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The Yamato is sent on a trip to Iskendar, 183,000 light years away. Their goal is to reach Iskendar and return to Earth before a year has elapsed. Due to the distance from earth, this will require faster-than-light travel to make it back in time. However, taking into consideration of how fast they are travelling time dilation would cause their return to be thousands of years later, after the invasion has already completed and earth has been terraformed.

How can they make it to their destination and back within their one year deadline?

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    Time dilation doesn't work like that. Time dilation wouldn't make a faster-than-light round trip take thousands of years. (Relativistic effects make a faster-than-light trip impossible, but the space-folding stuff you describe in your answer is impossible too. The natural extension of GR to hypothetical faster-than-light particles produces nothing like what you say time dilation would do.) – user2357112 supports Monica Jan 3 '16 at 10:23
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Time dilation is not an issue here because the Yamato is travelling using warps.

Krazer explained in chat that the Yamato warps work by folding space.

The theory of time dilation applies if you travel at faster-than-light speeds. However, in this case the Yamato is making short jumps or warps through space. The idea is similar to if you had a piece of paper with two points on it. Normally, the shortest path from Point A to Point B is to draw a straight line between the two points. However, one can cheat and fold the piece of paper to make the distance between these two points shorter in 3D space.

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This is the same thing the Yamato is doing when it is warping. Since they are not actually travelling at a faster speed, just getting to the destination faster, they can attempt to make it back in time without worrying about time delation.

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    "How did you get there in time?" "That nothing, just time dilatation, foldin' space and stuff y'know!" – Ikaros Jan 3 '16 at 8:02

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