|MadSci Network: Physics|
It's also pretty hard to answer. The problem is this: contrary to what you can read on the Internet, antigravity isn't just technologically impossible. According to our current understanding of physics, it's also physically impossible. So it's kind of hard to put it into the context of physics.
But suppose there was some kind of matter -- call it N-matter for fun -- which, instead of being attracted to other matter, was instead repelled by it (NB: the antimatter we know about doesn 't work like this). How would it behave? It would try to get away from other matter. But since there's matter everywhere, eventually our N-matter would wind up someplace where it was repelled equally from objects in different directions. Then it would just sit there.
Would it end up in the center of the universe? No, because there is no such place. Seems to me that the N-matter would have to participate in the general expansion of the universe. The expansion involves all of space-time, so there's nowhere else for the N-matter to go.
Better for science fiction might be some kind of repulsive force that only works over short distances. This way a spaceship or floating city, or whatever, could be repelled by the Earth but not, say, by the Sun. Then things would work more or less as science fiction describes them. (There is an astrophysical precedent of sorts for this: the cosmological constant, which is kind of like a repulsive force that only works at large distances.)
I would love to recommend some reading on this topic, but I don't know of any "serious" (as opposed to "crackpot") physics treatments of antigravity. Maybe in my spare time I will work on it..
[Moderator note: Negative mass is a fun concept. A piece of N-matter is repelled by and repels a piece of normal matter, however N-matter moves in the opposite direction of the applied force. This means it starts to chase the piece of normal matter! I found this link with a casual search: Negative Mass]
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