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There's a solution to the Einstein equations which works something like a hyperdrive: https://en.wikipedia.org/wiki/Alcubierre_drive … Unf it's not clear it can be made physical - it has an unusual & probably unphysical stress-energy tensor. There's also more, ahem, conventional solutions like wormholeshttps://twitter.com/paulg/status/1122993861301764096 …
TBC, there are quite a few objections to the Alcubierre drive, as discussed at the link. Still, it's striking IMO, and one wonders exactly what the Einstein equations will allow that naive intuition would consider "impossible".
Gödel's solution to the Einstein equations could be an example: the universe like a rotating cylinder with time travel allowed. Freeman Dyson tells Gödel used to ask the astronomers if they had proofs of this solution.
Walter Russell believed that the reason we cannot unite quantum mechanics and general relativity is bc the equations we have are sufficient to describe our universe, but fundamentally wrong. He had some interesting theories, it’s worth looking into.
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