The Schwarzschild radius of the mass of the observable Universe is 10B light years. So, several billions years ago, when observable Universe had 10B radius, it would be a black hole (though i think that in less expanded space of the earlier Universe the constants like "c" had different values and thus that Schwarzschild radius was less). We can imagine it in another way - increase 125 times (the observable Universe has 40+B light years radius) the amount of matter, ie. galaxies, stars, etc... inside the 10B radius ball around us, and you'd get the black hole with 10B light years Schwarzschild radius (and i don't think we would ever notice the change - only with time the galaxies's movement will be affected). Obviously, the gravitational field on the surface of that imaginary 10B radius sphere and inside it would increase somewhat (like 125 times on the surface, 125 times 0 pretty much 0) - not even close though to any values to affect space curvature or to prevent anything from crossing it from inside. Of course, anything that would cross it from inside would return back eventually.

The Schwarzschild radius does have the property that, if you plug it into the classical equation for escape velocity, you get the speed of light. That doesn't mean arbitrary classical analogies (like the "thrown stone" example) hold.

In particular, consider that you can escape the earth while never achieving escape velocity. Just keep firing your rockets to counteract gravity. But if this were possible with black holes, then they would not be very interesting!

> The various local effects like time dilation, light path curving, etc... are defined by the value of gravitation force, not gravitational potential.

These are not local effects! Locally, there is no time dilation, and no curvature of light. I see my watch tick at the same rate, no matter where I am, because my watch and my eyes are in the same local reference frame.

In order to see effects like time dilation or curvature of light, you must compare events separated in spacetime. For example, we can look at the paths of light emitted by distant stars. And since the light had to get to our eyes, we have to account for the entirety of the path that it took. So the time dilation I observe for an event depends on the entirety of spacetime along the path from the event to me, not just the local curvature for the event.

In mathematical language, if you want to move a vector from one point to another on a curved manifold, you can't just specify the start and end point. The path you take affects the resulting vector.

> The potential defines the fact that anything originating at or below the horizon would never escape completely the gravitational field, i.e. never reach the infinity.

So objects originating outside the event horizon reach infinity in a finite time? Huh?

No, it can't; the "acceleration due to gravity", which is the acceleration required to "hover" at a constant altitude above the horizon, diverges as the horizon is approached.

What can be made as small as desired by making the hole's mass large enough is tidal gravity at the horizon.

several billions years ago, when observable Universe had 10B radius, it would be a black hole

No, it wouldn't. A black hole is a stationary spacetime. The spacetime of the universe is not stationary. The spacetime model that describes the universe is very different from the spacetime model that describes a black hole; the fact that you can plug the mass of the observable universe into the Schwarzschild radius formula and get a number out does not mean that number has any physical meaning for the universe.

exactly. There is no stationary spacetime in our Universe. Black hole is pretty artificial model where pure mathematical artifacts of singularity at the horizon is taken for the real thing.

> the fact that you can plug the mass of the observable universe into the Schwarzschild radius formula and get a number out does not mean that number has any physical meaning for the universe.

Taking a big chunk of space and calculating escape speed from its gravitational field - how is that not a physical meaning? At what specific size of the chunk you think it becomes not meaningful?

The universe as a whole is not stationary, nor even close to being so; but portions of the universe are very close to being stationary. The solar system, for example. Black holes do not have to be exactly stationary; if they are as close to being stationary as the solar system, that's plenty close enough.

Black hole is pretty artificial model

The exactly spherically symmetric solution is an idealization, yes; but there are plenty of numerical simulations that show that non-symmetric spacetimes still form event horizons.

where pure mathematical artifacts of singularity at the horizon is taken for the real thing.

There is no physical singularity at the horizon. Some coordinate charts have a coordinate singularity at the horizon, but that's easily fixed by just using a different coordinate chart. The only physical singularity is at r = 0.

Taking a big chunk of space and calculating escape speed from its gravitational field

Escape to where? You can't escape from the universe as a whole. The concept of "escape speed" has no meaning for the universe as a whole.

take a 1B light years radius ball, populate it with density of our Milky Way - that ball will have 1B Schwarzschild radius. Calculation of gravitational potential (escape speed) from it to the rest of the Universe makes sense, doesn't it?. And due to this gravitational potential it will be bona fide black hole from the point of view of the rest of the Universe.

Yes, but that's not the same as escaping from the universe as a whole.

And due to this gravitational potential it will be bona fide black hole from the point of view of the rest of the Universe.

So what? What does that have to do with assigning a gravitational potential to the universe as a whole?

Nor mine. You're correct, any object in the interior of the hole, even one that is moving radially outward at the speed of light, gets closer to the singularity (i.e., its r coordinate decreases) with time, which means it can't get closer to the horizon (that would require increasing r).