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Twin Paradox

There is a famous so-called paradox in Special Relativity that has confused many students at one time or another – including myself. In this thought experiment, an identical twin makes a high-speed journey into space, and on returning to Earth finds that he has aged less than his twin who stayed behind.

 

The apparent paradox is because the effects of Special Relativity are symmetrical and reciprocated between two inertial observers, as we’ve already explained. Hence, since the Earth-bound twin can be considered moving relative to the astronaut twin then neither should see a difference when they meet again.

 

The argument is further complicated by the popular view that Special Relativity implies time slows down when you’re moving close to the speed of light – which it doesn’t.

 

The explanation of the effect – which has been verified by experiment – is that the astronaut twin is distinguished by undergoing an acceleration or deceleration during the turn-around. It doesn’t matter whether you consider that the Earth was initially at rest or moving since the effects of an acceleration or decelerations would be indistinguishable to the astronaut twin. However, whereas all inertial motion is relative, acceleration and deceleration are absolute, and only one twin would have experienced them. It’s not the acceleration or deceleration itself that causes the aging difference – it’s the fact that the returning twin changed from one inertial frame to another during that turn-around. Before the turn-around, both twins see the other’s clock as recording time more slowly, just as we’ve explained already for two inertial frames of reference. After the turn-around, though, the astronaut twin has a different inertial frame of reference. The relative simultaneity between the new frame and the Earth frame is then different and the astronaut sees a sudden change in the age of his Earth-bound twin.

 

In summary, the effect is not due to either twin simply travelling close to the speed of light, or due to one of them experiencing forces during the turn-around. It is due to there being three inertial frames involved rather than two. The two frames for the travelling twin are inclined differently to the Earth frame in Minkowski space-time, and jumping from one to the other changes the relative simultaneity of the two twins.
 
 

 

In this diagram, the ‘Planes of Simultaneity’ are lines in the Earth frame (S) corresponding to a constant time t′ in the astronaut frame (S′). The equation relating t′ to S is:

 

t - vx/c2 = constant

 

In other words, the equivalent simultaneity in S depends on the location.

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Tony Proctor,
24 Nov 2011, 09:53