I'm trying to figure out a concept for some sort of super-advanced space propulsion system that works by bending spacetime. As I understand it, planets' orbits are actually straight paths, but they move through a spacetime sufficiently curved that they travel in ellipses around their system's center of gravity. Taken to the extreme, stuff falls into a black hole because, if you get in close enough, spacetime is so warped that any direction you move in is toward the singularity.

Is it possible, within our current understanding of physics, to bend the spacetime around an object so that it moves relative to nearby objects? Like, something that translates the movement toward a center of gravity like a planet or star into directional movement?