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Special Relativity - The Twin Paradox

The so-called 'twin paradox' is one of the most discussed and misunderstood results of special relativity. I say 'so-called' because if relativity really did lead to a paradox, then the theory would have to be abandoned or modified. However, the twin 'paradox' is not a paradox at all - it just requires some careful consideration.

Lets start by recalling the Lorentz Transformations introduced in Lesson 3:

special relativity lorentz transformations

The important equation for this discussion is the last one. This equation explains how time is perceived in different frames of reference. This equation describes what is known as 'time dilation' and it is this time dilation that forms the basis of the twin paradox.

Consider the follow experiment. Alex and Bob are twins. One day, Bob decides to board a high speed rocket for a round trip journey to a distant star. Apart from quite short periods of acceleration and deceleration near to Earth, Bob travels away from Earth with a velocity v (which is close to the speed of light).

Now suppose the distant star is 20 light years away. It therefore takes Bob a little over 40 years to complete his round trip journey. Alex, who has remained on Earth, waits 40 years for Bob's return. However, as Alex knows about special relativity, he can apply the time dilation equation to calculate that time has run more slowly for Bob. Alex calculates that Bob will actually have only aged by, say, 4 years on his journey. Just to make it clear - from Alex's point of view - Alex will have aged 40 years, but Bob will have aged only 4 years.

So far so good. Strange - but not paradoxical. The so-called paradox arises when we consider the same journey from Bob's point of view. From Bob's point of view, he remains stationary and the Earth and Star move relative to him. In other words, from Bob's viewpoint it is Alex moving with velocity v. Using the same logic as before, Bob experiences 40 years of time and calculates that Alex has experienced only 4 years. There is now an apparent paradox - when the twins are reunited back on Earth, who has aged 40 years, and who has aged 4 years? They can't both be right!

Luckily for special relativity, there is a flaw in the above thought experiment. The paradox rest on the assumption journeys taken by the twins can be considered as equivalent. However, the journeys undertaken by the twins cannot be considered as equivalent. Whereas Alex remained stationary on Earth, Bob underwent three periods of rapid acceleration at the beginning, middle and end of his journey. It is these periods of acceleration that screw up our calculations. During the periods of acceleration, Bob is in a non-inertial frame of reference, and the laws of special relativity do not apply. Special relativity only deals with transformations between inertial frames of reference. That is to say, special relativity only considers frames of reference that are not accelerating relative to each other. Special relativity does not tell us how to deal with transitions from one non-inertial frame to another.

On the other hand, Alex has remained in a single inertial frame for the whole experiment. In this case, we can apply special relativity. The (correct) result of the experiment is therefore that Alex has aged 40 years and Bob has aged only 4 years. No paradox.

Dodgy thought experiments are a mainstay of the 'Einstein was wrong' school. I suggest that readers are very wary of such thought experiments, as the flaws can be quite subtle. I hope that the twin paradox serves as an example of this.

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