@vibayjain: that is for a star - for the Sun to become a black-hole it would need to be super-dense first. That is correct. However, a supermassive black hole need not be so dense. You can see this from the math - the Schwarzschild radius is proportional to the mass, but the radius of a spherical lump of matter is proportional to the cube-root of mass.
This means that, for a given density, one can find an amount of mass whose physical radius is the same as the Schwarzschild radius.
It won't be that simple IRL - since you won't get a uniform density. That's just to simplify the thought experiment - just like we do a lot of thought experiments with non-rotating BHs even though these probably don't exist IRL (find a star that doesn't spin - still, I understand it is plausible).
One way SMBHs form is by having big popIII "seed" BHs absorb each other and stars and anything else they can and so join up to make a really big one. If you double the mass, you double the schwarzchild radius, but this increases the volume by eight times ... so the density decreases: it's a quarter what you started with. YOu can keep going and get the density as low as you like.
Or do it the other way - because this suggests that a black-hold could form from a low density accumulation of matter.
BA talks about this [#10]:
"
A billion solar mass black hole (big, but we see them this big in galaxy centers) would drop that density by a factor of 1 x 1018. That would give it a density of roughly 1/1000 of a gram per cc… and that’s the density of air! ... A billion solar mass black hole would have an event horizon 3 billion km in radius — roughly the distance of Neptune to the Sun. ... See where I’m going here? If you were to rope off the solar system out past Neptune, enclose it in a giant sphere, and fill it with air, it would be a black hole!"
(Of course - that mass would collapse unless it was also pretty hot - back-of-envelope suggests 8-1000K, which seems awfully cool to me so I don't trust it. Still, see below.)
We could argue that, since the event horizon is right there the space-time geometry means that as a diffuse contracting gas cloud approaches (BA example) air density, it suddenly radiates all it's remaining energy and collapses to a super dense state at the center of it's event horizon.
We could argue that a gas cloud that size and mass cannot possibly sustain itself against it's gravity - there is no possible equilibrium. The mass is so large that it's gravity will eventually overcome even it's pauli pressure.
I'd like to do the math on this.
When a stellar BH forms it has to radiate away a lot of the energy that was previously supporting it in hydrostatic equilibrium. But if the energy could not escape the surface of the star for some reason, wouldn't the star be able to support itself indefinitely?
There are a couple of big ifs here, and I'd like to do the math on them.
1. the contracting cloud does not undergo a sudden collapse when approaching the critical density, allowing an event horizon to appear while the gas is still at quite low densities.
2. no energy escapes the resulting black hole - so the enclosed system is purely adiabatic
So we may ask, if there is a balance point where the mass is big enough for a low-density SMBH but low enough to allow a balance when there are no losses - and, where no equilibrium is possible, how long one may expect the low density state to exist.
We could look at it the other way around too ... what is the space-time geometry inside a spherical matter field of a given density and extent. What happens to this as the extent is increased, keeping the density the same?
At first I thought that such large gas fields could not balance their gravity thermally, so they would have to use up energy in nuclear reactions etc - then I saw that ISM about SMBHs was very close to thermal in the early universe. So it does not seem so far-fetched that the SMBH itself formed from material in a similar state.
I thought that small variations in the density within such a large gas cloud would make bits of it form stars, and stellar-size black holes, so this would never happen. But it turns out that, while this almost always happens, there are ways that
a large gas cloud can contract isothermally without forming stars
"
In the absence of H2 molecules, the primordial gas in early dark matter haloes with virial temperatures just above Tvir >~ 104K cools by collisional excitation of atomic H. Although it cools efficiently, this gas remains relatively hot, at a temperature near T ~ 8000 K, and consequently might be able to avoid fragmentation and collapse directly into a supermassive black hole."
... that's just one model.Probably Drakkith is correct and total collapse is inevitable. From the pov of an observer outside the nascent SMBH, there would be a huge burst of light to make ordinary supernovas look weak and pale. (The energy release is a consequence of the collapse.) The explosion would clear out the ISM by the SMBH - except maybe an accretion disk for quasars.
I'd still like to see the math.
That's all I'm asking about.
I'm really surprised at the amount of hostility I'm getting I mean: what's wrong with doing the math? I'd have thought something like this would be an exercize for senior undergrad college students somewhere like the BH formation math is in basic GR courses. However, it seems that nobody reading this thread knows of any such thing so I'll have to look elsewhere, maybe do it myself.
Thanks anyway folks.
