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And that's not all. If you fly from the first time zone to the second one, your time will in fact go slower than if you had remained on the ground.
Even so, if you go fast enough, then the time will go slower for you (whas somebody quoting Einstein? ;)). If an astronaut (who travels at a high speed) is one month travelling at 60.000 km/h, s/he should age only 29 days. It's not accurate, is only a simplification. I'm not sure of the relation between time and speed diferences, but is something like the example. I readed it once and found it interesting.
Greetings,
Herenvardö