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Forgive my 'amazing' MS-Paint skills, but this is along the lines of what i'm wondering:

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Is there any situation where Scenario A occurs, or is space always dragged back out as in Scenario B?

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    $\begingroup$ I think this is so grossly simplistic that no answer applicable to the real universe and the way things actually work can be given. $\endgroup$ – zephyr Sep 20 '16 at 16:48
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    $\begingroup$ This is a case of taking the rubber-sheet analogy too far. Spacetime does not literally extend into some other dimension; this just serves a way to try to imagine the curvature of spacetime. But the premise here is false. $\endgroup$ – HDE 226868 Sep 20 '16 at 17:34
  • $\begingroup$ I'm not suggesting it extends to some other dimension, i mean, i didn't even mention dimensions. But instead i was wondering if space can 'pile up' in one spot, like dragging the fabric of an endless sheet towards you - eventually there will be a pile. However, the only answer seems to be that space is in fact let go. $\endgroup$ – Logan Sep 21 '16 at 9:16
  • $\begingroup$ This is a lovely question. We could have black holes working to counteract the expansion of spacetime in their immediate vicinity. Unfortunately it sort of implies the existence of something like the Luminiferous aether, which might be sucked into a black hole. $\endgroup$ – Wayfaring Stranger Sep 21 '16 at 14:08
  • $\begingroup$ To the people saying "space isn't a real thing", I'm not sure it's quite that simple, at least, the spinning black hole theory suggests a dragging of space-time is possible, phys.org/news/2011-02-black-hole-space-time-visible-earth.html and while that may or may not be accurate I think questions on stretched or squashed space-time have validity. That said, I think Zephyr is right as well. The 4 scenarios and space being "pulled in and kept there" "Piling up around" - doesn't make a whole lot of sense the way it's asked. Space-time is more of a continuous state, not a thing. $\endgroup$ – userLTK Sep 22 '16 at 22:12
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Black holes orbit things and galaxies move within space, so it would be safe to assume that black holes move through space-time and do not hold it in any particular state or position. They simply drag their "depression" along with them as they go.

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Space is not a real thing, it is kind of a conceptual picture. However, there are standard ways to apply that picture that work reasonably well. One such picture is that all gravitational objects "pull in" space, and keep it as long as the object remains an intact massive body. Not just black holes, even the Earth. You can think of the space as being like it is made of rubber and is being contracted whereever there is mass. This is a poor man's version of the concept of curvature of spacetime, but it works quite well. Note that if black holes eventually dissipate by Hawking radiation, then they will "let go" of that space. But it won't re-expand until it rejoins the "Hubble flow" of the Big Bang, and dark energy gets a hold of it-- it seems that empty space is just empty space, it does not have a memory of its own history. More formally, what we mean by space is just a coordinate system for talking about relationships like distance and motion. Whether or not space "moves through" an object is really just an issue of reference frame and coordinate system.

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