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Twin ParadoxThe 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): | idth="60%" bgcolor="#ffffff" style="border:1px solid #c6c9ff;padding:1em;padding-top:0.5em;"| 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) | 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 thecalculated 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. Alternative resolution of paradox Consider a space ship going from Earth to the nearest star system; 4.45 light years away; at 0.866 c. The above image shows the the ship with its 0.5 length contraction. To an observer on Earth the trip will take 5.14 years. Allowing for a 5 year stay time, produces a round trip time of 15.28 years. However to the ship's crew the stars and the distance between them will be shortened in the direction of motion to 0.5 of what is observed on Earth, resulting in a travel distance of only 2.23 light years and a corresponding 2.57 year travel time. This produces a round trip time of only 10.14 years. Thus the ships crew would experience less time than those on Earth. If one of the astronauts on the ship had a twin that stayed on Earth. he would return home to find his brother 5 years older than himself. 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).
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