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Notes -
Laser sail/Bussard brake. Antimatter ramjet. (The Bussard idea doesn't work as far as we know, because of the scoop's drag and the difficulties getting p-p fusion to happen in a scramjet throat, but it works as a brake - or injecting antimatter into a scramjet throat would certainly get it to burn.) Baryon number nonconservation does also seem allowed, which implies that non-antimatter-based total conversion engines could be possible. There is, indeed, trouble with keeping a ship intact in the intergalactic void given the cosmic rays, relatively-relativistic dust, and lack of new material to replace that blown off - but that's mostly an issue of building a bigger ship.
General relativity does allow for FTL in the broad sense of "get from A to B faster than light conventionally could" - the Alcubierre metric and wormholes being the most obvious. And there are some plausible answers to the time-travel problem (there's a hypothesis that attempting to convert a wormhole into a time machine would collapse it, for instance). Whether FTL's possible is an open question.
Okay now I'm getting into things I'm not too certain on (obviously IANAP), but from what I understand apparent FTL that entails the warping of spacetime is one of these things that we're not 100% sure is impossible but does pose a lot of problems. Apart from the whole "closed timelike curve" problem that these apparent FTL methods seem to create (which, granted, as you noted one can try to resolve through all kinds of difficult-to-verify chronology protection conjectures), there's also the fact that both Alcubierre drives and traversable wormholes alike require unobtainum exotic matter that at best isn't impossible but there's no evidence for its existence and at worst violates an energy condition.
So they're not exactly impossible per se, but there's reasons to believe they probably are.
Clarke's First Law is a decent heuristic, and there's no clear no-go theorem (Earnshaw's theorem is the obvious example of a theorem with a lot of important loopholes). I recall reading about somebody trying to build an Alcubierre metric using the Casimir effect, though I'm not sure how it turned out and that's well beyond my own paygrade.
Overall I'd say it's in the "maybe" category; I'm leery of saying it can be done, but at least as leery of saying the opposite.
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