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Special relativity teach us there is no way to deduce which body is moving and which one is standing still and that it is all frame of reference dependent. But in the case of the two twins that are moving at different speeds we can say both are moving away from each other and then getting close or we can say one was standing still while the other moved away and then got close to his brother. And all would be reasonable if just one of the two wouldn't become older for a fair amount of time. Does it imply that the one who became older was the one standing still? And that an observer from a frame of reference looking on them remained younger for just an amount of time depending of his own velocity?

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The twin paradox consists precisely in spotting why the argument given here in your question is incorrect. It is all about showing why the two people's journeys are indeed different in an absolute way.

The answer is that one of the twins follows a non-straight path in spacetime, while the other follows a straight path in spacetime. In order to say which path in spacetime (called a worldline) is straight, one needs a way to define inertial motion, and this too can be done in an absolute way. Inertial motion can be defined by setting up a framework of rods and clocks, and then checking whether Newton's first law holds for test particles moving freely, with distance and time measured by the rods and clocks. If all such test particles don't accelerate relative to the rods and clocks then the frame is an inertial frame.

In the twin paradox, the twin following the straight path in spacetime, as measured by an inertial frame, accumulates the most amount of time along their worldline.

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