Traveling at light speed will cause time to go by faster. So if I travel at light speed around Earth will I end up in the future with people from the future, or will I just have went through time faster with no real affect?
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Relativity. You're going faster, so time's going to move slower for you. The people on Earth will experience what you experienced at a much greater extent in terms of time.
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Uh. . . what? The closer you get to lightspeed, the more time slows for you. I don't know the exact ratio, but depending on your speed you might age one year while the Earth ages several decades.
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The latter.
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ming=blown