Do things move towards each other in space? I watched a video once explaining space-time curvature, and it said that one of the things that made Einstein think of it is because objects in free fall move radially inward, so two objects falling side by side are actually moving towards each other, while objects floating in space are not moving relative to each other. Space-time curvature solves this problem, because the curvature of space-time means that two objects in free fall aren't moving towards each other, they only seem to be because space-time is curved. But wouldn't objects in space also be moving towards each other, either because of their gravity or because they're actually in free fall to some planet nearby, as some answers to other questions I saw while looking this up suggest?
 A: Well, yes. Technically we can never turn off gravity completely. But it's useful to pretend that we can do so, and so when we talk about "objects floating in space", we're implicitly assuming that these objects are small enough that the gravitational force between them is negligible, and that they're far away from any astronomical body. Under these conditions, we can pretend that two objects placed in space will remain where they are.
A: Considering increasing Entropy and that we observe the universe to be expanding exponentially and the space between matter is doing the expanding we must conclude at certain distances between objects gravity is not the prevailing force. At such a hypothetical zero gravity point and perhaps anti-gravity point I would have to say it is very important to consider what other forces would be acting on these objects. If there is no observable force nearby increasing entropy and an expanding universe should prevail and in time these objects will be farther and farther apart.
However in a special case as a black hole I could see them falling in on themselves once they breach the event horizon. In this case gravity is definitely the prevailing force and space time curvature should play a role in their radially inward motion as they free fall. 
Additionally if they happen to be drawn in by gravity into a decaying orbit of a planet or sun they would experience a space-time curvature that would change their velocity and cause a "free fall with a radially inward motion to the larger masses center that is acting on them.
I think it is very important to note that if the two bodies are in very empty space with no black wholes, no suns, no planets, no large gas nebula, no matter of any consequence, that the gravitational influence of empty space will approach a zero point and may even be negative {perhaps due to dark matter} and that the law of entropy and an exponentially expanding universe will over time result with the two objects being infinitely separated over an infinite time. This result should also be evident by examining a quantum probability distribution of two particles laid out by the Schrodinger wave equation on a macro scale with universe set as the boundary conditions.    
