Can one define a flow of spacetime?

Question

One often reads things like, 'At the event horizon, the flow of spacetime exceeds the local speed of light.' But is this actually correct? Can you mathematically define some sort of spacetime flow which is then directed inwards for black holes? Or are sentences like the above actually nonsense? After all, spacetime for non-rotating, neutral black holes is described by the Schwarzschild metric which does not depend on coordinate time, so taking the temporal derivative of its component yields zero everywhere.

Answer 1

My understanding is that it can be stated in a way such that no self-contradiction arises.

The corresponding mathematical model is called 'the river model' of spacetime.

Article by Andrew J. S. Hamilton, and Jason P. Lisle (last revised 2006): The river model of black holes

What that model does is that given a metric it assigns a velocity vector to every spacetime coordinate.

Of course, as a matter of principle there is no experiment that can provide a measurement of such a velocity value.

However, that does not exclude the possibility that a velocity attributing model can be devised, in 1-on-1 correspondence with a given metric, with that model being mathematically consistent.

Hamilton and Lisle discuss the river model, and they argue that it is a useful tool in that it can help guide thinking.

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[Later edit]
About the sentence that you paraphrase (and that you are doubtful about):
'At the event horizon, the flow of spacetime exceeds the local speed of light.'

I take that expression as a form of physics shorthand.

Comparison:
Two physicists can in conversation refer to 'sunrise' and 'sunset'. Both know that in fact the Earth is rotating, but in casual conversation there is no need for either to remind the other of that. Both know what 'sunrise/sunset' is.

I visualize the river model as follows:
Acceleration and velocity stand in the same relation to each other as force and potential energy.

Potential energy:
Gravity falls off with distance, so a natural choice is to set the value of the potential to zero at infinity. Difference of potential is defined as the amount of change of kinetic energy as a test mass moves from one point to another point.

Velocity:
For the flow in the river model a natural choice to is to set the value of zero velocity at infinity. For a point at distance 'r' to the center of a gravitational potential: integrate acceleration from infinity to 'r'; that is the assigned velocity.

What I find appealing about the river model is that it explicitly puts all the parts of a region of spacetime in a relation to each other.

We organize our understanding of motion in the Universe in levels of scale of gravitationally bound systems.

Gravitationally bound systems, in ascending scale:
-Earth-Moon system
-The Solar system
-Our Galaxy
-Local Group (Andromeda Galaxy, the Galaxy, plus about 100 smaller galaxies)
And so on...

To evaluate the motions of the celestial bodies of the Solar System we apply a coordinate system at the appropriate scale: a coordinate system that is co-moving with the barycenter of the Solar System, and that encompasses all of the space that contains gravitationally bound objects.

(Conversely: to use a coordinate system from the wrong scale of representation (a scale that is zoomed in one or more levels), is in effect an act of deception.)

The flow of spacetime (in terms of the river model) represents the shape of spacetime (both spatial shape and as a function of time)

As we know: the celestial bodies moving through that spacetime do not couple to the velocity of the spacetime flow, they couple to the acceleration of the spacetime flow. Objects in inertial motion in spacetime are co-accelerating with the local spacetime. (The local spacetime is accelerating with respect to the barycenter of the bound system you are in the process of evaluating)

Physicists are totally accustomed to assigning values for potential energy. We do know that we have to be cautious, because unlike, say, temperature: potential energy does not have an intrinsic zero point. The choice of zero point of potential energy is arbitrary. The only way to define potential energy at all is in terms of difference of potential energy. As we know, potential energy is a very powerful and expressive concept. The fact that potential energy does not have an intrinsic zero point does not count against the concept of potential energy.

In terms of the river model of spacetime the choice of point of zero velocity is arbitrary. That should not count against the river model.

Answer 2

A light cone is defined at every point and represents in some sense the flow of time or causality, but of course it doesn't flow relative to something else (a background).

On a closed trapped surface there is an overall inward flow of time/causality, but it's a nonlocal property of the whole surface.

I don't think it's reasonable to call either of those a flow of spacetime or space.

'At the event horizon, the flow of spacetime exceeds the local speed of light.'

If anything, the local speed of light defines the flow. The only thing that exceeds the speed of light (i.e., is outside the light cone) is a coordinate direction that they chose arbitrarily to call $t$.

I'm not a fan of the river model. Its proponents call it "mathematically sound", which it is, in the same sense that, e.g., a geocentric model of the solar system is mathematically sound. You can always write a theory in different coordinates, and attach some philosophical interpretation to those coordinates (e.g., that they are the true coordinates used by the computer that runs the universe), as long as you don't change any actual physical predictions.

The river model makes gravity look like a "sucking" force, which it very much isn't. Gravity is time-reversal symmetric, and has no inherent tendency to make systems smaller (in the absence of friction). The exterior gravitational field of a black hole is like that of an ordinary gravitating body, so black holes no more "suck" than the objects that collapse to form them.