The twin paradox

Let's expand the relativistic journey to a star with the popular twin paradox. We assume that observers A (gray color) and B (blue color) are twins who are the same age when the spacecraft leaves the Earth (origin event) and that the spacecraft starts its way back to Earth when it reaches the star (event E1). Both twins measure during the journey that the time of the other twin passes more slowly both on the way out and on the way back (reciprocal time dilation), so the paradox that arises is: when they meet again on Earth, both twins should expect to find the other twin younger, which cannot be. Which twin will be the youngest?

As we're going to see in the diagram, there's no such paradox and it can be easily understood with special relativity. You simply have to take into account the relativity of simultaneity, which is a consequence of the space-time continuum. Let's assume that the spacecraft changes its trajectory at the star almost instantly, so that we can perform the calculations within special relativity. To solve the paradox, it's not necessary and may be confusing to enter general relativity (acceleration effects on the ship).

Diagram 17: twin paradox

During the journey, twin A on Earth measures that time passes slower on the ship. He measures the same time dilation on the ship both on the way out and on the way back (reciprocal time dilation).
During the journey, twin B on the ship measures that time passes slower on Earth. He measures the same time dilation on Earth both on the way out and on the way back (reciprocal time dilation).
Twin A on Earth measures that the spacecraft takes 5 years to reach the star and another 5 to return to Earth, making a total of 10 years.
Twin B on the ship measures that it takes 4 years to reach the star and another 4 to return to Earth, making a total of 8 years.
When they meet again on Earth, twin A sees it normal for twin B to be younger. Throughout the journey, the time on Twin B's ship has passed slower.
When twin B reaches the star (event E1) he changes his trajectory and reference frame to return to Earth. He goes from reference frame (ct ', x') to frame (ct '', x ''). This has a surprising consequence: its "now" suddenly goes from being simultaneous with Earth at T = 3.2 years from the origin event (0, 0), to being simultaneous with Earth at T = 6.8 years from the origin event. This is the reason why he arrives younger on Earth despite the fact that throughout the journey, B measures that twin A's clock on Earth run slower than his.
Therefore we can see that really there's no paradox. The paradox only arises if we don't take into account the relativity of simultaneity, which is a consequence of the space-time continuum.