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yep, according to special relativity a moving clock appears to go slow. So from the point of view of the people on Earth, the spaceship clock ticks slowly and passengers age more slowly than themselves. But the odd thing is this: from the point of view of the the passengers, the clocks on Earth are the ones moving and running slow! (this is known as the Twin Paradox btw)
As David says, it's the acceleration that takes it out of SR's domain and breaks the symmetry between Earth and the spaceship people, so it the passengers who stay young.
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