I was trying to reason about how could quantum mechanics be related to the space-time curvature, and I have ended up in an apparent contradiction, which puzzles me. It would be nice if someone could point out if I am mistaken.
Let's say one wants to determine a distance, for instance, the position of a particle, with high precision. Then, according to the uncertainty principle, one has to sacrifice accuracy on how well the momentum of the particle can be known, so trying to resolve a distance more precisely involves an increase in momentum uncertainty. On the other hand, according to general relativity, the curvature of spacetime is related to the energy and momentum of whatever matter present, so, if curvature is dependent on momentum, increasing momentum uncertainty should lead to increasing "curvature uncertainty" (although I don't think I have ever heard this term used). This is where I'm having trouble: since the distance between two points depends of the curvature between them (that is, the intrinsic curvature of a surface depends of the distance relationships that hold within it), then an increase in curvature uncertainty would imply an increase in distance (hence position) uncertainty, which seems in contradiction with the initial assumption.
I suppose the apparent paradox comes from introducing this "curvature uncertainty" in my argument, which does seem a bit odd. Is the whole argument incorrect, or could it be used as a way to show that infinitely small distances can't be measured?