I had a feeling that all of you would put holes in this theory. I looked for intormation on this at another site and someone has a theory that all of Mars' water is now on Earth.
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Q: To a person on that rocket that traveled 10 trillion Km in that year, would it be a year to them?
A: Yes, it would be one year to them.
Q: If a rocket left Earth at close to the speed of light and went straight for a year, it would be 10 trillion km away.
A: Yes, the distance would be how far it could travel in one year.
Q: If this were to be possible would it seem like the rocket crew lived to be hundreds of years old to everyone on Earth?
A: Yes. I think that would be true, were it possible. (I think that was the premise of the first "Planet of the Apes" movie. The subsequent movies went into time travel, which does not appear to have much basis in science at this point in time.
The "trick" here is to dispel any notion that there is some sort of master clock for the Universe. Time is not a constant. For a particular body, time is a function of the conditions that body is subject to.
I found it difficult to think this way. From a Human perspective, time seems to be one of the constants we can rely upon. Unfortunately, we can't, and that's been proved again and again.
(Sorry for the Q: A: inserts, but it seemed like one way to minimize confusion.)