Space falling faster than light after it falls inside the horizon? Typing my question directly so people know what I am asking, afterwards providing background and context.
Q: What does it mean when space is falling, faster than light?
(I am  specifically wondering about just that "space falling"  which is what is confusing me the most).
I mean it does this after it falls inside the horizon of a black hole. (A Schwarzschild black hole to be specific, so a "static",
no electric charge & no spin)
this is stated in both the documentary, and in the book(both provided below).
Citing the book

space is falling into the black hole. Outside
the horizon, space is falling less than the speed of light; at the horizon space is falling at the speed of light;
and inside the horizon, space is falling faster than light, carrying everything with it. This is why light cannot
escape from a black hole: inside the horizon, space falls inward faster than light, carrying light inward even if that light is pointed radially outward.


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*General Relativity, Black Holes, and Cosmology by Andrew J. S. Hamilton 4 December 2021 Available here at chapter 7. "Schwarzschild Black Hole" section '7.6 Horizon' - page 137


And, I include a screenshot of this Wikipedia link about the Event horizon of a black hole, which also illustrates this:



Background and what I have tried
Background

I have no university level education in Physics yet, but I know some terms and have read some articles and books(such as Wikipedia and *edu sites, like jila.colorado.edu).
And I am reading books(like Sean Carroll's "spacetime and geometry" intro to general relativity, and Taylor/Wheeler's book as well, and many others). And so I am not completely "new" to Physics but, not at all an expert.
What I have tried

Before asking this question I thoroughly tried to find my question in various, numerous sites(including this one), and books. But didn't find anything that would explain the stating that "space" would fall. (excluding the one I typed above)
Before this question was asked
I read other questions and answers on this site, and specifically this question really inspired me how I should write my question, as well as of course first of all I read the following:

*

*don't-ask

*how to ask
I want to be very clear that I "Keep an open mind", and I try to be on-topic and specific.
Which is why I typed the question directly above, and, to be sure it doesn't become a vague or more discussion kind of question, I try to keep things short. Since this is also, my first question.
Citing the Documentary
This documentary about black holes - at timestamp 33:13 Hamilton, Andrew J S  says the following:

space is falling faster than light

My question is about just that, and I do know this is from YouTube, which is why I provided the link to the book and checked the topic discussed about in here.
Resources I have tried

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*If spacetime can expand faster than the speed of light, then can a black hole do that too?

*

*why didn't that article help?

*the question is about "spacetime", whereas my question is about "space" itself.




Related and External links

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*Andrew Hamilton's Homepage - jila.colorado.edu

*Andrew's book - jila.colorado.edu

*black_holes - math.ucdr

*Can space expand with unlimited speed? - Physics.stackexchange and sub-links

*Frame-dragging - Wikipedia

*Thirring_effect - Wikipedia

*Schwarzschild_metric - Wikipedia

*Black_hole - Wikipedia

 A: This is the "river" picture of black holes, as Dale said, but I disagree strongly with his statement that it is a "nice heuristic".
Rivers flow at a certain speed. If you fall into a river, friction with the water will accelerate you until you're moving at the same speed as the water.
Gravity doesn't work like that at all. Gravity doesn't give you a certain velocity, but a certain acceleration. The time reversal of a river is a river flowing in the opposite direction, but the time reversal of a gravitational field is a gravitational field in the same direction. If you film a ball being tossed in the air and play the film backward, the ball is still attracted toward the earth in the reversed film. Gravity is a conservative, not a dissipative, force. The river picture fails to capture any of those properties.
The river picture leads people to wonder why black holes don't just suck up everything around them, and how the space that flows into them is replenished. Those would be reasonable questions if the river picture made sense, but it just doesn't.
The river picture is inconsistently used only for black holes when it could equally be applied to any other gravitating body. Black holes are unique in that time symmetry is broken at the event horizon and inside it, but that's irrelevant to the observable behavior of astronomical black holes, which depends only on the physics outside the horizon. The physics outside is time-symmetric, and doesn't fundamentally differ from that of other gravitating bodies. The river picture tries to explain the broken time symmetry by breaking it everywhere, even outside the horizon. But it isn't broken outside the horizon. The explanation is wrong.
A: This is referring to the so-called “river model” of spacetime. This model is explained in detail here:
“The river model of black holes” — at arXiv.org
In the river model space flows into the black hole and the motion of particles is affected by the flow of space. The event horizon is where the flow of space exceeds $c$.
The issue with the river model is that it does not generalize to arbitrary spacetimes. It does work for the Schwarzschild and the Kerr spacetimes, but not other spacetimes in general. So while it can be a nice heuristic it cannot be thought of as the way gravity works.
