If gravity is just the effect of warping space, then why does it cause acceleration? For example, an object is dropped from a height and moves towards the ground. When the object reaches height/2 another object is released from that location. Since they both exist in close proximity relative to the earth, shouldn't they both be in the same warped space? Hence, they should be moving at the same speed because the warping of the space for each object would be the same.
I could see that the closer to the earth, the more the warping of space and consequently some "apparent" acceleration would occur. But an object released should immediately move at the speed of the warped space. IOW, the apparent acceleration would be infinite because the object isn't actually accelerating but simply sitting still in space that is moving relative to the massive object.
Of course this isn't what actually happens. But it would help to better understand the actual mechanism behind gravity if I knew why it doesn't work like in my question.