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.

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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.