cosmology - Looking for help in understanding how black holes can move

1. I saw Dr. Kip Thorne in a documentary on black holes state that there is no longer any matter in a black hole once it forms. He said something like, the matter was there, but it has been crushed out of existence and basically has been transformed into the energy that produces such a great curvature of spacetime.

Isolated black holes are indeed vacuum solutions to general relativity. Thus in particular, the mass and energy density is identically zero everywhere in spacetime. Because of some technical issues, this does not imply that black holes have no energy or mass; rather, it means that we can't directly think of them as an integral of mass or energy density. See also this question.

3. I also saw Dr. Andrew Hamilton in another documentary state that within the event horizon of a black hole (again paraphrasing), space is falling toward the singularity so fast that it effectively drags everything with it (including light).

For an isolated black hole in a particular frame field, this is a valid picture. For example, the Schwarzschild spacetime in Gullstrand–Painlevé coordinates can be interpreted as Euclidean space falling into the black hole. However, the spacetime is itself is stationary (even static) in the geometrical sense—there is a timelike field representing a direction in which spacetime geometry is left unchanged (this corresponds to Schwarzschild time, in fact). Similarly, rotating black hole spacetimes are stationary.

But generally speaking, you really can't think of black holes as some sort of suck-holes for space. It's just an analogy that applies to simple situations (isolated black holes, with nothing else in spacetime) and then only in a particular frame or coordinates. That's not to say it's useless--e.g., acoustic horizons in fluids are interesting analogies to event horizon--but don't take it too literally. See also this question.

4. Based on 1 (no matter remains in the black hole) and 2 (space is being pulled towards the gravitational singularity from all directions), I try to picture the black hole moving, and it seems to me that if, say, the black hole moves in one direction, I can see how its event horizon is also moving, bringing that bit of space into it.

So what's the problem?

But how can the other end of the black hole let space "escape" in order to complete the motion?

For realistic black holes, there is no "other end". If you really want to tie everything together, then interpret Dr. Thorne's comment about stuff falling into the singularity as crushed out of existence as also applying to space. It doesn't come out of any "other end". It stops existing.

But really, that's past the point where the fluid analogy make sense anyway. I think Dr. Hamilton might note that the full maximally-extended Schwarzschild spacetime is a black hole, so one may be able to think as it "coming out" of the corresponding white hole, but he would also tell you that this white hole is a mathematical artifact that doesn't have anything to do with actual astrophysical black holes.

But I want to understand why it is that space itself can cross the event horizon when other things can't.

Of course they can. In the infalling-fluid analogy, they're carried along by the space that's being sucked in. Perhaps you're thinking of the fact that it takes an infinite Schwarzschild time for an object to reach the event horizon. But that doesn't imply that things can't cross the horizon; rather, it's just a symptom of Schwarzschild coordinate chart not covering the horizon.

Intuitively, think of a coordinate chart as a "grid" drawn on a patch of spacetime. That patch may be the entire spacetime, or it might be just a piece of it. In the case of the usual Schwarzschild coordinates, it's just a piece... one that simply doesn't cover the horizon.

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Regarding the last part, I meant space crossing the event horizon the other way - out of the sphere demarcating the event horizon - whereas material objects go one way - in but not out, spacetime itself does not seem so restricted, at least not in the sense of "pinning" a black hole to one part of itself.

OK, your edited question provides some more context. But the answer is completely the same: what Dr. Hamilton is talking about in the video is the maximal analytic extensions of the rotating black hole solution, which does contain a passage into a completely different region (sometimes called different 'universe'). This is very much analogous to the maximally extended Schwarzschild spacetime, which contains a wormhole that comes back out of a white hole in another spacetime region, except that the extension of the rotating Kerr black hole spacetime contains an infinite chain of such connected regions.

I haven't watched the entire video, but it's clear that's what they're talking about when they say "in a ship propelled by pure mathematics", because the causal structure of such spacetimes is well-known. Once again, I direct your attention to Dr. Hamilton's own page, linked above, that explains that this dual waterfall picture is an artifact of mathematical idealization and not something that actually happens in reality.

However, even if you take the maximal analytic extension of such black hole spacetimes overly seriously, it's important to emphasize that you don't come back out into the same region, but rather a different one connected by a wormhole inside the horizon. If you're up to it, I also recommend Dr. Hamilton's conformal diagrams of said black hole spacetimes, which make it quite clear that's what going on is a whole "chain" of black holes and white holes.