I'm not an expert in these things. But my recollection is that the solutions of general relativity that behave like warp drives all violate various energy cond

I'm not an expert in these things. But my recollection is that the solutions of general relativity that behave like warp drives all violate various energy conditions and hence are only possible in the presence of what physicists politely refer to as "exotic matter". The Alcubierre drive, for example, requires matter with a negative energy density. We're reasonably certain that the class of quantum field theories we use to describe matter (QED, QCD, the Standard Model itself,..) are not compatible with these requirements.