One that’s on the fringe of what you’re asking is warp drives. Right now it looks like you need ridiculous amount of energy and matter that may or may not exist… But General Relativity is okay with it on principle at least
Not really. It would require negative mass which as far as we know does not exist. And it would generate so much radiation in front of the warp bubble that it would decimate anything nearby when you stopped. There are tons of other major issues with it but those are just 2 I remember off the top of my head.
Putting aside the issue that it requires a negative energy density, there’s still the issue that it will necessarily violate causality, which is the reason FTL travel is considered problematic in the first place. Maybe it’s ultimately okay, but it may also mean that warp drives are fundamentally impossible.