From my understanding of relativity, gravity is not a force, but a result of the curvature of spacetime. If Object1 moves past Object2, even though it's moving in a straight line, its direction may change due to the distortion caused by Object2's mass.
However, what about the situation where Object1 is not moving? How can there be an attraction between two objects that are at rest relative to each other? i.e. what makes them move towards each other?
Answer
Two objects that are initially at rest with respect to each other have initially parallel world lines.
However, the curvature of spacetime means that world lines that are initially parallel do not remain so. This is called geodesic deviation.
In the above image, the geodesic segments are parallel at the equator but, nonetheless, converge at the pole.
Imagine the time direction (and remember, every object is relentlessly "moving" forward through the time direction even when "at rest" in space) is along lines of longitude and the spatial direction is along lines of latitude.
If the surface were, instead, a plane, the two geodesics would remain parallel and the objects associated with those world lines would not move towards or away from each other.
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