# Twin paradox

The**Twin paradox**is a thought experiment in special relativity (SR): of two twin brothers, one undertakes a long space journey while the other remains on Earth. When the traveller finally returns to Earth, it is observed that he is younger than the twin who stayed put.

This outcome is predicted by special relativity ("time dilation of moving clocks") a phenomenon which has been verified experimentally. 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 comfirmed time dilation when it compared the effect of speed on two atomic clocks, one based on earth the other abord a supersonic plane. Both were out of sinc slightly afterwards, the one on the plane being behind slightly.

The apparent paradox arises if one takes the position of the travelling twin: from his perspective, his brother on Earth is moving away quickly, and eventually comes close again. So the traveller 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 travelling 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 travelling 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 travelling 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 (if we ignore the comparatively small acceleration resulting from Earth's mass and movement) and he is therefore able to distinguish himself from the travelling twin.

The confusion arises because 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 travelling 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 U-turn, when the traveling twin switches frames, that 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 notion 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 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.

In resolving the paradox, it is sometimes claimed that special relativity cannot be applied to accelerating bodies, and that general relativity has to be used, but this is not correct. For instance, the age of both the Earthbound and travelling twin can be correctly calculated by integrating the spacetime interval (or *proper time*) over the spacetime paths they make in any inertial frame (these paths are known as the twin's *worldlines*). Similar methods can be used to calculate the relativistic behaviour of an accelerating spacecraft (see relativistic rocket). SR only becomes inapplicable when the effect of gravity is non-negligible, in which case general relativity must be used.

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