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.
We'll look at time travel in the next section.