>"otherwise motionless"

There's the trick to it - in any context where you are talking about space's curvature, those "motionless" rocks are going to be moving through time. The geodesics that they will follow are parametric curves of the form f:(s)->(x,y,z,t).

While there are many possible choices of parameter s (as with any parametric curve), one that physicists particularly like is proper time: the time as shown on the watch of the walking man.

So, the only "truly" motionless object would be one that for every second of time it experienced, it would find itself zero seconds in the future. Quite impossible - which is why you can't "pause" yourself along gravitational geodesics and float.