The Twin paradox is a thought experiment in special relativity. Of two twin brothers one undertakes a long space journey with a very high-speed rocket at almost the speed of light, while the other remains on Earth. When the traveler finally returns to Earth, it is observed that he is younger than the twin who stayed put. Or, as first stated by Albert Einstein (1911):
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If we placed a living organism in a box ... one could arrange that the organism, after any arbitrary lengthy flight, could be returned to its original spot in a scarcely altered condition, while corresponding organisms which had remained in their original positions had already long since given way to new generations. For the moving organism the lengthy time of the journey was a mere instant, provided the motion took place with approximately the speed of light. (in Resnick and Halliday, 1992)
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This outcome is predicted by Einstein's special theory of relativity. It is an experimentally verified phenomenon called time dilation. One example is with muons produced in the upper atmosphere being detectable on the ground. Without time dilation, the muons would decay long before reaching the ground. Another experiment confirmed time dilation by comparing the effects of speed on two atomic clocks, one based on earth the other aboard a supersonic plane. They were out of sync afterwards, with the one on the plane being slightly behind.
The apparent paradox arises if one takes the position of the traveling twin: from his perspective, his brother on Earth is moving away quickly, and eventually comes close again. So the traveler can regard his brother on Earth to be a "moving clock" which should experience time dilation. Special relativity says that all observers are equivalent, and no particular frame of reference is privileged. Hence, the traveling twin, upon return to Earth, would expect to find his brother to be younger than himself, contrary to that brother's expectations. Which twin is correct?
It turns out that the traveling twin's expectation is mistaken: special relativity does not say that all observers are equivalent, only that all observers in inertial frames are equivalent. But the traveling twin jumps frames when he does a U-turn. The twin on Earth rests in the same inertial frame for the whole duration of the flight (no accelerating or decelerating forces apply to him) and he is therefore able to distinguish himself from the traveling twin.
There are not two but three relevant inertial frames: the one in which the stay-at-home twin remains at rest, the one in which the traveling twin is at rest on his outward trip, and the one in which he is at rest on his way home. It is during the acceleration and deceleration of the departure and arrival to Earth and similar accelerations at the U-turn when the traveling twin switches frames. That's when he must adjust the calculated age of the twin at rest. This is a purely artificial effect caused by the change in the definition of simultaneity when changing frames. Here's why.
In special relativity there is no concept of absolute present. A present is defined as a set of events that are simultaneous from the point of view of a given observer. The notion of simultaneity depends on the frame of reference, so switching between frames requires an adjustment in the definition of the present. If one imagines a present as a (three-dimensional) simultaneity plane in Minkowski space, then switching frames results in changing the inclination of the plane.
In the spacetime diagram on the right, the first twin's lifeline coincides with the vertical axis (his position is constant in time). On the first leg of the trip, the second twin moves to the right (black sloped line); and on the second leg, back to the left. Blue lines show the planes of simultaneity for the traveling twin during the first leg of the journey; red lines, during the second leg. During the U-turn the plane of simultaneity jumps from blue to red and very quickly sweeps a large segment of the lifeline of the resting twin. Suddenly the resting twin "ages" very fast in the reckoning of the traveling twin.
It is sometimes claimed that the twin paradox cannot be resolved without the use of general relativity, since one of the twins must undergo acceleration during the U-turn. This is false, for two reasons. First, most simply, the acceleration can easily be made to be a negligible part of the trip by making the inertial legs long enough. Second, it is no problem, in principle, to describe the effects of acceleration in special relativity as long as one does so using the laws of physics formulated in an inertial frame of reference — general relativity is only needed to make the laws of physics in the accelerated frame the same as in an inertial frame with a gravitational field. As Hermann Bondi once quipped on this question (French, 1968), "it is obvious that no theory denying the observability of acceleration could survive a car trip on a bumpy road," and special relativity certainly does not deny acceleration.
The Twin paradox and time dissemination
In the following discussion the same names will be used as in the Usenet Physics FAQ . The twin half who stays at home is called Terence, and the twin who travels into space is called Stella.
The essence of the Twin scenario is that it is a scenario of time dissemination. To see this, imagine a fleet of space-ships, travelling in interstellar space. The ships are cruising in formation (no relative velocity) so the clocks of all the ships count time at the same rate. They want a procedure to ensure all the clocks are synchronized. They can use radio-signals for that, in which case they need to take the transition time of the radio-signals into account, or they can use a separate ship that visits the other ships one by one, disseminating time. Either with radio-signals or a portable clock, the transition time must be taken into account.
The following adaptation of the twin paradox preserves the character of the scenario, while making graphical representation clearer. At the beginning Stella is already in her space-ship, and she cruises past Terence at a constant velocity. As they pass each other they both set their clock to zero. Stella cruises on with the same velocity, until at some point she makes a U-turn, and then she travels back with a constant velocity. When she flies past Terence for the second time the two compare clock readings. It is the synchronisation and later comparison of clock readings that counts.
Another simplifying step that preserves the character of the scenario is to allow yet another sibling. At some distance away from Terence Stella passes a sister, we shall call her 'Allets'. As Allets cruises past Stella she synchronizes her clock with the clock of Stella and after that Allets cruises on to the point of rendez-vous, where she and Terence compare clock-readings.
This facilitates an interpretation of the Twin paradox in purely geometric terms.
In space, Allets retraces the (straight) path that Stella had taken moving away from Terence. In the space-time diagram, that looks like a triangle.
A remarkable feature of the geometry of time dissemination is that the difference in time at the point of rendez-vous is independent of the velocity of either Stella or Allets. Suppose that the space journey is 300.000 kilometers out into space and back again. It would take radio signals 2 seconds to make that round trip. When there is a time dissemination relay by space-travellers then it takes overall more time to close the loop, but in the end the difference in counted time is the same 2 seconds.
This independency on the travel velocity of the time dissemination relay can be seen in the space-time diagram. To focus on the most simple case, we put the events somewhere in interstellar space, where the gravitation of surrounding suns cancels almost completely. Terence moves inertially all the time.
The space-time diagrams for different velocities of travel are different in shape, but in the end the difference in counted time is 2 seconds if the trip is 300.000 kilometers out into space. In terms of geometry this is Pythagoras' theorem.
When time is disseminated, then the path the time-disseminating relay takes matters. Distance is time. a space ship must accelerate In order to cover distance, but it is not the acceleration itself that causes the difference in time, only the length difference of the paths taken matters. When Stella's worldline in the space-time diagram is not straight, when she is accelerating now and then, then still only the length of her path with respect to the inertial frame matters.
This property of space-time was discovered in the field of physics, but it was not immediately seen that it could be interpreted geometrically. The mathematician Minkowski showed that if it is assumed that there is an upper limit to velocity, then it is unavoidable to formulate what today is known as Minkowski space-time. Galilean relativity can only hold good in a universe with absolute space and absolute time, and no upper limit to velocity. Can the principle of relativity of inertial motion and an upper limit to velocity co-exist? They can, if the geometry of space and time is Minkowski space-time geometry. 2
To understand the Twin paradox it must be observed what the minimal requirements are to elicit the Twin paradox. Those minimal requirements are:
- There is dissemination of time, either by signals, or by way of a portable clock.
- A loop is closed; the time dissemination relay takes two different paths.
References
- A. P. French, Special Relativity (W. W. Norton: New York, 1968).
- Robert Resnick and David Halliday, Basic Concepts in Relativity (Macmillan: New York, 1992).
External links