Joe observes that Jane's on-board clocks (including her biological one), which run at Jane's proper time, run slowly on both outbound and return leg. He therefore concludes that she will be younger than he will be when she returns. On the outward leg, Jane observes Joe's clock to run slowly, and she observes that it ticks slowly on the return run. So will Jane conclude that Joe will have aged less? And if she does, who is correct? According to the proponents of the paradox, there is a symmetry between the two observers, so, just plugging in the equations of relativity, each will predict that the other is younger. This cannot be simultaneously true for both so, if the argument is correct, relativity is wrong.
The naive interpretation--the reason why the situation is called a paradox--is to assume that the situation is competely symmetrical. If that were the case, Jane's diagram would simply be a mirror image of Joe's. But Special Relativity applies only to the relations between inertial frames of reference. In this regard, the situations of the twins are definitely not symmetrical. Joe is in one inertial frame throughout.(We discuss the partial symmetry below.)
Jane certainly knows that she has not been in the same inertial frame for the whole trip: in order to stop the outward journey and to commence the return, she had to turn the ship around and fire the engines hard and long. During that time she knew that she was not in an inertial frame
n the second of Jane's inertial frames (the homeward trip), she receives a lot of anniversary messages from Joe. If she pretends that she has been in this same frame of reference all along (the dashed line extrapolation of her returning world line), i.e. if she assumes that she has been travelling towards Earth at constant v for six of her years, she would conclude that Joe had been sending them for eight of his years (follow the dashed lines). Now this is a strange assumption
It it better to go to the site and look for yourself. It has drawings and even a simple animation to help you understand this concept.Let's now assume that Jane is not naive, that she knows about relativity, that she remembers the acceleration, that she remembers being with Joe at the beginning of the trip and that she uses this knowledge in analysing her version of the space time diagrams. First, once she has left the Earth, accelerated and is travelling without acceleration towards her destination, she can apply Special Relativity. She observes that the distance between the Earth and her destination has shrunk.(See Relativistic time dilation, simultaneity and length contraction for an explanation.) It has shrunk by the factor 1/γ =(1 − v2/c2)1/2 = 0.75, so she now only has to travel for three of her years to get there. Similarly, in her return trip (another inertial frame so she can use Special Relativity again) the distance is also shorter, so she only has to travel for three years to get there. So Jane's space time diagrams are those shown at right.(We repeat the diagram.