I'm not a physicist so my question might be silly.
We don't. The density of a black hole is not even a well-defined concept; black holes are not like ordinary objects.
> My confusion lies in how we can derive the mass at the hyper-quantum state at the center of a black hole
There is no such thing. A black hole is vacuum inside. The "mass" of the hole is not due to "stuff" sitting inside it; it's a global property of the hole's spacetime geometry, that maintains itself because of the nonlinearity of the Einstein Field Equation.
> from the gravitational waves we observe as it loses energy.
An ordinary black hole in our universe will be gaining energy, not losing it--if for no other reason that CMB radiation is shining on it. But the black holes we see all have matter falling into them as well.
As for detecting gravitational waves, except for violent events like black hole mergers, we don't detect any; what few such gravitational waves there are are far too weak for us to detect.
I thought LIGO and VIRGO had been detecting gravitational waves for the last 4 years pretty regularly (i.e. once a week).
If it's perfectly non-spinning, and nothing is falling into it, then it emits no gravitational waves at all.
If matter or energy falls in, and does so in a manner that is not spherically symmetric (which will most likely be the case), then some very weak gravitational waves will be emitted. But they will be much, much too weak for us to have any hope of detecting them. Even a gravitational wave detector like LIGO in the same stellar system would have difficulty detecting them.
From that you can compute a density number, but I agree it's not quite the same concept as the density of water.
Yes, but not because of "stuff" inside it. As I said, the mass is a global property of the spacetime geometry.
> and a volume defined by its event horizon.
No, it has a surface area defined by its event horizon. But you can't compute a meaningful volume from this surface area. You can, of course, plug the surface area into the formulas of Euclidean geometry, but the number you get will have no physical meaning for a black hole. A black hole has no well-defined volume at all--indeed, one can find spacelike surfaces inside that are infinite in extent, even though it has a finite surface area.
In simpler terms, let’s say a star has a mass just below the limit of making it a black hole. Then, a planet collides and it crosses the threshold. Are you saying the “stuff” disappears ?
Because the solutions of the Einstein Field Equation that describe black holes are vacuum solutions. "Vacuum" means "no matter or energy".
> let’s say a star has a mass just below the limit of making it a black hole.
There is no lower limit to the mass of a black hole.
There are maximum mass limits for white dwarfs and neutron stars, and stars more massive than those limits (after they have undergone supernova explosions and probably shed a large portion of their original mass) will collapse to black holes.
As for what happens to the matter in the original object that collapses to a black hole, it reaches the singularity at the center of the hole and is destroyed. But anyone falling into the hole after the collapse will never encounter that collapsing matter; all they will see is vacuum.
That's false; GR does make a definite prediction: that curvature invariants increase without bound as the singularity is approached, but that they are finite everywhere in the actual spacetime (see further comments below). (Actually it's more nuanced than that; there are cases where there are incomplete geodesics, which is how "singularity" is actually defined in GR, but not unbounded invariants. But those are edge cases that aren't relevant to what we're discussing here.)
The Stack Exchange thread's claim about geodesic incompleteness is misleading. The actual singularity--heuristically, the "point where things become infinite"--is not included in the spacetime manifold. It is an abstract "boundary point" that is not part of spacetime. What is actually included in the manifold is perfectly well-defined, with nothing infinite anywhere, and is a perfectly self-consistent mathematical model that makes perfectly valid predictions for everything in it.
The Stack Exchange commenter basically doesn't like the fact that GR says "sorry, your model ends here, you can't extend it any further", which is fine, but it's not the same as saying GR must be wrong when it says that. It's just something many physicists are uncomfortable with and so they are looking for a model that doesn't have that property, like quantum gravity. But that's no guarantee that they will find one, nor is it a guarantee that GR must end up being overridden in this regime. It's just a best guess of many physicists at our current state of knowledge. Nor does it mean that GR doesn't make definite predictions; it just means GR's predictions are ones that most physicists would not like having to be stuck with if that's how it turns out. But nature doesn't care what humans like or don't like.
https://www.forbes.com/sites/startswithabang/2019/10/24/this...
This is wrong in two ways. First, light at the hole's horizon does not "redshift to nothing"; it just stays at the same radial coordinate because of the curvature of spacetime. The "redshift to nothing" view is an illusion, created by a bad choice of coordinates; that illusion was corrected in the late 1950s and early 1960s by the discovery of better coordinates.
Second, gravity does have an equivalent to light: gravitational radiation. Gravitational waves travel on null geodesics, just like light, and any gravitational waves at the hole's horizon would stay at the same radial coordinate just as light does.
From an outside observer, we never see the photon. It can’t make it out. But — the photon does exist, and it is traveling at the speed of light over a finite (though deeply warped in spacetime) distance. How does the photon not reach us then? Is it not going at the speed of light?
The redshift model applies here; the photon is redshifted until it has no energy from our frame.
Gravity doesn't have to get out of the hole. The gravity you feel outside a black hole is due to the global spacetime geometry, not to any "force" coming out of the hole. If you want to attribute it ultimately to the presence of matter, it is the matter in your past light cone, which originally collapsed to form the hole, before it fell below the event horizon.
See this Usenet Physics FAQ article:
https://math.ucr.edu/home/baez/physics/Relativity/BlackHoles...
This does trap light inside, but speed has nothing to do with it.
What's the connection to dark matter?
Not even that. The "volume" of a black hole is not even well-defined. A black hole doesn't work like an ordinary object.
> lensing and orbiting bodies would behave the same regardless of whether it was a point mass or low density sphere.
First, a black hole is definitely not a "point mass". Second, the density of a black hole is also not well-defined.
Lensing and orbiting bodies do exhibit unique properties close enough to the black hole's horizon, but unfortunately we would have to be relatively close to the hole (meaning, in the same stellar system, not light years away) to distinguish them.
Black holes are ordinary objects, if they are anything at all. Take any normal object and give it the mass of a black hole, and what do you have?
To claim that a black hole “doesn’t work like normal matter” seems to ignore that these entities have come to exist “normally”, or rather, through a natural course of events.
It’s not that the black hole “doesn’t work like normal matter”, but rather that you refuse to accept that normal matter with the mass of a black hole (and hence, the observable properties we associate with black holes) are still “normal objects”.
When objects of a certain class display certain properties, it’s not accurate to say “that’s not normal.” Instead, it would be accurate to say, “that is the normal behavior of objects once they reach this class or state.”
No, they aren't. Their spacetime geometries are very, very different from those of ordinary objects.
> Take any normal object and give it the mass of a black hole
There is no such thing as "the mass of a black hole". What distinguishes a black hole from other objects is not its mass, but its spacetime geometry. That's particularly true for the holes studied by this paper, which are far smaller than the ones we actually see with our astronomical observations.
The rest of your post is simply wrong. A black hole is not made of normal matter; it's vacuum. I'm sorry, but you simply don't understand the physics of black holes. A good free online reference is Sean Carroll's lecture notes on GR:
https://arxiv.org/abs/gr-qc/9712019
It has an entire chapter on black holes.
Fair enough. Primordial black holes are a good theoretical example of environmental density conditions being perhaps a larger factor than mass even. My point was that it was still matter. It doesn’t matter if you squish it all down - it’s still matter.
> A black hole is not made of normal matter; it's vacuum.
When we talk about black holes, some might be referring to the event horizon, and others the singularity which makes the event horizon possible. But I’m not sure what you mean when you say that a black hole is made of vacuum. Are you saying that the singularity of a black hole is a vacuum? And not matter?