I The initial density of an object and its compression into a black hole

[i walk away]

TL;DR Summary

Could you explain whether the initial density of matter affects the rate of its compression into a black hole, and if so, how exactly?

The other day, a friend and I had a discussion about black holes, namely how the density of a body affects the process of its transformation into a black hole.

My friend and I have too little collective knowledge in the field of theoretical physics regarding black holes, so the discussion has reached an impasse. I thought it would be great to ask for help from someone more knowledgeable. I've been emailing a professor from the Department of Space Physics at my university, but he doesn't seem quite willing to respond. I've asked this question at many places and to many different people, but got absolutely ignored. So i decided to ask here.

The discussion began with a thought experiment:

There are two objects of the same mass, but of different densities: an ideal sphere made of pine and an ideal sphere made of iron. The mass of these spheres is sufficient for their own gravity to eventually compress them to the Schwarzschild radius, forming a black hole.

Since the mass is the same, the Schwarzschild radius of the two spheres will also be the same, and the question was: which of the spheres will shrink to the Schwarzschild radius faster, pine or iron? Or will both do it at the same time?

My friend's intuition told him that the pine sphere would win this race, because its initial density is lower and it supposedly "resists compression less." I thought about it and tried to find confirmation of this guess.

After several days of studying the issue, I came to the following conclusion. The pressure of a degenerate electron and neutron gas plays a key role in countering gravitational compression. In turn, the pressure of a degenerate gas, due to the Pauli principle, increases with increasing density, since particles occupy higher energy states in order to avoid the prohibition of being in the same quantum state. Thus, for a denser substance, the degenerate gas will have a higher pressure - which means that the body will resist gravitational compression more strongly. Accordingly, the iron sphere will later turn into a black hole.

However, my intuition tells me that I am missing some important factor in my reasoning. After all, my competence, experience and knowledge are too small. I wonder if the initial density has any noticable effect, given that the sheer gravity of the object will compress it anyway, thus making it very dense, with the initial density playing but a negligible role in the process.

-----------
To clarify my question: could you explain whether the initial density of matter affects the rate of formation of a black hole, and if so, how exactly?

I will be extremely grateful and glad if you can shed light on this issue and possible "holes" in my reasoning! Thank you in advance for your time.

 

[Baluncore]

Pine density ≈ 1.0 ; Iron density ≈ 8.0
Both have the same mass.
The pine sphere has about twice the radius of the iron sphere.
Make two black holes from the two spheres, or drop the two into a BH?

The outside of the pine sphere has further to fall, and is further from its centre of mass, so it will take longer to collapse.

Initial and final conditions?
Do both spheres start inside their Schwarzschild radius?
Or is the race for their radius to reach the Schwarzschild radius?

 

[Hornbein]

Spheres of ordinary materials so large they spontaneously collapse into black holes. Do the spheres instantaneously materialize in a vacuum with uniform density or do they form by gradual accretion? I'll assume the former. It seems to me the process would be complicated. Fusion would occur in the center, the radiation pressure slowing the collapse. It seems to me that iron would yield much less radiation from fusion and hence collapse more rapidly. It would also have the advantage of a smaller initial diameter. So iron goes first.

But I'm not at all sure about this. Maybe a supernova explosion would occur with the pine, blowing away most of the mass and creating a black hole more rapidly. Is that cheating?

 

[i walk away]

Pine density ≈ 1.0 ; Iron density ≈ 8.0
Both have the same mass.
The pine sphere has about twice the radius of the iron sphere.
Make two black holes from the two spheres, or drop the two into a BH?

The outside of the pine sphere has further to fall, and is further from its centre of mass, so it will take longer to collapse.

Initial and final conditions?
Do both spheres start inside their Schwarzschild radius?
Or is the race for their radius to reach the Schwarzschild radius?

Sorry if i was unclear about my though experiment. I'll now clarify as precise as i can.

Make two black holes from the two spheres, or drop the two into a BH?

We instantaneously materialize the two spheres in a vacuum. We assume that they do not interact with each other in any way and are infinitely far away from each other.

Do both spheres start inside their Schwarzschild radius?

No, the spheres are in a state of equilibrium density. In an absolute vacuum they are not affected by any external forces, their volume is determined only by the internal bonds between the atoms of which they consist. This formula can be used to find a radius of a sphere given its mass and density of the material:

Or is the race for their radius to reach the Schwarzschild radius?

Yes.

 

[Ibix]

We instantaneously materialize the two spheres in a vacuum.

This is impossible. If you try to describe doing it in GR then you can get to 1=0 in two lines of maths. However, I suspect you can at least consider a case with a sphere of constant density as an initial condition and just not ask too many questions about how it got there. In that case the iron sphere wins by several million years at least since the pine contains elements light enough to undergo exothermic fusion reactions and it turns into a star, while the iron does not. And I would tend to doubt that we have accurate knowledge about the behaviour of pine under such extreme stresses.

There is an exact solution known for the collapse of spheres of non-interacting dust, which is called Oppenheimer-Snyder collapse. In this case the denser object collapses faster (there are a lot of complications around what "when it reaches its own Schwarzschild radius" means, but other measures that are easier to understand are available). You specified equal-mass spheres, but the same is true of equal-radius spheres of different densities. This model isn't realistic for the exact scenario you want to discuss because it ignores resistance to collapse (fine for something that's already a neutron star and then collapses, less so for something that isn't).

Generally, this is a nasty problem because so many other things than just density matter to the result, and it's not doable so we can't refer to experiment. So anything is basically a guess unless you want to build a realistic model in a computer (expensive and slow, and still reliant on assumptions about the behaviour of ordinary materials in impossibly extreme circumstances). If I had to guess I would come down on the side of the denser sphere collapsing faster, but it's more than possible that there are factors I haven't thought of.

 

[Ken G]

I will presume that your curiosity is not really around what would happen to those two objects, it is you want to use those objects in a thought experiment to try to understand how and why black holes form. The first thing you should understand there is that if you put together an iron ball or a pine ball with more than the Chandrasekhar mass of 1.4 solar masses, or more than the maximum neutron star mass of about 2 solar masses, it would not right away collapse into a black hole, any more than a massive star does. So why would it make a black hole at all? That's what you really want to know, and the time it takes to make a black hole will depend on how long it takes for all the things that must happen first to happen. That will control the total time (the actual collapse to a black hole, once everything else needed has occurred, is very fast).

So you talk about the degeneracy pressure of neutrons, so what you need to realize is that there is not any special new pressure that appears when neutrons are degenerate. It's the same pressure that was always there, which comes from the kinetic energy of the neutrons (their fast motion), it's just that as systems get to low entropy (which happens after they have lost a lot of heat, since change in entropy depends on heat lost), they start to resist losing any more entropy. (This is like saying, if your office is a complete mess you can easily start organizing it, but once everything is in their proper place you have to look around quite a bit to find something out of place!) Degeneracy is the condition like a well-ordered office space, and it resists losing heat because it resists lowering its entropy.

So now you understand one part of the crucial process of making a black hole: loss of heat. Ultimately what determines the time it will take your ball of iron and ball of pine to become a black hole is how much heat they each need to lose, and how fast they can lose it. So that is going to require knowing a lot about things like the thermal conductivity of iron and pine, and how they go through various other stages as they contract and get hotter. They will first ionize into a plasma, and later the electrons will get captured into neutrons, all that happens before you can make a black hole and it takes a very long time. Also, as mentioned above, the pine contains nuclei that can release energy when they fuse (the iron cannot), to that's a big complication that will slow the pine down a lot. If you don't want to worry about fusion, you can imagine turning that off (it doesn't play an important role in understanding the formation of black holes). Basically, I don't think you really care about all these complications in how long it will take to make the black hole, since they deal more with energy transport than gravity, I think you just want to understand how a black hole happens. Here are the stages (ignoring fusion, say for iron):
1) heat loss leads to contraction,
2) contraction leads to gravitational energy release and rising temperature
3) rising temperature leads to turning the material into a plasma
4) further heat loss causes the electrons to become relativistic, which means they contract even more when they lose heat
5) faster contraction leads to higher energy in the electrons, allowing them to enter nuclei and essentially join with protons to become neutrons
6) neutrons continue to lose heat and contract, achieving higher energy until they go relativistic also
7) relativistic neutrons contract even more when they lose heat
8) the Schwarzschild radius is ultimately achieved.

So those are the stages, and the time it takes is more in the early stages than the later ones. A massive stellar core does something quite similar. More massive stars do it faster, but that's not because their gravity is stronger (it tends to be weaker), it is simply because they lose heat faster, so those early stages go by faster (though they take millions of years).

 

[i walk away]

I will presume that your curiosity is not really around what would happen to those two objects, it is you want to use those objects in a thought experiment to try to understand how and why black holes form. The first thing you should understand there is that if you put together an iron ball or a pine ball with more than the Chandrasekhar mass of 1.4 solar masses, or more than the maximum neutron star mass of about 2 solar masses, it would not right away collapse into a black hole, any more than a massive star does. So why would it make a black hole at all? That's what you really want to know, and the time it takes to make a black hole will depend on how long it takes for all the things that must happen first to happen. That will control the total time (the actual collapse to a black hole, once everything else needed has occurred, is very fast).

So you talk about the degeneracy pressure of neutrons, so what you need to realize is that there is not any special new pressure that appears when neutrons are degenerate. It's the same pressure that was always there, which comes from the kinetic energy of the neutrons (their fast motion), it's just that as systems get to low entropy (which happens after they have lost a lot of heat, since change in entropy depends on heat lost), they start to resist losing any more entropy. (This is like saying, if your office is a complete mess you can easily start organizing it, but once everything is in their proper place you have to look around quite a bit to find something out of place!) Degeneracy is the condition like a well-ordered office space, and it resists losing heat because it resists lowering its entropy.

So now you understand one part of the crucial process of making a black hole: loss of heat. Ultimately what determines the time it will take your ball of iron and ball of pine to become a black hole is how much heat they each need to lose, and how fast they can lose it. So that is going to require knowing a lot about things like the thermal conductivity of iron and pine, and how they go through various other stages as they contract and get hotter. They will first ionize into a plasma, and later the electrons will get captured into neutrons, all that happens before you can make a black hole and it takes a very long time. Also, as mentioned above, the pine contains nuclei that can release energy when they fuse (the iron cannot), to that's a big complication that will slow the pine down a lot. If you don't want to worry about fusion, you can imagine turning that off (it doesn't play an important role in understanding the formation of black holes). Basically, I don't think you really care about all these complications in how long it will take to make the black hole, since they deal more with energy transport than gravity, I think you just want to understand how a black hole happens. Here are the stages (ignoring fusion, say for iron):
1) heat loss leads to contraction,
2) contraction leads to gravitational energy release and rising temperature
3) rising temperature leads to turning the material into a plasma
4) further heat loss causes the electrons to become relativistic, which means they contract even more when they lose heat
5) faster contraction leads to higher energy in the electrons, allowing them to enter nuclei and essentially join with protons to become neutrons
6) neutrons continue to lose heat and contract, achieving higher energy until they go relativistic also
7) relativistic neutrons contract even more when they lose heat
8) the Schwarzschild radius is ultimately achieved.

So those are the stages, and the time it takes is more in the early stages than the later ones. A massive stellar core does something quite similar. More massive stars do it faster, but that's not because their gravity is stronger (it tends to be weaker), it is simply because they lose heat faster, so those early stages go by faster (though they take millions of years).

This is easily the most educating answer i've had for my question, and for that i am very grateful to you.
Ultimately, the core of my question was about the role that initial density alone plays in the process of the formation of a black hole. But i've failed at bringing this fact up clearly enough to make people throw away other variables in their replies. However, i'm glad that i did, in fact, fail to simplify this thought experiment to such extent. Because i've learned a whole lot from every answer that considered intiially unrequired conditions.
Summarizing every received answer, i think i now have a much better understaing of black hole formation overall.
You all have my gratitude gentlemen.

 

[Drakkith]

4) further heat loss causes the electrons to become relativistic, which means they contract even more when they lose heat

Can you expand a little on this, Ken?

 

[Ken G]

Can you expand a little on this, Ken?

Sure, you are probably referring to the seeming paradox that the electrons go relativistic when the system loses a lot of heat. This follows from force balance, expressed most easily in terms of the nonrelativistic virial theorem. The virial theorem states that when any ball of gas in force balance loses an amount of heat equal to Q, the internal kinetic energy of the particles increases by Q (the missing 2Q is provided by gravity, as the system must contract). So the more heat the system loses, the higher the internal kinetic energy.

When the system is also electron degenerate, this implies that most (or nearly all) of the internal kinetic energy is in the electrons. Hence, the Q goes to the electrons. When their average kinetic energy exceeds 511 keV, they go highly relativistic. This changes the nature of the virial theorem, such that now when Q is lost, the kinetic energy goes up by more than Q (possibly even a lot more as the system goes more and more relativistic). So it's a kind of "leveraging of heat loss" that causes the rapid contraction leading to core collapse and supernova, once the electrons gain enough kinetic energy to enter the nuclei and essentially merge with protons to become neutrons.

 

[bland]

You do not have to complicate matters by having the two spheres be iron and wood, one can be a solid ball of iron and the other can be a larger ball of iron foam.

 

[i walk away]

You do not have to complicate matters by having the two spheres be iron and wood, one can be a solid ball of iron and the other can be a larger ball of iron foam.

you are right

 

[PAllen]

A different, related question is an initial state of the universe (thus avoiding difficulties of how such could arise) having a ball of whatever, with its 'normal' density, large enough to be inside its Schwarzschild radius -with the rest of the universe empty, and given by the vaccuum Schwarzschild solution. For example, a ball of air the radius of the milky way is already well inside its Schwarzschild radius. In this case, per general relativity, there would be extremely rapid collapse to a singularity, without need or possibility of loosing heat or radiating anything - any outgoing directed light would be collapsing almost as fast as matter.

This brings up the idea black holes are not necessarily features of extreme density. The more massive the BH (formed from a single collapse), the less dense a state just before the horizon crosses the surface.

 

[ohwilleke]

TL;DR Summary: Could you explain whether the initial density of matter affects the rate of its compression into a black hole, and if so, how exactly?

The other day, a friend and I had a discussion about black holes, namely how the density of a body affects the process of its transformation into a black hole.

My friend and I have too little collective knowledge in the field of theoretical physics regarding black holes, so the discussion has reached an impasse. I thought it would be great to ask for help from someone more knowledgeable. I've been emailing a professor from the Department of Space Physics at my university, but he doesn't seem quite willing to respond. I've asked this question at many places and to many different people, but got absolutely ignored. So i decided to ask here.

The discussion began with a thought experiment:

There are two objects of the same mass, but of different densities: an ideal sphere made of pine and an ideal sphere made of iron. The mass of these spheres is sufficient for their own gravity to eventually compress them to the Schwarzschild radius, forming a black hole.

Since the mass is the same, the Schwarzschild radius of the two spheres will also be the same, and the question was: which of the spheres will shrink to the Schwarzschild radius faster, pine or iron? Or will both do it at the same time?

My friend's intuition told him that the pine sphere would win this race, because its initial density is lower and it supposedly "resists compression less." I thought about it and tried to find confirmation of this guess.

After several days of studying the issue, I came to the following conclusion. The pressure of a degenerate electron and neutron gas plays a key role in countering gravitational compression. In turn, the pressure of a degenerate gas, due to the Pauli principle, increases with increasing density, since particles occupy higher energy states in order to avoid the prohibition of being in the same quantum state. Thus, for a denser substance, the degenerate gas will have a higher pressure - which means that the body will resist gravitational compression more strongly. Accordingly, the iron sphere will later turn into a black hole.

However, my intuition tells me that I am missing some important factor in my reasoning. After all, my competence, experience and knowledge are too small. I wonder if the initial density has any noticable effect, given that the sheer gravity of the object will compress it anyway, thus making it very dense, with the initial density playing but a negligible role in the process.

-----------
To clarify my question: could you explain whether the initial density of matter affects the rate of formation of a black hole, and if so, how exactly?

I will be extremely grateful and glad if you can shed light on this issue and possible "holes" in my reasoning! Thank you in advance for your time.

A black hole forms essentially instantly at the moment when its mass is all within the Schwarzschild radius of that mass (named after the scientist who first calculated it in the year 1916), which is given by the following formula:

For a so called "stellar black hole" this happens when you have an object with the density of roughly a neutron star (i.e. close to that of an atomic nucleus) and a mass equal to a cutoff somewhere in the vicinity of more than 2 but less than 3 solar masses (i.e. using the mass of our Sun as a unit of mass), and happens when enough mass-energy is added to a neutron star to take it over that threshold. The exact cutoff has not be determined, but there is a name for this cutoff, which is the Tolman–Oppenheimer–Volkoff limit (sometimes abbreviated as the "TOV limit"). The best estimate is that this is 2.2 to 2.9 solar masses, and theoretical calculations of this are limited mostly by our lack of a complete understanding about how the matter in a neutron star behaves at such extreme gravitational pressures and our lack of knowledge of whether any kinds of matter other than neutrons are found in neutron stars (and if so, how much of what and with what high pressure properties).

A neutron star is the collapsed core of a massive supergiant star. It results from the supernova explosion of a massive star—combined with gravitational collapse—that compresses the core past white dwarf star density to that of atomic nuclei.

The collapse that forms a neutron star is also extremely rapid.

As a practical matter, neutron stars form mostly from the hydrogen and helium which make up most of a supergiant star, with some trace heavier elements (all of which count as "metals" in the sense that astronomers use the word). In the real world, a collision of compact objects made of more dense atomic elements or molecules, like water or iron, doesn't actually give rise to objects dense enough to form neutron stars or black holes.

While general relativity doesn't actually require that black holes be formed when a neutron star gains enough mass to cross the threshold of mass within a Schwarzschild radius, we don't know of any other process in the real world that actually give rise to black holes (at least at the energy scales that have prevailed since the first stars began to form).

So, after the very rapid supergiant star collapse that forms a neutron star, the time period that passes before that neutron star can become a black hole (which happens essentially instantaneously), is the question of how long it takes to form a black hole upon reaching the mass which is the Tolman–Oppenheimer–Volkoff limit depends entirely on how fast the neutron star gains mass-energy less the rate at which it losses mass-energy (mostly by emitting light). This depends mostly upon what mass happens to be in its vicinity, and essentially not at all upon what that mass is made of before it gets close to the neutron star and is torn to bits. It could happen in a few minutes, it could take billions of years or never happen at all.

The duration between the formation of a neutron star and its graduation to black hole status is based upon how much mass-energy there is in its vicinity to absorb and bring it over the transition mass threshold.

The initial mass of a neutron star can impact how long it takes to form a black hole, because a less massive neutron star had to add more mass to become a black hole than a less massive neutron star, and all other things being equal, adding more mass takes more time than adding less mass. The minimum mass of a neutron star is somewhere between 0.7 and 1.4 solar masses.

But what particular kinds of stuff the neutron star is absorbing is basically irrelevant. The free fall rate of 100 tons of gold and 100 tons of hydrogen gas into the neutron star from the same starting point is the same, and has the same effect in terms of moving the neutron star closer to the transition point to a black hole. Once atoms hit a neutron star, the gravitational pressure of the neutron star at near contact range will rip the atom to pieces into its constituent protons and neutrons, and will tend to fuse the atom's electrons into its protons to form neutrons.

A black hole could start out bigger than this, but this would require two bigger objects not dense enough to be black holes, at least one of which is close to neutron star density, to collide. As a practical matter, this kind of collision would be very rare in the current conditions of the universe.

The size of a black hole's event horizon, once you cross that threshold and a black hole forms, is proportionate to its mass. More massive black holes have larger event horizons, less massive black holes have smaller ones. (The simply formula for the Schwarzschild radius is tweaked a little if the black hole is spinning or has an electromagnetic charge, but the basic relationship of the event horizon size being proportional to the mass of the black hole remains intact.)

Somewhat counterintuitively, as a black hole gets larger in mass, the mass of the black hole divided by the volume of space within its event horizon falls. For a supermassive black hole, the mass of the black hole divided by the volume of space within its event horizon is on the order of the density of water.

It isn't quite proper to talk about this ratio of black hole mass to event horizon volume as the "density" of the black hole, however, since we don't know how mass is distributed within the event horizon. For example, it could have an extremely dense core that gets progressive less dense as you approach the event horizon.

Nothing outside the event horizon tells us anything about what is going on inside the event horizon.

 

[Ibix]

A black hole forms essentially instantly at the moment when its mass is all within the Schwarzschild radius of that mass

A black hole includes the entire past domain of dependence of this region. That is, it exists for some time before the matter has collapsed behind its own horizon.

In the real world, a collision of compact objects made of more dense atomic elements or molecules, like water or iron, doesn't actually give rise to objects dense enough to form neutron stars or black holes.

I would say this misses the point. In the real world, we do not see stellar-mass objects made of heavier elements at all - it's not lack of density, it's lack of size.

While general relativity doesn't actually require that black holes be formed when a neutron star gains enough mass to cross the threshold of mass within a Schwarzschild radius, we don't know of any other process in the real world that actually give rise to black holes (at least at the energy scales that have prevailed since the first stars began to form).

I found this sentence a bit difficult to follow. I think it's meant to say that GR doesn't put any minimum mass limit on black holes, but we know of no process except stellar collapse that can make black holes, so we don't see (e.g.) planet-mass holes. If so, I agree.

A black hole could start out bigger than this, but this would require two bigger objects not dense enough to be black holes, at least one of which is close to neutron star density, to collide. As a practical matter, this kind of collision would be very rare in the current conditions of the universe.

Depends what you mean by "rare". LIGO has detected several neutron star/neutron star collisions in the few years and fairly narrow sensitivity range it's been looking for. They're certainly not happening at a huge rate, but they're not so rare that we never see them.

Somewhat counterintuitively, as a black hole gets larger in mass, the mass of the black hole divided by the volume of space within its event horizon falls.

I don't think this is true, since the interior volume of a black hole isn't really a well-defined notion. The mass divided by the volume of a Euclidean sphere of radius equal to the Schwarzschild radius falls as the mass increases, which is where the "density decreases with mass" thing comes from as far as I know, but the whole point about black holes is that they're very much non-Euclidean.

It isn't quite proper to talk about this ratio of black hole mass to event horizon volume as the "density" of the black hole, however, since we don't know how mass is distributed within the event horizon.

In fact, the mass isn't anywhere in standard classical black hole solutions. They're vacuum everywhere. Even models like Oppenheimer-Snyder collapse are (at least in some senses) vacuum once the matter reaches the singularity.

 

[ohwilleke]

A black hole includes the entire past domain of dependence of this region. That is, it exists for some time before the matter has collapsed behind its own horizon.

I'm not certain that I'm following what you are trying to say here.

I found this sentence a bit difficult to follow. I think it's meant to say that GR doesn't put any minimum mass limit on black holes, but we know of no process except stellar collapse that can make black holes, so we don't see (e.g.) planet-mass holes. If so, I agree.

My point is that while GR doesn't itself limit how black holes can be formed as long as there is a sufficiently concentrated mass, that stellar collapse is the only known process by which this happens in the current era (leaving aside the possibility of primordial black holes in the early universe by some other process).

Depends what you mean by "rare". LIGO has detected several neutron star/neutron star collisions in the few years and fairly narrow sensitivity range it's been looking for. They're certainly not happening at a huge rate, but they're not so rare that we never see them.

Even two neutron stars can't be much more than 5.8 stellar masses, and could be quite a bit less than that. I was imagining a neutron star colliding with something (other than a black hole) that is much more massive than another neutron star to create a fairly big intermediate sized black hole at the outset. So far as I know, we aren't aware of any stellar black holes being formed in that manner, but I could be wrong.

I don't think this is true, since the interior volume of a black hole isn't really a well-defined notion. The mass divided by the volume of a Euclidean sphere of radius equal to the Schwarzschild radius falls as the mass increases, which is where the "density decreases with mass" thing comes from as far as I know, but the whole point about black holes is that they're very much non-Euclidean.

I think I was actually quite clear on the point that black hole density is not a well-defined notion and does not exactly correspond to density in the ordinary sense.

But, the volume of a sphere of radius equal to the Schwarzchild radius and the mass of a black hole are well-defined, even though they don't correspond exactly to the concept of density. Since this relationship is not well known to the general public, and is quite counter-intuitive, it bears mentioning.

 

[Ibix]

I'm not certain that I'm following what you are trying to say here.

The event horizon forms inside the star and expands outwards. So the black hole finishes forming when the last of the matter falls through, but it started forming earlier.

stellar collapse is the only known process by which this happens in the current era

Agreed, then.

I think I was actually quite clear on the point that black hole density is not a well-defined notion and does not exactly correspond to density in the ordinary sense.

Maybe. But what you said was that it isn't quite proper to talk about this ratio of black hole mass to event horizon volume as the "density" of the black hole, however, since we don't know how mass is distributed within the event horizon. The point is not that we don't know how the mass is distributed, it's that the volume depends on your choice of coordinates and may be chosen to be infinite, so the average density is a matter of choice and may be zero. In fact, in simple models of eternal black holes the whole thing is vacuum. Even in collapse models like Oppenheimer-Snyder the interior volume can be chosen, and the average interior density is whatever you choose. In the non-vacuum region the mass density is time varying at least, rising to infinity as you approach the singularity.

So saying we don't know the mass distribution is missing the point. If we assume general relativity is exactly correct and one of the simplistic models is in play and we know all its parameters then we know exactly what the mass distribution is - and we still cannot define the density in a non-arbitrary way.

But, the volume of a sphere of radius equal to the Schwarzchild radius and the mass of a black hole are well-defined

The volume of a sphere in Euclidean space is well defined, sure, but a black hole spacetime does not obey Euclidean geometry. I'm not even sure if it's possible to find a Euclidean foliation of something like Schwarzschild spacetime. If you can find one the interior volume of the black hole will be time dependent because the only time-independent one is the regular Schwarzschild foliation which is non-Euclidean.

So you can certainly divide the mass of the hole by the volume of a Euclidean three-sphere, but the question is why would you do that? It's a bit like dividing the mass of wine in a bottle by the volume of glass used in its construction. Sure it's a well-defined process, and you'll probably even get a fairly clear relationship between bottle size and this number that might be of interest to an economist, but it isn't the density of anything. Similarly, dividing the mass of the hole by the volume of something else doesn't give you the density of anything.

 

[ohwilleke]

So you can certainly divide the mass of the hole by the volume of a Euclidean three-sphere, but the question is why would you do that?

Because it provides you with intuition about what is necessary to form a black hole. It helps you to make sense of the question:

Given a certain mass distributed in a certain way in space that is not at this time a part of a black hole, is that mass distribution close to reaching the threshold of becoming a black hole or not?

The pre-black hole density of a neutron star has to be on the same order as an atomic nucleus at a mass of 2-something solar masses to be on the brink of forming a black hole.

But, in principle, you could have a much less dense mass distribution, with a much greater total mass, and it could also be on the brink of forming a black hole.

Conversely, for a mass distribution less than 2-something solar masses to form a black hole, the density of the mass distribution on the brink of forming a black hole must be progressively greater than the density of an atomic nucleus, which in turn, presents the open question of whether there could even be a way to squeeze matter that tightly (and, if so, how).

This, in turn, brings home the point that what distinguishes a non-black hole mass distribution that is close to becoming a black hole from one that is not, is not, as we commonly say when trying to describe this in plain English, truly about the density of the pre-black hole mass distribution. Instead, it is something quite different expressed by the Schwarzschild radius formula.

Developing an understanding that black hole physics has what feels like scale dependence in an unexpected way, is a useful way to grasp what is going on in general relativity's non-linear functions in a way that would otherwise be more difficult to grok.

This may pose all sorts of logical and definitional conundrums post-singularity in terms of definitions and meanings, but the inside of a black hole post-singularity isn't something we ever have to worry about. We will always like outside the black hole and it is sensible to want to have a good idea about what conditions are necessary to form them in theory, even if the "in practice" part is much more singular.

 

[ohwilleke]

The event horizon forms inside the star and expands outwards. So the black hole finishes forming when the last of the matter falls through, but it started forming earlier.

I guess that's one way to think about it.

I would be inclined to think that the black hole forms as soon as there is an event horizon inside the star, basically instantaneously, and that the expansion outwards is a further accretion of matter to a black hole that already formed, i.e. that it is no different in principle from the black hole sucking up some interstellar hydrogen gas that it happens to pass thorough at some arbitrary time in the future.

But, the distinction seems to just really boil down to the definition of what a "black hole forming" means and is arbitrary, so it doesn't really matter.

In any case, while the process of a newly formed black hole absorbing the entire star wouldn't be instantaneous with that definition of black hole formation, it would, presumably, still be rapid to the point of being almost explosively rapid. My intuition is that this would be a matter of seconds or minutes, at most. It wouldn't take Earth months or Earth years to happen the way that it can take that kind of time for a black hole to gradually feed off of a companion star in a binary system.