Can a system of non-black holes become a black hole?

Question

As we know, the Schwarzschild radius and mass are proportional: $r_s=\frac{2GM}{c^2}$. If the density of a substance is constant, the mass and radius of an object made of that substance are proportional to the cube. Therefore, if the radius of an object doubles, the Schwarzschild radius increases by a factor of 8. In other words, can a system made of multiple neutron stars rotating around each other become a black hole? Please let me know if I am wrong.

I thought that a black hole with a very large Schwarzschild radius could have a very low density. If you replace $M$ in the Schwarzschild radius formula with density, you get the Schwarzschild radius formula depending on the density: $r_s = \sqrt{\frac{c^2}{8.37Gρ}}$. If you substitute the density of water here, the Schwarzschild radius becomes $711334662 \,\text{km}$. So, a black hole with a radius of about 10000 times that of the sun doesn't have to have a very high density (the density of water). If we extend this size to the entire universe, our universe could be a black hole and there would be no need for a singularity. Please correct my foolish thinking.

Answer 1

I think you are imagining that the density of matter that goes into forming a BH is what causes the singularity. But what causes the singularity is that for any point in spacetime inside the event horizon, any trajectory whatsoever must point radially inward. The radial direction becomes timelike, and no material object at that point could continue at its current radius without moving inward – any more than right now you could remain at 12:00 pm without progressing to 12:01, and later to 1:00 pm.

Extended matter cannot exist in this type of spacetime (at least matter as we currently understand it), because in order to be "extended" it would have to exist at various radial locations indefinitely. Inside a Black Hole, all matter and light have a destination: the center. And all will inexorably reach it. This point, if it were to exist, would be where everything that ever existed in the BH now resides, colocated, at $r=0$. So we inelegantly say that point would be "infinitely dense."

In reality though, it is nearly certain that general relativity and our current understanding of matter fail inside a BH, so you should take this description of a singularity lightly. It is as if we made a very accurate simulation of the universe in a virtual reality like a video game, and most everything matched the real world extremely well, but suddenly you opened a door and just saw

00010111111000111

00011110100010001

#ERROR

inside.

That doesn't mean the real world looks like that. It means our code crashed under those conditions, and we need a better code engine to simulate the contents of that room.

It's currently under development, but progress is slow.