This post is my notes for reading an online article published on Scientific American about resolving the twin paradox purely in the special relativity framework. Refer to Ref. [1, 2] for the original article.
In the article, the Lorentz transformation relation was applied to the whole process corresponding to the traveler launching from the earth to landing back to the earth. The final conclusion is both the homebody and traveler agree that their clock reading is different – the traveler is younger than the homebody. The discussion was divided into two sections - the traveler traveling towards to the remote star and the traveler traveling back from the remote star to the earth. However, the fundamental rationale that resolves the twin paradox is actually not the way being used to do the calculation. As pointed out in the article, it is the asymmetry between the homebody and the traveler that makes the reading of their clocks are meant to be different. In another word, the statement that the homebody can be thought of as being traveling relative to the traveler and therefore the homebody can also be rationalized to be younger than the traveler following exactly the same logic actually does not stand at all.
At this point, we may need some clarification, based on the discussion presented in the article. First, as mentioned earlier, although the calculation can be done in two sections (which is sort of tedious), the end result is very easy to obtain. Taking the homebody as the reference, we can easily use either length contraction or time dilation to arrive at the conclusion that the traveler will be younger than the homebody. It is just that the article provides a more straightforward picture to view this whole process. Second, in the article, it was mentioned two types of asymmetry. From my understanding, it is actually only the second asymmetry that really matters. The first asymmetry says that the traveler is really traveling, i.e., his position is really changing from one place (the earth) to another (the remote star) but that is not the case for the homebody who has been always staying on the earth without moving at all. However, since we know that according to relativity theory, there does not exist any absolutely static reference. Then we can say that both the traveler, the homebody, and the remote star are all traveling relative to another third-party reference. Then the traveling of the traveler just makes himself stay still in the third-party reference. In this case, it is actually the homebody who is really traveling and therefore if really it is the first asymmetry mentioned in the article that resolves the paradox, the logic is not self-explaining. The second asymmetry is therefore what I think as the fundamental asymmetry that resolves the asymmetry. Although, we can have the imagined third-party reference as mentioned here, it is always that case that it is the traveler meets the remote star (i.e., they are in the same location at the same time) that defines the event-3 (see the original article for details). Therefore, the two brothers can only agree upon the clock reading of the traveler but not vice versa.