Well we're nearly at the end of our discussion of special relativity, and we have seen that various strange phenomena can arise when we start observing objects or phenomena in moving frames of reference.  However, the effects of time dilation, length contraction and mass increase only become significant when the speeds involved are very high.  Nevertheless, such high speeds can occur, especially in the atomic world, and in fact we need our results on time dilation and length contraction to explain why cosmic muons are observed at the surface of the earth.

The time dilation, length contraction and mass increase results are amazing in themselves, but we have to note two important factors about these results, which are:

1.  The expressions for the magnitudes of these effects only apply when considering inertial frames of reference.  That is, frames of reference that are moving at a constant speed relative to each other.

2.  The effects are reciprocal, i.e. in the examples we have considered so far, an observer in EITHER frame of reference sees the various effects in the other frame of reference.

The second point is important, and arises because either observer sees a frame of reference moving relative to his own.  We used this fact (without really stating it) in our explanation of cosmic muons reaching the earth's surface.  This is illustrated in the following diagram.

The two points about inertial frames and reciprocity lead to a number of apparent paradoxes and contradictions when they are forgotten or not applied correctly in some situations.  Perhaps the best-known paradox in relativity, of this type, is the so-called "twin paradox".

The twin paradox works like this (adapted from Beiser's treatment in his book Concepts of Modern Physics):

1.  Consider two twins, each aged 20 years.  Let's call them John and Sue.

2.  Suppose that Sue gets into a rocket ship, and makes an outward journey of 20 light years at 80% of the speed of light.  (A light year is a unit of distance, equal to the distance that light travels in one year, which is about 6,000,000,000,000 miles ...)

3.  Sue then turns around and comes back, also at at a speed of 80% of the speed of light.

Now, let's analyse the times required for the journey, and how they are perceived by John and Sue.  As you might guess, the effects of time dilation are going to come into play here!  Note that both John and Sue have biological clocks, and the effects of time dilation are therefore applicable to the life cycle of our twins.  John's frame of reference is the surface of the earth, and Sue's frame of reference is the rocket ship.

4.  To John, Sue is in a moving frame of reference, and so appears to be living more slowly that John, because of time dilation.  In fact, application of the time dilation formula shows that to John, the total trip (20 light years out and 20 light years back) seems to take 50 years, and so when Sue comes back, John is aged 70 years (20 at the start + 50 year journey).

5.  However, to John, Sue's clock has been running more slowly (because of time dilation), and in fact applying the time dilation formula shows that her biological clock has advanced only 30 years - and so Sue is aged 50 years (20 at the start + 30 years during the journey).

Now for the interesting bit, and where the paradox arises!

6.  As we noted above, the time dilation effect is reciprocal, so that as far as Sue is concerned, John is in a frame of reference that is moving away from her, and so to Sue, John appears to age more slowly, and on her return it might be expected that John is only 50 years old and she is 70 years old, i.e. the exact opposite of what John expects!!

This is the twin paradox, possibly the most discussed paradox of them all.  The resolution is actually quite simple.  We noted above that the time dilation formula that gives the magnitude of the time dilation effect is only valid for inertial frames of reference.  Now, only one of our twins, John, is present in an inertial frame of reference at all times.  So, only John is entitled to apply the time dilation formula for the whole journey, and thus deduce that Sue ages 30 years while her ages 50 years.

Sue is only in an inertial frame for the outward part of her journey.  This is because in order to come back, she has to slow down, turn around and then get back up to speed again.  This involves changing from one inertial frame to another.

The correct conclusion from the space trip, therefore, is the one observed by John, namely that John is aged 70, and Sue is aged 50, on her return.

The moral of this story is that there is in fact no paradox about the twin paradox, only a faulty application of the principles of relativity and the associated formulae.

THE END