The Twin Paradox using Simultaneous Events

Simultaneity (or lack thereof) is a terrific tool for understanding many of the paradoxes associated with SR. And, if I am to be thorough, simultaneity must be considered for all SR events between separate frames of reference. Let's re-visit the twin paradox (John travels out 12 hours at 60% the speed of light and returns at the same speed). Basically, there are three frames of reference to consider. First, the twins are on the earth with no relative velocity between them. Second, John embarks on the outgoing leg of his trip. Thirdly, John (after instantaneously turning around) embarks on his return leg of his trip. I am using the same example as before, except I am using numbers from the Lorentz Transforms as opposed to the Relativistic Doppler Shift to explain the observed phenomena.

1st frame:

Hunter and John each agree on everything they observe. This should be easy to understand since there is no relative velocity between the two twins. They are in motion together.

2nd frame:

John travels out 12 hours by his clock. With the two postulates in mind, we realize that Hunter observes time dilation for John's outgoing trip. Thus, if John records 12 hours, Hunter will record 15 hours. Remember that at 60% the speed of light, the time dilation will be 80%. Therefore, if John records his time to be 12 hours, this is 80% of what Hunter records - 15 hours. But what does John observe for Hunter's time? He observes the time dilation as effecting Hunter; therefore, he measures his trip to be 12 hours, but he observes 9.6 hours (80% of his clock's time) for Hunter's time.

2nd frame totals:

Hunter measures his time to be 15 hours, but John's time to be 12 hours. John measures his time to be 12 hours, but Hunter's time to be 9.6 hours.

Obviously, the event, which is the end of the outgoing trip, is not simultaneous. John thinks Hunter's time is 9.6 hours but Hunter thinks his time is 15 hours. On top of that, they both think that John's time is 12 hours, which doesn't agree with either of the first two times.

3rd frame:

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.

Trip totals:

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.

Many people have trouble with the twin paradox because of the way in which the frame change is handled. In this case, the jump on John's clock for Hunter after the frame change (9.6 to 20.4 hours) is the problem. There really is no problem here. If you want to integrate the acceleration to use various inertial frames during the turn around, it can be done (with the same results). Another common approach is to imagine someone else in space that passes John just when he reaches the point of his turnaround. This person is heading towards Hunter at the same speed that John was travelling, so there is no need to consider John any further. The key fact is that if we then went back in the substitute's frame and looked at his clock for Hunter, it would show that some amount of time had already been recorded when the substitute began his trip towards Hunter. How far back should we go? Since John traveled out 12 hours on the outgoing trip, we should go back 12 hours in the substitute's frame. At this starting point for the substitute, his clock for Hunter would read 10.8 hours. This is extremely important. It clearly shows that both twins or the twin and the substitute observe the other as having slower times. The big shift occurs when the frame of reference is changed. This means that both observe the other to have a slower time during the actual outgoing and return trips, but there is a shift during the frame change that more than makes up for John's account of Hunter's slowly running clock. After the frame change, the damage has been done. John will still observe Hunter's clock to run slow, but it will never slow down enough to compensate for the 10.8 hours that were perceived during the frame change. Is this time jump a physical occurrence? No. The time jump occurs because when John changes frames, he is no longer using the same event as a reference. When John made his turnaround, the event in Hunter's frame that John thought was simultaneous with his turnaround changed. John's frame change caused this confusion because his new frame uses a different time for the event in Hunter's frame. More clearly, the turnaround event in Hunter's frame has a different time value for the outgoing leg and the return leg, as perceived by John. Keep in mind that in the above references to Hunter's frame, I'm really talking about what John thinks Hunter's frame time would be. This time difference is only apparent to John because it is his frame change that causes the discrepancy. In Hunter's frame, nothing changes for Hunter when John changes frames. Here again, by realizing that the two events are not simultaneous, the paradox is resolved. The point I am trying to emphasis is that there are a variety of ways to handle the paradox. All of the methods yield the same result, but if you actually consider the simultaneity of the situation, then the how's and why's become more clear.