As the traveler reaches the star he reads his clock at eight years as mentioned, but he sees the homebody?s clock as it was six years ago (the amount of time it takes for the light from the earth to reach him), or at four years (10-6). So the traveler also views the homebody?s clock as running half the speed of his clock (4/8).
On the trip back, the homebody views the traveler?s clock going from eight years to 16 years in only four years' time, since his clock was at 16 years when he saw the traveler leave the star and will be at 20 years when the traveler arrives back home (event 3). So the homebody now sees the traveler's clock advance eight years in four years of his time; it is now twice as fast as his clock. On the trip back, the traveler sees the homebody?s clock advance from four to 20 years in eight years of his time. Therefore, he also sees his brother?s clock advancing at twice the speed of his. They both agree, however, that at the end of the trip the traveler?s clock reads 16 years and the homebody?s 20 years. So the traveler is four years younger. The asymmetry in the paradox is that the traveler leaves the earth?s reference frame and comes back, whereas the homebody never leaves the earth. It is also an asymmetry that the traveler and the homebody agree with the reading on the traveler?s clock at each event, but not vice versa. The traveler?s actions define the events.
The Doppler effect and relativity together explain this effect mathematically at any instant. The interested reader will find the combination of these effects discussed in The Fundamentals of Physics, by David Halliday et al. (John Wiley and Sons, 1996). Paul Davies also does a nice job explaining the Twin Paradox in his book About Time (Touchstone 1995, ppf 59.) My explanation follows Davies?s closely; I hope my graph adds further clarity. The reader should also note that the speed that an observed clock appears to run depends on whether it is traveling away from or toward the observer. The sophomore physics problem, mentioned earlier, is a special case as it applies only when the motion of the traveler passes the observer?s reference frame with no separating distance in the direction of motion.
For those with a little more formal physics background, a spacetime diagram also explains the paradox nicely. It is shown with the supporting calculations for the Doppler effect on the observed time. Proper time is time in the frame of the observer.?