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I remember back in my uni days i attended a special physics lecture about coming up with ways to travel FTL. One of the interesting ideas that they talked about was taking advantage of the fact that matter bends spacetime. The idea being that you could bend spactime in such a way that the spaceship would be able to travel on wave of sorts. The wave would then travel FTL even though the ship's local speed wasn't.
The problem came when trying to solve the equations for the mass that would allow this to occur. The simple solution always lead to negative mass! There was no known positive solution because it was too difficult to solve. But obvously since we're talking bending spacetime it would be significant.
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