Lack of Simultaneity
From Hunter's perspective, nothing new has happened. He remained in his initial frame of reference and John returned at the same velocity he left with. Therefore, Hunter measured the return trip to take 15 hours for his frame (same as the outgoing trip) and observes the trip to take 12 hours for John. From John's perspective, he encountered a major change. He actually changed frames from one of traveling out to one of traveling back. Now, at the start of the return trip, when John looks at his clocks, he observes his clock to read 12 hours and Hunter's clock to read 20.4 hours. Think about this. John now shows that Hunter's clock has jumped ahead from 9.6 hours to 20.4 hours. How can this be???? When John changed from the 2nd frame to the 3rd frame, the established symmetry between Hunter and John was broken. Thus, each views their own time as having no change. And since John was the one that actually changed frames, he showed more elapsed time for Hunter. From here on out, it is business as usual. The return trip is clocked at 12 hours by John, but he observes 9.6 hours for Hunter. Again, let's clean this up…
3rd frame totals:
Hunter measures his time to be 15 hours, but he measures John's time to be 12 hours. John measures his time to be 12 hours, but he measures Hunter's time to be 9.6 hours. Remember, this 9.6 is only for the return trip after the frame change.
Hunter measured his time to be 15 hours for the outgoing trip + 15 hours for the return trip…30 hours.
Hunter observed John's time to be 12 hours outgoing + 12 hours return …24 hours.
John measured his time to be 12 hours outgoing + 12 hours return…24 hours.
John observed Hunter's time to be 20.4 hours (after outgoing trip and frame change) + 9.6 hours for the return trip…20.4 + 9.6 = 30 hours.
Can you find any events in which both John and Hunter agree on the time for both themselves and the other? No, you can't. The lack of simultaneity is the key to the paradox. Both twins are measuring and observing. Unfortunately, they are not measuring and observing the same events. It is impossible for them to consider something like the end of the first leg as simultaneous when they each view it occurring at different times for Hunter. It's interesting to note that the results are the same as the Relativistic Doppler shift results. Is there a pattern here? SR allows for various methods to be employed to resolve the problems. For this case, use of space-time diagrams (there's those words again) would clearly show every point that we have talked about. I have merely used the Lorentz transforms in combination with the Relativistic Doppler effect.
We'll look at problems with the twin paradox in the next section.