Does space expand between two bodies moving away from each other?

Question

Space itself is expanding on very large scales. But since space and mass influence each other, I wondered if an expansion of space could also occur on smaller scales, given the right circumstances. Since massive bodies locally curve spacetime, I think that the space between two bodies might "relax" if these bodies are moving away from each other.

So, if two massive bodies move through space away from each other, does this also cause a metric expansion of space in between them?

My feeling is that it should, but I also know that general relativity can be tricky...

Update: To clarify, I was thinking about two distant bodies that are not tightly bound by gravitation or other forces. Let's say there is an empty patch of space in the universe with only two distant galaxies or two distant black holes in it that are moving away from each other due to some initial motion. Even though the gravitational attraction between them is very small due to the large distance, does the increasing distance between the two bodies also cause a (very small) metric expansion of space?

Answer 1

The concept of "expanding space" can't be defined in the tensor calculus of general relativity.

If you have some objects moving away from each other in a sufficiently symmetrical way, then the local gravitational field will have the appropriate shape to be covered (approximately) by FLRW coordinates. The metric will be the FLRW metric, with a scale factor $a(t)$ obeying the Friedmann equations. In these coordinates, the objects are at rest, and the time variance of the physical distance between them is modeled by the function $a(t)$. The existence of this coordinate system has no bearing on the physical nature of the system. It's still as much a system of relatively moving objects as it ever was.

Your example of just two objects is not very symmetric. But the real world is just as asymmetric at small scales. If space itself is expanding in the real world then it must be expanding at small scales, since physics is local, so there's no reason to suppose it doesn't also expand in your setup. But it's a moot point since expansion of space just isn't a thing in general relativity.