Does the total volume of space change in a n-body problem? Say you have a few massive objects like planets which bend and warp space-time in an otherwise empty space. They move about in a region of size R, perhaps like a solar system. Then you calculated the volume of space say in a region 100R. Will this volume change at all? I mean, the planets will stretch and curve space. But since no energy is going in or coming out, will the total volume of space stay the same?
I would measure the volume up to the surface of each planet.
What I'm trying to get at is if the volume of space varies. And if space is made of plank-volumes, whether these increase or decrease or stay the same.
You would have to choose a specific foliation of space-time to measure your volume.
 A: 
They move about in a region of size $R$ …

This generally is not sustainable even in Newtonian gravity of $N$ point-like particles, as the system would tend to occasionally eject some of the bodies from itself. The escaping bodies (or small tight clusters of bodies) would have asymptotically constant velocities, so a convex hull around all the bodies would eventually be growing in volume approximately as $t^3$. 
Mostly the same behavior would remain if we consider realistic isolated bodies with dynamics governed by GR: some bodies get ejected, while the remaining would be orbiting closer together, until bodies either start colliding, coalesce into a black hole, or we end up with a dispersing cloud of gravitationally unbound isolated bodies and binary systems.
If a black hole is formed, then one could choose space slicing in which the volume inside the black hole is a quantity growing with (outside) time. See for example this paper:


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*Christodoulou, M., & Rovelli, C. (2015). How big is a black hole?. Physical Review D, 91(6), 064046, arXiv:1411.2854.


From the abstract:

The 3d volume inside a spherical black hole can be defined by extending an intrinsic flat-spacetime characterization of the volume inside a 2-sphere. For a collapsed object, the volume grows with time since the collapse, reaching a simple asymptotic form, which has a compelling geometrical interpretation. Perhaps surprising, it is large. The result may have relevance for the discussion on the information paradox. 

