#Deep Space 9
Well, see, of course the science breaks down. That's why the FICTION part
of the title in Science Fiction.
Obviously, in an Einstein-ian Universe, travel at greater than C is
impossible. We know that, and the people who make Star Trek know that.
However, a weekly show in which the ship is still enroute to it's next
adventure for several weeks, and we watch an hour of "transit" wouldn't be
very interesting, and so they bend the rules. They make a
"pseudo-scientific" explanation, that sounds good, but that a real
physicist could punch holes in quickly. The writers, for the most part,
aren't physicists, and just use the jargon that was used in previous shows
and hope it holds up (which is where the fans come in. WE go to great
pains to "fix" the mistakes the writers make, and show how they weren't
mistakes. I remember a several week (month?) long discussion about how the
Warp figures in ST:TNG are different than the Warp figures in the original
series. In one early episode, they gave a distance and a time to travel at
Warp 9. From that, the math was simple, but it didn't match similar data
from 20 years earlier).
In any case, the "Warp" process bends space, and thus the Enterprise
doesn't actually travel faster than C, but arrives in an amount of time
such that it appears to have travelled faster than C. Higher Warp, you get
there exponentially faster, and use more fuel.
Where does the anti-matter come from? Good question. They have a magnetic
field to contain it, and they control it with Dilithium Crystals
(techno-babble, meaning they made it up), but they never discuss where it
comes from. But! Remember the original series, Kirk had to constantly sign
Fuel reports. So, you are right, it must come from SOMEWHERE! But
logistics doesn't make such a good TV show, either, so they never delve
into it (though we HAVE seen Dilithium mines).