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John Wheeler said “Matter tells spacetime how to curve and spacetime tells matter how to move.”

So, when objects fall to Earth they travel along the spacetime curves that they and all the surrounding matter create.

For the Schwarzschild black hole model the matter is all contained at the singularity. This means that spacetime everywhere else within its gravitational field is in the vacuum state.

We have the matter in the singularity unable to move because all paths lead to it and the surrounding spacetime curved to the greatest extent possible.

Since the vacuum state is not purported to be zero though but just ‘the lowest possible energetic state’ does that mean all of spacetime is essentially matter also?

Therefore, I would ask if spacetime (being matter) falls towards the singularity?

If it does fall (at greater than c below the event horizon) is it not adding energy to the black hole - making it grow?

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    $\begingroup$ I find it useful to think of spacetime as being squished more densely together near a gravitating object. $\endgroup$ – niels nielsen Nov 2 '20 at 3:13
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    $\begingroup$ “For the Schwarzschild black hole model the matter is all contained at the singularity.” - It is not. There is no matter anywhere in the Schwarzschild spacetime. It is a pure vacuum solution. A Schwarzschild singularity is not an object or location in space where matter could be, but a moment in time when empty space disappears. $\endgroup$ – safesphere Nov 2 '20 at 17:38
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    $\begingroup$ “We have the matter in the singularity unable to move because all paths lead to it and the surrounding spacetime curved to the greatest extent possible.” - This makes about as little sense as saying that the moment of noon “cannot move”, because it is “surrounded” by morning and afternoon. Except there is no “afternoon” in case of a black hole where time ends at the moment of singularity. $\endgroup$ – safesphere Nov 2 '20 at 17:45
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As a matter of principle there is no such thing as assigning a velocity vector to spacetime. Acceleration vector yes, velocity vector no.

Other than that, check out the river model of black holes
2004 article by Andrew J. S. Hamilton and Jason P. Lisle

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