Stellar-mass black hole formation sequence

[sevenperforce]

I feel like this could go in quite a few of the Physics subforums (Quantum Physics, Beyond the Standard Model, Special and General Relativity, or High Energy, Nuclear, Particle Physics) instead of Astronomy and Cosmology, but hopefully this will work. This is my first question I've posed here, btw, so please forgive (but feel free to correct) any missteps.

(Background: B.S. in physics, pretty good grasp of GR and SR, wide familiarity with astrophysics, less robust grasp of QM; this question is personal interest alone.)

As far as I've been able to tell, the sequence of events in a massive star's collapse aren't very well-understood yet. It's not known whether the iron core collapses first to an unstable neutron star and then into a black hole, or if the collapse goes directly to a black hole. Moreover, there's a gap between the largest theoretically possible neutron stars (~3 M_☉) and the smallest observed stellar-mass black holes (~5 M_☉); presumably, this would be the result of some aspect of the collapse sequence, but that's not really known either.

There is a great deal of discussion about the theoretical minimum-mass black hole, usually in the context of quantum gravity and the potential for quantized black holes. The entropic argument for black holes (that the event horizon of a black hole must be quantized to the Planck area in order to limit information to discrete bits) doesn't seem to hold up well and has been generally dismissed by the larger physics community, but at the same time it seems quite reasonable to assume that some minimum to the mass of a black hole must exist. If there was no lower limit on the mass of a black hole, then the power output of Hawking radiation would go to infinity, which means either there's some minimum on black hole mass or the Hawking radiation equations break down at quantum levels.

On the other side of things: all the discussions/models/depictions I've ever seen of stellar-mass black hole formation feature a macroscopic event horizon, with the Schwarzschild radius and the surface radius growing closer (either because the Schwarzschild radius is increasing due to added mass or because the surface radius is shrinking due to increased density) until they finally meet and the black hole is formed.

However, in the collapse of a massive star where neutron degeneracy pressure is insufficient to arrest collapse and a black hole forms, it doesn't seem that there would be anything preventing the density at the center from increasing without bound. The only pressure would be radiation pressure from the massive amounts of energy received, but due to the gravitational field the incoming radiation would be dramatically blueshifted while outgoing radiation would be dramatically redshifted, so that the pressure gradient goes to infinity as you approach the center. Is it possible, then, that the exponential increase in density at the center will reach the necessary conditions for the formation of a minimum-mass black hole before the conditions for a black hole are met at a larger radius, such black holes originate at the center and propagate outward?

If so, that might lead to some interesting predictions about the rest of the black hole formation sequence, including a possible explanation for why the gap exists between neutron stars and black holes as well as a more promising basis for black hole quantization.

Of course, I fully recognize that trying to model a minimum-mass black hole using GR and the characteristics of Hawking radiation may well be comparable to trying to model a thermonuclear explosion using a 2nd-grade chemistry set...but I figure it's worth a shot anyway.

 

[Simon Bridge]

Not all the mass of a star need form a super-dense object. i.e. neutron star formation.
You are asking, I guess, if the huge pressure inside a very massive collapsing body could produce such high density inside that a black hole region could form which would not otherwise form by collapse from it's own mass alone? Off the top of my head: sure.
However, I suspect you are also wondering if this means that maybe such a "light" black hole could remain afterwards ... that would be a no. Once formed, the light hole would quickly gobble more mass until you had a quite conventional black hole. However, this could be seen as a stage in the transition from regular degenerate matter to a black hole... probably very rapid if the transition to a Neutron star in a supernova is anything to go by.

But I didn't look anything up - there's probably a wrinkle I didn't think of right away. There may be no need, particularly in the quantum regime, for there to be an intermediate stage at all.

This is a highly speculative area after all.

 

[sevenperforce]

Not all the mass of a star need form a super-dense object. i.e. neutron star formation.

Indeed. In fact, this is one of the things that makes the observed mass gap so odd. Neutron stars can be produced by O+Ne+Mg-core stars in the 8-10 M_☉ range, by Fe-core stars in the 10-25 M_☉ range, or by larger-mass progenitors if the stellar metallicity is high enough, but in no circumstance does the degenerate remnant exceed 3 M_☉. In contrast, black holes can form from progenitors above 25 solar masses with lower stellar metallicity, sometimes involving the fallback of matter onto an initial neutron star, but in no circumstance is the black hole less than 5 M_☉. There must be some aspect of the collapse process in which collapsing stars which meet the conditions for creating a neutron star remnant "blow away" all but <3 M_☉ to escape velocity, but collapsing stars which meet the conditions for creating a black hole remnant always manage to retain >5 M_☉.

For two collapsing progenitors of equal mass but different metallicity, such that the first will leave a neutron star remnant and the latter will leave a black hole remnant, the former's collapse sequence must necessarily be markedly more energetic than the latter. There do not seem to be any cases in which a neutron star grows to its maximum mass and then collapses into a black hole of 3-5 M_☉ thereafter.

You are asking, I guess, if the huge pressure inside a very massive collapsing body could produce such high density inside that a black hole region could form which would not otherwise form by collapse from it's own mass alone? Off the top of my head: sure.

Well, yes, but in a less general case. Specifically, can the density gradient go up rapidly enough that a black hole of arbitrarily low mass forms?

The density/mass threshold for black holes looks like this (log plot):

If the tremendous pressure inside the center of a collapsing body causes the central density to increase more rapidly than the inverse-squared relationship between density and mass, then the black hole will start at the center at an arbitrarily low mass. I'm just trying to figure out if that's possible. It seems possible; after all, in a collapse scenario where degeneracy pressure is no longer able to function, radiation pressure would be the only remaining force other than gravitational force, and the strong wavelength shifting of the gravitational field would presumably make inward radiation pressure exponentially greater than outward radiation pressure.

I suspect you are also wondering if this means that maybe such a "light" black hole could remain afterwards ... that would be a no. Once formed, the light hole would quickly gobble more mass until you had a quite conventional black hole.

That's not quite what I was wondering, but close. If the black hole forms at an arbitrarily low mass, then presumably this would be the minimum possible mass of a black hole. Since this would likely be near the Planck scale, the decay lifetime of such a black hole would be almost infinitely brief (if, in fact, Hawking's predictions hold for this scale), too brief for it to "gobble up" any more mass. Which could perhaps lead to some interesting predictions. But I know that I'll first need to figure out whether the initial concept (exponential density gradient causing an initial black hole to form with arbitrarily low mass) is even a possibility.

 

[Bernie G]

... It's not known whether the iron core collapses first to an unstable neutron star and then into a black hole, or if the collapse goes directly to a black hole. ... Moreover, there's a gap between the largest theoretically possible neutron stars (~3 M☉) and the smallest observed stellar-mass black holes (~5 M☉); presumably, this would be the result of some aspect of the collapse sequence, but that's not really known either. ... Is it possible, then, that the exponential increase in density at the center will reach the necessary conditions for the formation of a minimum-mass black hole before the conditions for a black hole are met at a larger radius, such black holes originate at the center and propagate outward? ... There must be some aspect of the collapse process in which collapsing stars which meet the conditions for creating a neutron star remnant "blow away" all but <3 M_☉ to escape velocity, but collapsing stars which meet the conditions for creating a black hole remnant always manage to retain >5 M_☉.

Isn't the maximum observed neutron star mass only about 2.01 M☉? To me this indicates neutron stars greater than this are unstable and eject mass. Just my opinion. Instead of a collapsing star blowing away all but <3 M☉maybe neutron stars >2.01 M☉ blow away mass by themselves. Its a good idea but maybe you are incorrect about there being an exponential increase in density at the center during collapse so as to form a mini black hole. In a stable gravitational structure the core density is typically only 2.5 times that of the average star density. I think you are raising important issues that should be discussed more. Q: What happens to an existing accreting neutron star when its mass exceeds 2.01 (or 3 if you prefer) M☉?

 

[Simon Bridge]

That's not quite what I was wondering, but close. If the black hole forms at an arbitrarily low mass, then presumably this would be the minimum possible mass of a black hole. Since this would likely be near the Planck scale, the decay lifetime of such a black hole would be almost infinitely brief (if, in fact, Hawking's predictions hold for this scale), too brief for it to "gobble up" any more mass. Which could perhaps lead to some interesting predictions. But I know that I'll first need to figure out whether the initial concept (exponential density gradient causing an initial black hole to form with arbitrarily low mass) is even a possibility.

Except the hawking radiation stuff assumes there is vacuum outside the even't horizon - in this case the miniature hole is surrounded by other not quite dense enough material, so this reasoning won't hold.
You would basically have to do the maths. It may be something more like the supermassive black holes in a galactic core but on a very small scale.

 

[Chronos]

An obvious problem with current theories for stellar mass black holes is the mass gap between neutron stars and stellar mass black holes. The former has an upper observational limit of around 2 solar, and the latter has a lower observational limit of about 5 solar. This is very interesting. While the mass of both objects are difficult to quantify save in binary systems, it clearly suggests physics beyond our current understanding.

 

[sevenperforce]

Except the hawking radiation stuff assumes there is vacuum outside the even't horizon - in this case the miniature hole is surrounded by other not quite dense enough material, so this reasoning won't hold.

AFAIK, there are a few different/competing explanations for how Hawking radiation is produced. One possibility is that virtual particles arising from the quantum vacuum are boosted into existence by the gravitational field of the black hole and escape; another is that mass-energy inside the black hole can tunnel through the event horizon and escape in that way. The latter explanation would seem slightly more promising, as quantum tunneling tends to fit more neatly with the whole concept of the black hole becoming more energetic as its event horizon shrinks, but then again that's just a conceptual/qualitative intuition.

Applying conceptual intuition to these things is rarely fruitful, but it's worth a short.

You would basically have to do the maths.

Indeed.

Let's suppose, for the sake of argument, that it is possible for a strongly self-gravitating object to satisfy the condition R_S(r₁)/r₁ > RS(r₂)/r₂ with r₁ < r₂. In other words, regardless of how space is being curved (because that's going to mess with volume/density/etc.), the mass contained within r₁ is "closer" to satisfying the conditions for a black hole than the mass contained within r₂ even though r₂ > r₁. Let's further suppose that this condition is satisfied for all r₁ < r₂, so that R_S(r)/r increases monotonically as r decreases. If that is the case, then during a collapse a black hole will form at the center with the currently-unknown minimum possible mass for a black hole.

What might the maths look like? Well, such a black hole may be totally outside of classical behavior and bounds, but if it does fit, then we can put some constraints on it. Its decay lifetime and Schwarzschild radius need to be greater than the Planck time and Planck length, respectively, and ideally its mass would be lower than the Planck mass.

If Hawking's equations are applicable, then the power output of a black hole is going to go to infinity as the mass decreases, so at some point the remaining mass-energy is going to become too low to sustain the required power output within its remaining lifetime. That's where I think we can tentatively put the minimum mass of a black hole. As the black hole shrinks, its temperature will also go to infinity, requiring that the peak wavelength of its blackbody spectrum goes to zero. Since the energy of a particle is defined by its wavelength, this offers a possible way to relate the energy of the discrete Hawking radiation to the total mass-energy of the black hole.

At the extreme, the micro-black-hole emits two identical Hawking radiation particles in opposite directions (so as to satisfy conservation of momentum), each containing half its mass-energy. To satisfy the blackbody spectrum, these must be emitted at the peak wavelength λ = b/T_H, where b is Wien's displacement constant and T_H is the Hawking temperature. However, λ = b/T_H corresponds to the wavelength as seen by an observer at infinity, so we have to account for redshift. Each particle is effectively being emitted "from" the event horizon of the black hole but is leaving behind a mass distribution half the size of the black hole, and since the Schwarzschild radius is proportional to mass, the Schwarzschild metric dictates that the Hawking radiation at infinity will have a wavelength λ_f = sqrt(2)*λ_i, where λ_i is the emitted wavelength. Thus the wavelength of our particles is λ = b/sqrt(2)*T_H, and using the equation for Hawking radiation temperature (T_H = ħc³/8πGMk_B) allows us to put this in terms of the mass:

λ = 8πGMk_Bb/sqrt(2)ħc³

If each of these particles contains half the mass-energy of the black hole (which we can find quite easily with E = Mc²), then we can use E = hc/λ to find that λ = 2h/Mc, and we can combine this with our former equation then solve for M:

M = ħc*2^1/4/sqrt(2*Gbk_B)

A little math, and this evaluates to 1.627e-8 kg. A black hole with mass 1.627e-8 kg, about 3/4 of the Planck mass, would have a Schwarzschild radius of 1.5 Planck lengths and an evaporation lifetime of around 6700 Planck times. 6700 Planck times is probably not enough time for this micro black hole to "suck up" anything so it is going to evaporate into those two ultra-high-energy particles before it can grow. Each of those particles has enough energy that anything they come into contact with will probably collapse into a micro black hole as well, repeating the process in a chain reaction.

I don't want to speculate too far, but if this chain reaction were to progress until the entire 5+ stellar masses had been "consumed", then perhaps collective gravitation of the whole micro-black-hole cloud would contain them, time-dilate them, and redshift their Hawking radiation so as to match the classical characteristics of a typical stellar-mass black hole.

 

[sevenperforce]

An obvious problem with current theories for stellar mass black holes is the mass gap between neutron stars and stellar mass black holes. The former has an upper observational limit of around 2 solar, and the latter has a lower observational limit of about 5 solar. This is very interesting. While the mass of both objects are difficult to quantify save in binary systems, it clearly suggests physics beyond our current understanding.

Isn't the maximum observed neutron star mass only about 2.01 M☉? To me this indicates neutron stars greater than this are unstable and eject mass.

That's an intriguing possibility, but I'm not quite sure what mechanism could exist to allow a neutron star to eject mass. Perhaps the surface radiation or magnetic field of a heavy neutron star is energetic enough that any infalling matter above that threshold gets blasted into unstable particles which rapidly decay and escape? One problem would be that unlike black holes, neutron stars can have a wide range of temperatures and magnetic fields depending on their age, so this wouldn't necessarily hold true for a single mass limit. Also, this would suggest that infalling matter would produce spectacularly brilliant flashes of light which would probably be visible if you knew where to look for them.

Instead of a collapsing star blowing away all but <3 M_☉ maybe neutron stars >2.01 M_☉ blow away mass by themselves. What happens to an existing accreting neutron star when its mass exceeds 2.01 (or 3 if you prefer) M_☉?

That's a really good question. It should be noted that a black hole is actually the minimum potential energy configuration for a given amount of mass; a neutron star has dramatically more potential energy than a black hole of equivalent mass. That potential energy is bound up in the strong force interaction between the quarks that make up the neutron star's baryons as well as the gravitational potential energy of the object. It may be that once a neutron star becomes so heavy that the pressure at the center exceeds the limits of neutron degeneracy pressure, the release of potential energy exceeds the gravitational binding energy of the body and it simply blows itself apart. Such an event might be a candidate for the creation of certain gamma ray bursts, according to at least one source I found.

Neutrons are composed of two down quarks and one up quark, which have bare masses of 4.8 MeV/c² and 2.3 MeV/c², respectively. Neutrons, however, have a mass of about 940 MeV/c², meaning that fully 99.2% of the neutron's mass is strong-interaction-bound. Lattimer's speculative neutron star relativistic equations of state suggest that the relativistic gravitational binding energy fraction (ratio of binding energy to total mass-energy) of a neutron star can be found as 0.6β/(1 - β/2), where β = R_S/2R = GM/Rc². So for a neutron star of 2.01 M_☉ and a radius of roughly 11 km, β = 0.27 and so the star's gravitational binding energy is about 18.7% of its total mass-energy. Thus, if the initial collapse rapidly reduced at least 19% of the neutron star's neutrons to quark-degenerate matter, the release of strong-interaction-bound energy would exceed the binding energy of the neutron star and the whole thing would blow itself to bits.

I'm tempted to try and figure out how large the core would need to be in order to be 19% of the neutron star's mass, to see what it predicts for the overall size of the neutron star, but Euclidean geometry breaks down in the highly-curved space of strongly self-gravitating objects.

 

[Bernie G]

An obvious problem with current theories for stellar mass black holes is the mass gap between neutron stars and stellar mass black holes. The former has an upper observational limit of around 2 solar, and the latter has a lower observational limit of about 5 solar.

If neutron stars are limited by some process to about 2 M☉its probably due an internal mechanism in the star instead of an external mechanism that prevents further accretion.

 

[Bernie G]

... I'm not quite sure what mechanism could exist to allow a neutron star to eject mass.

Suggestion: Some neutrons collapse. As you indicated a neutron contains a few quarks with relatively small mass. Collider experiments show that when a nucleus collapses what is produced is from 1% quark type matter and 99% energy to 10% quark type matter and 90% energy. What if a small percentage of core neutrons collapsed into this stuff: some quark type matter and mostly radiation. Normally we think of “photons” as weightless but here there would briefly be zillions of tons of photons with a pressure of (rho)(c^2)/3. This explosive pressure would temporarily heat and support the neutron star, or blast out of the star if it had a channel. A magnetic solenoid is an easy way out. I'm suggesting that instead of forming outside of the star, the relativistic jets from neutron stars are radiation with a little quark matter blasting directly out of the core via the magnetic poles.

 

[phinds]

AFAIK, there are a few different/competing explanations for how Hawking radiation is produced. One possibility is that virtual particles arising from the quantum vacuum are boosted into existence by the gravitational field of the black hole and escape

No, that is not correct, even though it is what is always presented in pop-science. Hawking himself said that the whole "virtual particle" explanation is not at all what's really happening, it's just the only way he could think of to describe to laymen in English what really can only be described in math.

 

[sevenperforce]

Suggestion: Some neutrons collapse. As you indicated a neutron contains a few quarks with relatively small mass. Collider experiments show that when a nucleus collapses what is produced is from 1% quark type matter and 99% energy to 10% quark type matter and 90% energy. What if a small percentage of core neutrons collapsed into this stuff: some quark type matter and mostly radiation. Normally we think of “photons” as weightless but here there would briefly be zillions of tons of photons with a pressure of (rho)(c^2)/3. This explosive pressure would temporarily heat and support the neutron star, or blast out of the star if it had a channel. A magnetic solenoid is an easy way out. I'm suggesting that instead of forming outside of the star, the relativistic jets from neutron stars are radiation with a little quark matter blasting directly out of the core via the magnetic poles.

We already watch neutron stars pretty closely because most of them are pulsars, producing nice bright beams from their poles. If large neutron stars were occasionally flinging mass out of their poles we would have definitely noticed.

No, that is not correct, even though it is what is always presented in pop-science. Hawking himself said that the whole "virtual particle" explanation is not at all what's really happening, it's just the only way he could think of to describe to laymen in English what really can only be described in math.

Ah, I see; thanks. I know that the math behind Hawking radiation basically arises from the necessary temperature of the space just outside of the event horizon as measured by an infalling observer; I wasn't sure whether the virtual particle explanation was viable or just a convenient fiction. Does a notion of quantum tunneling fit more closely with what Hawking has said recently about super translations and the chaotic apparent horizon?

If the apparent horizon is actually a sparsely-populated spherical of black hole quanta in a probability distribution across the surface of the would-be horizon, then this would seem promising.

 

[Bernie G]

We already watch neutron stars pretty closely because most of them are pulsars, producing nice bright beams from their poles. If large neutron stars were occasionally flinging mass out of their poles we would have definitely noticed.

I'm suggesting the continuous nice bright beams and jets are the mechanism for mass ejection.

 

[sevenperforce]

I'm suggesting the continuous nice bright beams and jets are the mechanism for mass ejection.

Mechanism for or result of?

The polar jets from pulsars are categorized by three energy sources: loss of angular momentum, collapse of accretion disk, and magnetic field decay. It's possible that core neutrons breaking down into quarks+energy could offer a fourth energy source, perhaps to explain certain gamma-ray pulsars. If this is the case, then we would predict that all gamma-ray pulsars are at or near the same limiting mass. So far, we have identified about a hundred gamma-ray pulsars out of 1800 total known pulsars; I don't know if any of those are in binaries or offer any other way of determining their mass.

I'm still a little skeptical about a mechanism for getting something out of the core of a neutron star, solenoid or no solenoid. Existing models for the creation of pulsar jets all have to do with energy generation at the poles themselves, rather than deep within the most compact macroscopic objects in the universe.

 

[Bernie G]

I'm still a little skeptical about a mechanism for getting something out of the core of a neutron star, solenoid or no solenoid. Existing models for the creation of pulsar jets all have to do with energy generation at the poles themselves, rather than deep within the most compact macroscopic objects in the universe.

No problem. My best guess is: The basic process is matter collapsing into radiation at the core, and the jets are like a cylindrical flashlight beam exiting the rotation axis. If the magnetic field is not aligned it messes things up. A hard neutron crust or whirl pools are not a problem, a hard crust gets blown aside.

 

[Bernie G]

Rephrasing: I think core neutrons collapse into radiation and quark matter, and big jets are this stuff exiting the star at the rotation axis and/or magnetic poles.

 

[nikkkom]

An obvious problem with current theories for stellar mass black holes is the mass gap between neutron stars and stellar mass black holes. The former has an upper observational limit of around 2 solar, and the latter has a lower observational limit of about 5 solar. This is very interesting. While the mass of both objects are difficult to quantify save in binary systems, it clearly suggests physics beyond our current understanding.

I don't see the problem.
Core collapse happens in a quite small volume of the star - the core is something like 10000 km across. Core collapse creates a "cavity", a region of low density, when it collapses to a tiny object some 10 km across.

If core collapse creates a NS, almost all remaining matter gets expelled in the supernova: newborn NS is emitting insane amounts of gamma rays, something like 10 billion solar luminosities from each square meter of its surface. Total power is on the order of 10^40 watts. Matter would have very hard time falling onto such object, and one which does will likely immediately undergo thermonuclear fusion.

If core collapse creates a BH, the situation is very different. It emits no such "wind". It gladly "eats" any infalling matter and radiation. It may well be the case that in typical core collapses to BHs, several solar masses worth of matter fall into the hole immediately after its creation.

 

[nikkkom]

Neutrons are composed of two down quarks and one up quark, which have bare masses of 4.8 MeV/c² and 2.3 MeV/c², respectively. Neutrons, however, have a mass of about 940 MeV/c², meaning that fully 99.2% of the neutron's mass is strong-interaction-bound.

Not really, quarks inside hadrons move at relativistic velocities, thus a lot of this energy is their kinetic energy.

Lattimer's speculative neutron star relativistic equations of state suggest that the relativistic gravitational binding energy fraction (ratio of binding energy to total mass-energy) of a neutron star can be found as 0.6β/(1 - β/2), where β = R_S/2R = GM/Rc². So for a neutron star of 2.01 M_☉ and a radius of roughly 11 km, β = 0.27 and so the star's gravitational binding energy is about 18.7% of its total mass-energy. Thus, if the initial collapse rapidly reduced at least 19% of the neutron star's neutrons to quark-degenerate matter, the release of strong-interaction-bound energy would exceed the binding energy of the neutron star and the whole thing would blow itself to bits.

"Blowing itself to bits" would require quark-degenerate matter to revert back to neutrons, which takes the same energy. We have a problem with this scenario...

 

[Bernie G]

Chronos said: https://www.physicsforums.com/threads/stellar-mass-black-hole-formation-sequence.852285/goto/post?id=5347932#post-5347932
An obvious problem with current theories for stellar mass black holes is the mass gap between neutron stars and stellar mass black holes. The former has an upper observational limit of around 2 solar, and the latter has a lower observational limit of about 5 solar.

I don't see the problem.

It could well be an unanswered problem. If a large number of neutron stars have been around and accreting for billions of years why don't we see any greater than about 2☉?

 

[rootone]

Beyond that mass a neutron star can collapse further to become a black hole, (or the hypothetical quark star as an intermediate).
Neutron stars are small in size but observable, but an even smaller denser object may not be until it acquires additional mass through accretion,
At some point (around 5☉), it could become detectable again through indirect observation such as gravitational lensing.
So objects in the 2☉ to 5☉ may well exist, just we have no way to see them with presently understood physics and technology

 

[Bernie G]

So objects in the 2☉ to 5☉ may well exist, just we have no way to see them with presently understood physics and technology

Maybe but my guess at this time is probably not. There should be millions of old neutron stars in our galaxy alone. If some of them became 2☉ - 3☉black holes we probably would have seen some of them accreting.

 

[sevenperforce]

If core collapse creates a NS, almost all remaining matter gets expelled in the supernova: newborn NS is emitting insane amounts of gamma rays, something like 10 billion solar luminosities from each square meter of its surface. Total power is on the order of 10^40 watts. Matter would have very hard time falling onto such object, and one which does will likely immediately undergo thermonuclear fusion.

If core collapse creates a BH, the situation is very different. It emits no such "wind". It gladly "eats" any infalling matter and radiation. It may well be the case that in typical core collapses to BHs, several solar masses worth of matter fall into the hole immediately after its creation.

Yes, the formation of a neutron star is a far more energetic event (from an external POV) than the creation of a black hole.

This explanation does well to demonstrate why we would expect to see many black holes and many neutron stars but nothing in between...yet it fails to explain why something couldn't start as a neutron star and then accrete to greater than two solar masses.

Quarks inside hadrons move at relativistic velocities, thus a lot of this energy is their kinetic energy.

"Blowing itself to bits" would require quark-degenerate matter to revert back to neutrons, which takes the same energy.

Ah, makes sense. Looks like that explanation fails.

Beyond that mass a neutron star can collapse further to become a black hole, (or the hypothetical quark star as an intermediate).
Neutron stars are small in size but observable, but an even smaller denser object may not be until it acquires additional mass through accretion,
At some point (around 5☉), it could become detectable again through indirect observation such as gravitational lensing.
So objects in the 2☉ to 5☉ may well exist, just we have no way to see them with presently understood physics and technology

The trouble is that the very place where we'd be able to detect such objects (specifically, binary systems) are also the place which are most likely to allow accretion beyond two solar masses.

 

[PeterDonis]

the sequence of events in a massive star's collapse aren't very well-understood yet. It's not known whether the iron core collapses first to an unstable neutron star and then into a black hole, or if the collapse goes directly to a black hole

I'm not sure these are really different alternatives. But I agree that the details of the collapse are not well understood. The main reason, I believe, is that the equation of state of matter at neutron star and higher densities is not well understood, so we don't know how the pressure inside the collapsing matter behaves as the density increases. We can set some reasonable bounds on the behavior, but that still leaves a fairly broad range of possibilities--for example, AFAIK the maximum mass limit for neutron stars (analogous to the Chandrasekhar limit for white dwarfs) is only known to within a factor of 2 or so (it lies somewhere between 1.5 and 3 solar masses).

there's a gap between the largest theoretically possible neutron stars (~3 M☉) and the smallest observed stellar-mass black holes (~5 M☉);

Yes, and it is certainly an interesting question why this gap occurs. But your proposed explanations appear to me to be based on a misunderstanding of how black hole formation works.

in the collapse of a massive star where neutron degeneracy pressure is insufficient to arrest collapse and a black hole forms, it doesn't seem that there would be anything preventing the density at the center from increasing without bound

There isn't. According to classical GR, the density at the center of the collapsing matter does increase without bound, and the proper volume occupied by the collapsing matter decreases to zero. When it reaches zero, a singularity is formed.

What physicists think this is really telling us is that classical GR must break down at high enough densities (the usual heuristic is at densities comparable to one Planck mass per Planck volume), and quantum gravity effects must become significant. But we don't know what those quantum gravity effects are.

However, this issue, at least on what appears to be the current mainstream understanding, should not be important as far as understanding the formation of a black hole from the collapse of an object of a few solar masses or larger. For objects that large, the formation of an event horizon happens well before the density at the center of the object increases to scales where classical GR might become problematic. See further comments below.

all the discussions/models/depictions I've ever seen of stellar-mass black hole formation feature a macroscopic event horizon, with the Schwarzschild radius and the surface radius growing closer (either because the Schwarzschild radius is increasing due to added mass or because the surface radius is shrinking due to increased density) until they finally meet and the black hole is formed.

This is not a correct understanding of what happens. See my post in the other thread where this topic is being discussed:

"Gravitational Compression in Neutron Stars"

I would strongly recommend studying the idealized Oppenheimer-Snyder model of spherically symmetric collapse. A brief description is given here:

http://grwiki.physics.ncsu.edu/wiki/Oppenheimer-Snyder_Collapse

This model assumes zero pressure, so it is highly unrealistic; but the presence of nonzero pressure should, if anything, cause the collapse to be slower. And the statement I made above, about the central density vs. the time of formation of the event horizon, is based on this idealized zero pressure model.

there are a few different/competing explanations for how Hawking radiation is produced. One possibility is that virtual particles arising from the quantum vacuum are boosted into existence by the gravitational field of the black hole and escape; another is that mass-energy inside the black hole can tunnel through the event horizon and escape in that way.

These aren't different/competing explanations. They're different heuristic ways of trying to describe in ordinary language what the math actually says. The main lesson to be learned here is that ordinary language is not a good tool for describing these things.

 

[sevenperforce]

In the collapse of a massive star where neutron degeneracy pressure is insufficient to arrest collapse and a black hole forms, it doesn't seem that there would be anything preventing the density at the center from increasing without bound.

There isn't. According to classical GR, the density at the center of the collapsing matter does increase without bound, and the proper volume occupied by the collapsing matter decreases to zero. When it reaches zero, a singularity is formed.

However, this issue, at least on what appears to be the current mainstream understanding, should not be important as far as understanding the formation of a black hole from the collapse of an object of a few solar masses or larger. For objects that large, the formation of an event horizon happens well before the density at the center of the object increases to scales where classical GR might become problematic.

That's the point I'm questioning. Of course, my understanding of these metrics is only very slightly beyond an undergraduate level so I confess that I could be completely off-base.

This is not a correct understanding of what happens. See my post in the other thread where this topic is being discussed:

"Gravitational Compression in Neutron Stars"

I would strongly recommend studying the idealized Oppenheimer-Snyder model of spherically symmetric collapse. A brief description is given here:

http://grwiki.physics.ncsu.edu/wiki/Oppenheimer-Snyder_Collapse

This model assumes zero pressure, so it is highly unrealistic; but the presence of nonzero pressure should, if anything, cause the collapse to be slower. And the statement I made above, about the central density vs. the time of formation of the event horizon, is based on this idealized zero pressure model.

This, again, may be completely off-base, but what about radiation pressure? Wouldn't inward radiation pressure be highly blueshifted while outgoing radiation pressure was highly redshifted, causing the pressure gradient to greatly favor an exponential increase in central density?

 

[PeterDonis]

That's the point I'm questioning.

What point? That whatever quantum gravity effects become important at the singularity won't affect the formation of the horizon?

There are some speculations along the lines that quantum gravity effects can affect the formation of the horizon--but the speculations are that quantum gravity effects are strong enough to prevent a horizon from ever forming at all. These speculations have not fared well under criticism, so I would rate them as highly unlikely to be correct. But however that may be, they are only speculations; there is no firm theory at this point. The only firm theory we have is general relativity, and what I have been saying is what GR says.

what about radiation pressure? Wouldn't inward radiation pressure be highly blueshifted while outgoing radiation pressure was highly redshifted, causing the pressure gradient to greatly favor an exponential increase in central density?

No. You are still thinking about things from a static viewpoint--that's the viewpoint that says ingoing radiation gets blueshifted and outgoing radiation gets redshifted. But a collapsing configuration is not static, and your intuitions about a static viewpoint don't work.

Just as one example: a static observer very close to a static black hole's horizon sees incoming radiation highly blueshifted; but an infalling observer falling past that static observer sees incoming radiation redshifted, not blueshifted. This is still a static situation overall, so it doesn't fully capture what is going on in a collapsing star; but at least it illustrates that static intuitions can't be applied to infalling objects.

 

[sevenperforce]

That's the point I'm questioning.

What point? That whatever quantum gravity effects become important at the singularity won't affect the formation of the horizon?

That the horizon will invariably form (and by "form" I mean "have enough matter fall into it to consist of a black hole even if all the matter still-outside of it suddenly disappeared") at a size much larger than quantum gravitational effects could matter.

No. You are still thinking about things from a static viewpoint--that's the viewpoint that says ingoing radiation gets blueshifted and outgoing radiation gets redshifted. But a collapsing configuration is not static, and your intuitions about a static viewpoint don't work.

Just as one example: a static observer very close to a static black hole's horizon sees incoming radiation highly blueshifted; but an infalling observer falling past that static observer sees incoming radiation redshifted, not blueshifted. This is still a static situation overall, so it doesn't fully capture what is going on in a collapsing star; but at least it illustrates that static intuitions can't be applied to infalling objects.

Ah, I see. I think I'm getting a better picture of things.

Would it perhaps be accurate to say something like "the solutions to the Oppenheimer-Snyder model of collapse prevent a discontinuous event horizon; all the components of a gravitationally collapsing object will perceive the event horizon swallowing them up simultaneously"? Or something along those lines?

Of course, I'm not interested in the event horizon for its own sake; I'm really wanting to know at what point the conditions for the creation of Hawking radiation are met. Because Hawking radiation can't be teleological. I guess this is a restatement of the firewall problem...

 

[PeterDonis]

That the horizon will invariably form (and by "form" I mean "have enough matter fall into it to consist of a black hole even if all the matter still-outside of it suddenly disappeared") at a size much larger than quantum gravitational effects could matter.

You mean, at a density much smaller than that at which quantum gravity effects could matter. According to our best current understanding of quantum gravity effects, yes, this is true; the density at $r = 0$ when the horizon forms there is much, much smaller than the Planck density. To put it another way, if we imagine an observer sitting at $r = 0$ and watching the density increase with time, the event on his worldline at which, if he emits a light signal, it will be just at the event horizon, has to be significantly earlier, according to his own clock, than the event at which the density reaches the Planck density (which will be just before the entire collapsing object reaches zero volume and the singularity forms).

Would it perhaps be accurate to say something like "the solutions to the Oppenheimer-Snyder model of collapse prevent a discontinuous event horizon

I'm not sure what you mean by a "discontinuous event horizon". The event horizon, considered as a boundary between two regions of spacetime, has to be a continuous 3-surface between two 4-volumes; it's not possible for it to be disconnected. That isn't a feature just of the O-S model; it's a requirement for any event horizon, no matter how it is formed. (More precisely, it's a requirement for any event horizon formed by the collapse of a single isolated system; there can be multiple such systems in the universe that never come together. But we're just talking about a single isolated system here.)

all the components of a gravitationally collapsing object will perceive the event horizon swallowing them up simultaneously"?

No. This can't be true, because the event horizon is a null surface, and different parts of the object cross it at different events, so those events must be null separated. Null separated events cannot be simultaneous in any coordinate chart; only spacelike separated events can be simultaneous (and only if you choose the appropriate coordinate chart).

(Btw, if you think about it, you will see that this is part of the reason why the statement I made about the density at $r = 0$ must be true. The other part is that, once collapsing matter crosses the horizon, it's on a timelike worldline, so the event of it crossing the horizon and the event of it reaching the singularity are timelike separated.)

I'm really wanting to know at what point the conditions for the creation of Hawking radiation are met.

I think the current best guess for that is that Hawking radiation is generated at outgoing trapped surfaces--these are 2-spheres at which radially outgoing light stays at the same radius instead of moving outward. An event horizon does not have to be always associated with an outgoing trapped surface; in the O-S model, an outgoing trapped surface only forms when the surface of the collapsing object falls below the event horizon. From then on, assuming nothing else falls in, the event horizon and the outgoing trapped surface coincide. But before that, there is no outgoing trapped surface anywhere, at least as I understand the model.

 

[sevenperforce]

You mean, at a density much smaller than that at which quantum gravity effects could matter. According to our best current understanding of quantum gravity effects, yes, this is true; the density at $r = 0$ when the horizon forms there is much, much smaller than the Planck density. To put it another way, if we imagine an observer sitting at $r = 0$ and watching the density increase with time, the event on his worldline at which, if he emits a light signal, it will be just at the event horizon, has to be significantly earlier, according to his own clock, than the event at which the density reaches the Planck density (which will be just before the entire collapsing object reaches zero volume and the singularity forms).

So there cannot be a case in which core collapse takes place with a high enough central density that the 9/8 condition would be met at a very very low (subatomic) radius?

I think the current best guess for that is that Hawking radiation is generated at outgoing trapped surfaces--these are 2-spheres at which radially outgoing light stays at the same radius instead of moving outward. An event horizon does not have to be always associated with an outgoing trapped surface; in the O-S model, an outgoing trapped surface only forms when the surface of the collapsing object falls below the event horizon. From then on, assuming nothing else falls in, the event horizon and the outgoing trapped surface coincide. But before that, there is no outgoing trapped surface anywhere, at least as I understand the model.

I highlighted the point I was particularly interested in, because that's really where the crux of my question (and this whole thread) lies.

If Hawking radiation is generated from an outgoing trapped surface (OTS) as soon as an OTS forms, and an OTS forms as soon as an object falls within its own event horizon (presumably in its own reference frame), then what is the earliest point in the collapse of a star that Hawking radiation can be generated?

In particular, if the surface of the core has fallen within the core's own event horizon but the rest of the star is only beginning to collapse, can Hawking radiation be generated from an OTS at the core event horizon before the rest of the star has collapsed very much?

 

[PeterDonis]

So there cannot be a case in which core collapse takes place with a high enough central density that the 9/8 condition would be met at a very very low (subatomic) radius?

The 9/8 condition applies to a static object. A collapsing object is not static. You need to stop trying to apply static reasoning to a non-static situation.

what is the earliest point in the collapse of a star that Hawking radiation can be generated?

When the outer surface of the star reaches the event horizon. In other words, when all of the matter that is going to collapse into the forming black hole has reached the event horizon. There is no OTS until then.

if the surface of the core has fallen within the core's own event horizon but the rest of the star is only beginning to collapse, can Hawking radiation be generated from an OTS at the core event horizon before the rest of the star has collapsed very much?

Basically, it looks like you're envisioning a scenario where we have a core that's collapsing, then a significant region of empty space, and then the rest of the star. In that case, yes, you could view the collapse process in two separate stages: first the core collapses to a black hole, with a region of empty space around it, and Hawking radiation begins when all of the matter of the core has reached the horizon of this initial hole.

Then, some time later, an additional shell of matter falls into the hole. This is a different kind of process and a different kind of spacetime model than we have been talking about, and we should defer discussion of it until we've got the original model clear.

The question you appear to be asking is, how small could the core be in this type of scenario? And the answer is, unless it is larger than the maximum mass limit for a neutron star (1.5 to 3 solar masses), it won't collapse to a black hole. And Hawking radiation for a hole of that size is completely negligible. So Hawking radiation can't have a significant effect on such a collapse. If the object is below the maximum mass limit for a neutron star, it will form a neutron star (it could even form a white dwarf if it is below the maximum mass limit for that), and no Hawking radiation will be generated at all.

 

[sevenperforce]

The 9/8 condition applies to a static object. A collapsing object is not static.

Ah, yes, my bad. Forget I put it in those terms.

Basically, it looks like you're envisioning a scenario where we have a core that's collapsing, then a significant region of empty space, and then the rest of the star. In that case, yes, you could view the collapse process in two separate stages: first the core collapses to a black hole, with a region of empty space around it, and Hawking radiation begins when all of the matter of the core has reached the horizon of this initial hole.

Then, some time later, an additional shell of matter falls into the hole. This is a different kind of process and a different kind of spacetime model than we have been talking about, and we should defer discussion of it until we've got the original model clear.

Well, that's the process I've been aiming for, more or less. You don't have to have a region of empty space per se; it can simply be a region of lower density. All you need to meet this condition is that once neutron degeneracy pressure breaks down and the collapse begins, the core collapses more rapidly than the outer layers.

Is there some aspect of the Oppenheimer-Snyder model which would prevent the core collapse from "outrunning" the collapse of the rest of the star?

 

[PeterDonis]

You don't have to have a region of empty space per se; it can simply be a region of lower density.

I suggest taking some time to work out the math. It isn't as simple as you are assuming it is.

(As I note below, there is no known analytical solution to the differential equations for the case of nonzero pressure; but you can still look at the equations themselves and work out some qualitative features.)

Is there some aspect of the Oppenheimer-Snyder model which would prevent the core collapse from "outrunning" the collapse of the rest of the star?

Yes; but that aspect is that the O-S model assumes zero pressure, so it eliminates the only possible thing that could slow down the collapse of any part of the star.

AFAIK there is no analytical solution for the case of nonzero pressure, so the only way to study that case would be to do so numerically. I know such numerical simulations have been done, but unfortunately I'm not familiar enough with the numerical relativity literature to be able to point to specific results.