Ah, yes, the good old Twin Paradox. The key here is to realize that there aren’t two different frames of reference here; there’s three. You, here on Earth, have only one frame of reference the whole time, but your brother has two different frames, one when he’s moving from Earth to alpha Cen, and one when he’s moving from alpha Cen to Earth. This might be easier to understand if your brother doesn’t have his own spaceship yet, and has to hitchhike: On the way out, he’s traveling with Zaphod, who just keeps on going in the same direction before and after this leg of the trip, and on the way back, he’s traveling with Marvin, who also doesn’t change his speed or direction. So our three frames now are bikebloke’s frame, Zaphod’s frame, and Marvin’s frame.
You can pick any frame you want to do your calculations, but you have to be consistent about it. In your frame, you stay at rest, and your bro travels away from you at high speed, and therefore stays younger than you, and then comes back at high speed, and stays younger than you then, too. Net result is that when he lands again, he’s younger than you.
On the other hand, in Zaphod’s frame, for the first leg of the trip, your brother is at rest, and you’re moving at high speed, so for the first leg of the trip, you age less than your brother, in that frame. But then, your brother leaves Zaphod’s frame and goes into Marvin’s frame, which is moving very fast relative to Zaphod. For the second leg of the trip, Zaphod sees himself still at rest, you moving fast relative to him, and your brother moving much faster than you, so for the second leg of the trip, Zaphod sees your brother aging much more slowly than you. So much more slowly, in fact, that by the time your brother gets back home, you’ve caught up with him and then some, and so now you’re older than him again.
We could do the same thing in Marvin’s frame, but I think that’s enough to get the idea across.
