I don't really get GR. Why should curved space and time be a model for gravity? To me, curved space means a observers no longer measure distances as sqrt(x^2+y^2+z^2), but rather, given an x-ordinate, y-ordinate and z-ordinate, the length of the shortest path to that coordinate can be calculated by a different formula. Surely it means that two spatially separated objects which are moving parallel to each other, if they enter a bit of space that is not uniformly curved, will no longer be traveling parallel to each other. In this sense I get why light, for example, is bent by gravity in general relativity. But why do things in a gravitational field actually accelerate in the direction of the most negative gradient (I don't know if this is part of the maths, its just what I've understood from looking at those embedding diagrams or whatever they are called.) Is the previous statement even correct, or does SPACE accelerate into the direction of the most negative gradient, and objects with mass in that space continue to remain at rest in that space, which is moving? And would it be fair to say that mathematically, the curvature of spacetime needn't cause an acceleration of any sort, except that in general relativity, one is always accompanied by the other - you always have an acceleration of space in a gravitational field, and you always have a distortion of spacetime, thus in GR they are one and the same, or have I completely missed the meaning of curved spacetime? Thanks for your time.