twin 's older than the traveled. This is a consequence of time dilation.

The twin paradox as a result of time dilation

According to the relativistic time dilation is a clock that is moved between two synchronized clocks from A to B, after compared to the synchronized clocks. In addition, Albert Einstein showed in 1905 suggests that this is also the case when the clock returns to the starting point. This means that a clock that moves away from any point and return there pursues against a left behind at the starting point stationary clock. 1911 Einstein extended this idea to living organisms:

" If we have a living organism might bring, for example, in a box and run the same reciprocation left him as before the clock so you could reach it that this organism changed after an arbitrarily long flight arbitrarily little back to its original place returns, while all corresponding crafted organisms, which have remained dormant in the original locations have already long since made new generations of space. For the moving organism for a long time the trip was only a moment, if the movement was nearly the speed of light! This is an inescapable consequence of the principles laid down by us, which imposes the experience of us. "

The time dilation itself is symmetrical in accordance with the principle of relativity. That is, each should the clock of the other as moving and thus can be regarded as the transition rate slows. This raises the question of why the persisting at the same place clock does not follow from the perspective of the returning clock when meeting. This would result in a contradiction, because when meeting the pointer positions of both clocks can not pursue each to the other. This problem was answered correctly by Paul Langevin in 1911. 1911/13 succeeded Max von Laue, much more clear and vivid display the declaration of Langevin using Minkowski diagrams.

The paradox is based on intuitive, but invalid assumptions about the nature of time, such as the concurrency. In particular, the change of direction at the turning point of the journey is ignored. Through this reversal, the twins are not equivalent, but an observer changes its inertial frame, which changes the assessment of the simultaneity of events for him. In contrast, for the other twin, nothing changes, so that in this consideration of the pure time dilation returns the correct result. For the age difference of the twins next to the relative velocity is mainly the distance between the changes of direction distance of importance. Since the duration of the acceleration during the reversal compared to the travel time can be kept arbitrarily small with uniform movement, it is of negligible importance for the size of the age difference. So also provides the case where only the time is transmitted by radio signal to another, oncoming observer at the reversal point, the same result as in the case with negligible short duration of the acceleration.

In the theory of relativity space and time are combined to so-called space-time. Every traveler describes her as a curve whose length represents the time it takes for him there. In three-dimensional space is a straight line in space is the shortest distance between two points, which can cover a traveler. By contrast, in the four-dimensional space-time is a straight line between given points on a world line ( the proper time of each clock ), the longest time of all possible itineraries. For example, if both twins make changes in direction, the ratio of changes in direction and distances traveled must be considered on the world lines of the observer.

Experimental Evidence

The proof of time dilation is in particle accelerators already routine, and also a time difference in the nature of the twin paradox ( ie, with round-trip flight ) was in storage rings, where muons repeatedly came back on a circular path to the starting point, proven. And by comparing two atomic clocks, this effect could also be demonstrated in commercial aircraft in excellent agreement with the prediction of the theory of relativity. In this Hafele - Keating experiment, however, the Earth's rotation and effects of general relativity play a role.

Resolution of the twin paradox

For the resolution of the twin paradox in detail the following two questions must be answered:

• How is it that each twin sees the other of age slower?
• Why does the left on the earth after traveling twin turns than the older?

The mutually slower aging of the twins

To answer the first question, consider how the twin on earth ever finds that the flying slower aging. To this end, it compares the display on a clock that leads the flying twin with him, with two clocks at rest located at the beginning and end of a particular test route that passes through the flying twin. To this end, these two clocks must have been set, of course from the perspective of the stationary twin at the same time. The flying twin reads while at the same passages from Uhrstände as the rest, but he will argue that his view is going according to the clock at the end of the test section compared to the beginning. The same effect occurs when the flying twin analogue evaluated the aging of the earth with two watches.

Cause is the fact that there is no absolute simultaneity by the theory of relativity. The simultaneity of events at different locations, and thus the displayed time difference of two local clocks is perceived differently by observers moving at different speeds. An examination of the circumstances shows that the mutual assessment of a slowing of time does not lead to a contradiction. Are helpful to the comparatively vivid Minkowski diagrams over which this situation can understand graphically and without formulas.

Reciprocal slowdown is consistent with the principle of relativity, which states that all observers moving with constant velocity relative to one another, are completely equal. This is called inertial frames, in which those observers.

The different ages of the twins

Variant with acceleration phases

To answer the second question, the deceleration or acceleration phase is considered, which is required for the return of the flying twin. During this phase passes in the opinion of the twin flying time on earth more quickly. The remaining twin is aging while there after such an extent that he is in spite of the slower aging during the phases at a constant speed in the final outcome of the elderly, so that there is no contradiction even from the point of view of the flying twin. The result after the return is not in contradiction with the principle of relativity, since the twins due to the acceleration experienced by only the flying, respect can not be regarded as equivalent to the total trip.

Cause of this Nachalterung is again the relativity of simultaneity. During acceleration, the flying twin switches in a sense constantly in new inertial systems. In each of these inertial systems, however, there for the time that exists simultaneously on earth, a different value in such a way that the flying twin closes in on an Nachalterung of the earthly. The more the twins apart, the greater this effect.

The ratios are shown again back with 60% of the speed of light c in the shown path-time diagram of a journey from A to B and. The path of the remaining twin runs along the time axis, of A1 to A4, the flying takes the path B. Each horizontal line on the graph corresponding to events that occur from the point of view of the twin on the ground simultaneously. The flying twin contrast estimates during ascent all the events on the red lines as the same time and on the return flight, the blue on. Immediately prior to his arrival at the destination B is therefore the resting twin of flying believes in A2 and therefore seems less aged. During the reversal phase, which was assumed to be so short that they can not be seen in the chart, pivot the lines of simultaneity for flying twin, and his brother on earth is aging to the point A3 to. During the return trip to A4 of the twin on earth again seems slower to age. Since the slopes of the lines of simultaneity shown depend only on the travel speed before and after the reversal phase, the strength of the acceleration for the Nachalterungszeit is not relevant.

The earthly twin senses of its apparent Nachalterung nothing. It is, as described, to an effect in the framework of special relativity, which results from an account of the events in different coordinate systems between which the traveling twin changes.

Also, the traveling twin can observe the Nachalterungssprung the earthly twin described directly, but such only on the basis of incoming light or radio signals in conjunction with the knowledge of the distance, tap the relative velocity and the velocity of propagation of light. ( An on-board computer could take on this task and in each case as the same time deemed Show time on earth; during turning the display would advance by leaps and bounds. ) Direct is observable for the traveling twin only that the turning point B to the incoming light or radio signals changed in frequency and time interval have (see the following section " exchange of light signals "). Signals from the points A2 and A3 then meet after his turning point for him - in the uniform interval that applies to the incoming signals during the entire return flight phase.

Variant without acceleration phases

By introducing a third person can be a variant of the twin paradox formulated which does not need any acceleration phases. This variant ( "Three Brothers " approach ) was introduced by Lord Halsbury and others. This can be realized when rocket A passes the earth in a negligibly small distance and during which both synchronize their clocks with radio signals. A rocket then passes through a star at a constant rate and meets in negligible short distance to a second rocket B, simultaneously passes through the star with an equally large, but directed to the ground speed, where A transmits its radio signals to time B. Now, if B rocket arrives at the earth, the Raketenuhr here is also evidence to the earth clock. The mathematical treatment of this scenario and its end result is identical to that previously described. This variant demonstrated with three people, it is not the duration of the acceleration resolves the phenomenon of the twin paradox ( as these can be made arbitrarily small compared to the inertial flight time), but the fact that the events during the outward and return journeys in different inertial frames takes place in which the definition of the simultaneous failure necessarily different. It must always be present three inertial

• In which the earth clock rests.
• Where the Raketenuhr rests while going there.
• Where the Raketenuhr rests while coming back.

In the example with the same accelerations Raketenuhr changed from to. In the example without accelerations the rocket watches remain in their systems, and it is the information of the time at the point of coincidence of A and B, which changes from to.

Numerical example

For a round trip with 60% of the speed of light to a target in 3 light years distance, the following conditions are obtained (see graph above ): From the perspective of the twin on Earth every 5 years are required for the round trip. The factor of time dilation and length contraction is 0.8. This means that the flying twins on the way by only 5 × 0.8 = ages 4 years. This lower this time requirement can be explained with the fact that the distance has been shortened by the length contraction at its cruising speed of 3 × 0.8 = 2.4 light years. As the time passes more slowly on its assessment on the earth, on the earth seem to be directly = 3.2 years elapsed before his arrival at the distant star only 4 × 0.8. During the reversal phase but pass on the earth in his view additional 3.6 years. Together with the 3.2 years on the way back are thus also from the point of view of the flying twin on Earth passed a total of 10 years, while he himself is only aged for 8 years.

Exchange of light signals

Previously, it was shown what keep the observers, taking account of them known propagation speed of light for the real events. Below is described what both twins immediately see if they send once per year, a light signal to her brother. The paths of light signals in the above path-time diagram are straight lines with a slope of 45 °. Based on this example results in the diagrams opposite.

At first, the twins move away from one another so that the light beams are red-shifted by the Doppler effect. These light rays are shown in red in the diagram. Halfway through the journey, the twins move toward one another so that the light rays are blue-shifted, therefore, these light rays are shown in blue in the image. Due to the principle of relativity both observers measure the same time interval of 2 years between the red signals, and each half a year between the blue signals, which immediately becomes clear in the picture.

This results in the assumption that both twins would be the same old after returning, so that both twins would have received the same number of signals from the other, but now to a contradiction. For while the traveling twin immediately receives the time-compressed signals at the turning point and therefore after half the travel time to reach the earthly twin the stretched signals even longer. Due to the principle of relativity gets so the observer who gets in for a longer period blue-shifted signals, a total of more signals than the other. Thus, the traveling twin gets more signals than the twin on the earth, so that both consistently find that the traveling twin has aged more slowly.

In the numerical example in the opposite picture looks the traveling twin, due to a combination of relativistic effects and duration effects, the terrestrial age at first in 4 years to 2 years and in another 4 years to 8 years, for a total of 10 years. The earthly twin looks according to the traveler initially in 8 years age by 4 years, and then in 2 years to 4 years, making a total of 8 years.