Why does a stationary object start moving if there is no force acting on it in general relativity? I understand that gravity is not a force, but the function of curved space time.  But what gets an object in motion if no force acts on it?  For example, if I roll a ball, its path will curve towards a massive object due to the curvature of space time.  But, if a ball is suspended by a string, and I cut the string, why does it drop if there is no force pushing or pulling it?
 A: Newton's first law states that an isolated object (on which no forces act) moves at constant velocity, which more specifically means along a straight line at constant speed.
If we are not considering gravity as a force, but rather as a geometric constraint, then it turns out that this law can be applied to situations where particles move freely (with no other interactions) in a gravitational field (which is most relevant for your example), even in Newtonian mechanics, e.g. without relativity!
As spacetime curves, the meaning of the terms "constant velocity" and "straight line" change, to reflect this curving. It turns out that the straight line is now the trajectory followed by your particle falling down, and constant velocity corresponds to the velocity along this trajectory.
A nice introduction to this viewpoint on Newtonian gravitation can be found in this lecture.
The reason I bring up Newtonian mechanics here, is that GR plays a negligible role in your example of a ball suspended by a string (unless you happen to be in the vicinity of a black hole). But even in that example it is legitimate to not consider gravity as a force but rather as a geometric constraint.
