# A few questions about black holes

## Main Question or Discussion Point

A stellar mass black hole is around 10 solar masses, and some stars can be much massive than this. And doesn't a black hole have more gravity than anything in the universe? Does more mass necessarily mean more gravity? If a star is more massive, how can a black hole siphon off material from it? Does this just have to do with size? b/c I remember seeing white dwarfs siphoning material from their giant companion stars. I know in a binary system stars orbit each other, but what happens after one becomes a black hole? do they continue to orbit each other? Also, what causes some black holes to rotate and others not to? Why do black holes that do not rotate have double the diameter of black holes at the theoretical maximum?

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phinds
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2019 Award
A stellar mass black hole is around 10 solar masses, and some stars can be much massive than this.
yes

And doesn't a black hole have more gravity than anything in the universe?
no, it has more DENSITY than anything else in the universe other than a bigger black hole. A really large star can easily have more mass than a small black hole.

Does more mass necessarily mean more gravity?
yes

If a star is more massive, how can a black hole siphon off material from it?
It would likely just fall through the center of the more massive star and eat it much more quickly than just siphoning off material as it oscillated back and forth through the center.

Does this just have to do with size?
Does what have just to do with size?

b/c I remember seeing white dwarfs siphoning material from their giant companion stars. I know in a binary system stars orbit each other, but what happens after one becomes a black hole? do they continue to orbit each other?
don't know

Also, what causes some black holes to rotate and others not to?
depends on star rotation pre-black-hole formation rotation and angle of infalling mass

Why do black holes that do not rotate have double the diameter of black holes at the theoretical maximum?
Huh? There is no maximum for a black hole size that I've ever heard of

Does what have just to do with size?
Huh? There is no maximum for a black hole size that I've ever heard of
Sorry, I should of made that clearer. Does the ability of black holes/stars to siphon off material from other stars just have to do with size? And I didn't mean a maximum size. A black hole has a maximum rate at which it can spin, and for some reason, a black hole that is not rotating has double the diameter of a black hole spinning at the theoretical maximum. (Info from Astronomy magazine) And if a star can be much more massive than a black hole, and therefore have more gravity, can a black hole orbit a star as it slowly takes its material?

phinds
Gold Member
2019 Award
Sorry, I should of made that clearer. Does the ability of black holes/stars to siphon off material from other stars just have to do with size? And I didn't mean a maximum size. A black hole has a maximum rate at which it can spin, and for some reason, a black hole that is not rotating has double the diameter of a black hole spinning at the theoretical maximum. (Info from Astronomy magazine)
I'm not aware of any of that. I don't mean it's not true, I'm just ignorant of it. The "diameter" of a BH is the Schwartchild radius doubled and I was not aware that that's dependant on rotation, I thought it was just mass.

And if a star can be much more massive than a black hole, and therefore have more gravity, can a black hole orbit a star as it slowly takes its material?
I guess it could somehow happen with a binary system, but I'm out of my depth on that.

A stellar mass black hole is around 10 solar masses, and some stars can be much massive than this. And doesn't a black hole have more gravity than anything in the universe?
No

Does more mass necessarily mean more gravity?
If you have a 10 solar mass black hole, it will end up with the same amount of gravity as a 10 solar mass non-black hole, and it will suck up material in the same way that another 10 solar mass star will.

The difference is what happens when the material hits the object. A 10 solar mass star is large so the gravity at the surface is less. A 10 solar mass black hole is smaller so the material that gets sucked in will get accelerated to much higher velocities and generate X-rays.

stevebd1
Gold Member
A black hole has a maximum rate at which it can spin, and for some reason, a black hole that is not rotating has double the diameter of a black hole spinning at the theoretical maximum. (Info from Astronomy magazine)
A black hole technically has two radii, the coordinate radius and the reduced circumference. For a static black hole, the two are equivalent, for a rotating black hole, the reduced circumference takes into account frame dragging. So the equation for the coordinate radii for the (inner and outer) event horizons is-

$$r_\pm=M\pm\sqrt{M^2-a^2}$$
where M=Gm/c2 and a=J/mc

and the equation for reduced circumference is-

$$R=\frac{\Sigma}{\rho}\,\sin\theta$$

where R is the reduced circumference and

$$\rho=\sqrt{r^2+a^2 \cos^2\theta}$$
$$\Sigma=\sqrt{(r^2+a^2)^2-a^2\Delta \sin^2\theta}$$
$$\Delta= r^{2}+a^{2}-2Mr$$

While the coordinate radius is smaller, the reduced circumference is the actual path that light will take, which for a rotating black hole, is curved due to frame dragging and therefore longer than the coordinate radius. If you plug r+ (or even r- for that matter) into the reduced circumference equation to obtain R+ (or R-), you'll see that, in the equatorial plane at least, the reduced circumference at the event horizon matches the schwarzschild radius (2M) for an equivalent static black hole of the same mass.

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I'm torn whether to start a new thread or to add to this one, but I wanted to raise a question:

I've always had an issue with the term "infinite density" being applied to a black hole. Obviously black holes have a finite mass and a finite gravity. The "infinite" part comes from the observation that a black hole has zero volume giving us a density of x/0 where x is the black hole mass.
All well and good, but isn't it actually the case that black holes only have apparent zero volume due to spacetime curvature making its internal volume hidden rather than there being actual zero volume?

To wit: If some omnipotent being (regardless of impossibility) were to pass within the event horizon of the black hole, reach to its centre of mass and scoop up a handful and return, wouldn't this material simply be neutron degenerate matter?

Or to take a more possible example: if the black hole loses sufficient mass via evaporation/hawking radiation etc, could the original stellar remnant re-emerge as the total (but not apparent) density of the black hole drops to the point where spacetime can resume its "normal" curvature?

The conventional wisdom seems to suggest that once you have a black hole, you always have a black hole until it evaporates and spontaneous disappears (leaving some kind of topological defect?), even if its mass drops below the point where the original black hole formed - spacetime being plastic rather than elastic. I was wondering why this is - does spacetime have an elastic limit, and if so, what equation or experiment has determined this?

Perhaps I simply take the "rubber sheet" analogue too far, but I can't shake the idea that there is an internal volume to a black hole - constructed out of spacetime hidden by the event horizon.

bcrowell
Staff Emeritus
Gold Member
All well and good, but isn't it actually the case that black holes only have apparent zero volume due to spacetime curvature making its internal volume hidden rather than there being actual zero volume?
No, general relativity describes 100% of the volume inside a black hole's event horizon as empty space.

To wit: If some omnipotent being (regardless of impossibility) were to pass within the event horizon of the black hole, reach to its centre of mass and scoop up a handful and return, wouldn't this material simply be neutron degenerate matter?
No. Degenerate neutron matter is a different state than a black hole.

No, general relativity describes 100% of the volume inside a black hole's event horizon as empty space.
I have to say, that's not a very clear way of saying what I think you're trying to say: If there is any space (empty or otherwise) within an event horizon, then it DOES have internal volume and hence a non-infinite total density although, as stated, an apparent infinite density as observed from outside the event horizon.

No. Degenerate neutron matter is a different state than a black hole.
Then what form would it take? Neutron degenerate matter is what goes in, and there is effectively nothing more dense...

Perhaps this thought experiment would help clarify my thinking:

Imagine you had an isolated neutron star with almost exactly the mass required to collapse into a black hole. Then, a picogram of material (or whatever arbitrary mass) is added to the neutron star, causing spacetime to distort, hiding the neutron star behind an event horizon. ie a black hole forms.

I'm avoiding saying that the neutron start "collapses" into a black hole because it is not the neutron star that is collapsing, it is the surrounding spacetime.

Shortly, the neutron star would lose that added picogram in evaporation or Hawking radiation. Appreciating that time itself is very distorted in such a transaction and that the loss in density and mass (to below the Tolman-Oppenheimer-Volkoff limit) would only be apparent to a hypothetical observer inside the event horizon, could the internal/external spacetime return to its previous state, allowing the original neutron star re-emerge - assuming no further changes to the black hole occurred over an arbitrary period of time?

And if not, why not?

Drakkith
Staff Emeritus
I have to say, that's not a very clear way of saying what I think you're trying to say: If there is any space (empty or otherwise) within an event horizon, then it DOES have internal volume and hence a non-infinite total density although, as stated, an apparent infinite density as observed from outside the event horizon.
I think you are confusing the issue. The "total density" as you call it is definitely finite, that being the mass divided by the volume of the black hole. A supermassive black hole can have less density than water thanks to its extremely large volume behind the even horizon. However, if GR describes the space behind the event horizon as empty, the reason being that all matter has been crushed into the singularity in the center, then that matter no longer takes up volume and thus has infinite density.

Then what form would it take? Neutron degenerate matter is what goes in, and there is effectively nothing more dense...
There is no reason to believe this. Models show that there are several possible states of matter even more degenerate than neutron star material. For example there are some exotic types of stellar remnants proposed, such as quark stars. However, IF a singularity really exists at the center of a black hole, then your question is meaningless. Science cannot answer what a hypothetical omnipotent being would bring back since science says that nothing can come back in the first place. We can't even speculate on it. It just doesn't make any sense to ask about scooping up material that is infinitely compacted, it would take infinite force to do so.

As you can see we quickly run into nonsense in our math, since we can't work with infinities. It is precisely these infinities that give rise to the concept of a "singularity". The word actually refers to the point where we start to run into these infinities, not a physical state of something. That is, it's a mathematical singularity, the point where our model breaks down due to incomplete knowledge about the laws of nature in that environment.

Imagine you had an isolated neutron star with almost exactly the mass required to collapse into a black hole. Then, a picogram of material (or whatever arbitrary mass) is added to the neutron star, causing spacetime to distort, hiding the neutron star behind an event horizon. ie a black hole forms.

I'm avoiding saying that the neutron start "collapses" into a black hole because it is not the neutron star that is collapsing, it is the surrounding spacetime.

Shortly, the neutron star would lose that added picogram in evaporation or Hawking radiation. Appreciating that time itself is very distorted in such a transaction and that the loss in density and mass (to below the Tolman-Oppenheimer-Volkoff limit) would only be apparent to a hypothetical observer inside the event horizon, could the internal/external spacetime return to its previous state, allowing the original neutron star re-emerge - assuming no further changes to the black hole occurred over an arbitrary period of time?

And if not, why not?
Well, let's make an assumption or two. Let's assume that a neutron star that is *barely* over the mass limit to turn into a black hole doesn't collapse into another state of matter. In such a case nothing would happen to the neutron star really. The surface gravity is already so strong it might as well be a black hole in almost every aspect. Redshift would be so great that we already cannot observe anything about the neutron star. But be aware that at this time ALL stellar mass black holes and above absorb FAR more energy from the background radiation of the universe than they give off due to hawking radiation, so a realistic view is that once the neutron star absorbs just enough material to form a black hole, it stays that way for billions of years.

Now, if we assume that there ARE further states of matter, such as in a quark star, then the answer depends on the properties of this matter, such as the required mass of the star to form quark matter at it's core. It is possible that current high mass neutron stars already form quark material in their cores.

I think you are confusing the issue. The "total density" as you call it is definitely finite, that being the mass divided by the volume of the black hole. A supermassive black hole can have less density than water thanks to its extremely large volume behind the even horizon. However, if GR describes the space behind the event horizon as empty, the reason being that all matter has been crushed into the singularity in the center, then that matter no longer takes up volume and thus has infinite density.
Yes that's what I was driving at.

There is no reason to believe this. Models show that there are several possible states of matter even more degenerate than neutron star material. For example there are some exotic types of stellar remnants proposed, such as quark stars. However, IF a singularity really exists at the center of a black hole, then your question is meaningless. Science cannot answer what a hypothetical omnipotent being would bring back since science says that nothing can come back in the first place. We can't even speculate on it. It just doesn't make any sense to ask about scooping up material that is infinitely compacted, it would take infinite force to do so.
Of course, but this is skirting the thrust of the question which is: Does the matter inside a black hole change by virtue of being inside a black hole? As far as the matter is concerned, surely nothing has happened at all. For example - if you were passing into the event horizon of a black hole, you would not notice anything special happening at that moment (apart from the intense gravity and the tidal stresses that would depend on the mass of the black hole you were falling into, but that would be present from far outside the event horizon).

It comes back to my comment that when a neutron star reaches the TOV limit, it is space that collapses, not the star. Of course, there may be some process that happens once extreme masses are reached inside the black hole (ie supermassive) but assuming we have a simple stellar-mass black hole, is there any reason to believe that the matter inside will have experienced any change during the formation of the hole itself?

/edit

As you can see we quickly run into nonsense in our math, since we can't work with infinities. It is precisely these infinities that give rise to the concept of a "singularity". The word actually refers to the point where we start to run into these infinities, not a physical state of something. That is, it's a mathematical singularity, the point where our model breaks down due to incomplete knowledge about the laws of nature in that environment.
I agree with all of this of course - but it returns to my original question: Why is the mass in a black hole so often described as being of infinite density when describing it as such is both misleading and ultimately of no mathematical use?

Well, let's make an assumption or two. Let's assume that a neutron star that is *barely* over the mass limit to turn into a black hole doesn't collapse into another state of matter. In such a case nothing would happen to the neutron star really. The surface gravity is already so strong it might as well be a black hole in almost every aspect. Redshift would be so great that we already cannot observe anything about the neutron star. But be aware that at this time ALL stellar mass black holes and above absorb FAR more energy from the background radiation of the universe than they give off due to hawking radiation, so a realistic view is that once the neutron star absorbs just enough material to form a black hole, it stays that way for billions of years.

Now, if we assume that there ARE further states of matter, such as in a quark star, then the answer depends on the properties of this matter, such as the required mass of the star to form quark matter at it's core. It is possible that current high mass neutron stars already form quark material in their cores.
Again I would say that is side-stepping the question a bit - obviously there is no such thing as an "isolated" black hole or neutron star within the universe and naturally any black hole formed anywhere we can think of will gain mass for the time being due to capture of any form of mass. But I don't think it is unreasonable to ask what happens when a black hole loses mass so it is below the TOV limit (spherical chickens in a vacuum etc)? Obviously also this questions has implications in the nature of the distant future of the universe. The current thinking seems to imply that once a black hole is formed, even when its mass drops below its initial mass, it remains a black hole forever (causing the "black hole age" when heat-death approaches). I was wondering why this is?

Drakkith
Staff Emeritus
Of course, but this is skirting the thrust of the question which is: Does the matter inside a black hole change by virtue of being inside a black hole?
I would guess that no, nothing happens just because you increase the mass a tiny bit to form a black hole.

It comes back to my comment that when a neutron star reaches the TOV limit, it is space that collapses, not the star. Of course, there may be some process that happens once extreme masses are reached inside the black hole (ie supermassive) but assuming we have a simple stellar-mass black hole, is there any reason to believe that the matter inside will have experienced any change during the formation of the hole itself?
What do you mean "space collapsing"? Space does no such thing. It experiences curvature that increases as you add mass and energy into an area of space. There is no collapse, there is simply no event horizon one second, and then there is one the next second. Meaning that one second the curvature is very very high, but not quite high enough, and then you add mass/energy and the curvature increases to the point where there are no paths in spacetime that will lead out of a black hole. It is a smooth curve from an area with no mass, to an area with black hole mass and density.
I agree with all of this of course - but it returns to my original question: Why is the mass in a black hole so often described as being of infinite density when describing it as such is both misleading and ultimately of no mathematical use?
Because that's what our math tells us and we lack sufficient understanding to say either way. The math says that all matter is pulled towards the center of a black hole and no known force can withstand the increasing force provided by gravity.

Again I would say that is side-stepping the question a bit - obviously there is no such thing as an "isolated" black hole or neutron star within the universe and naturally any black hole formed anywhere we can think of will gain mass for the time being due to capture of any form of mass. But I don't think it is unreasonable to ask what happens when a black hole loses mass so it is below the TOV limit (spherical chickens in a vacuum etc)? Obviously also this questions has implications in the nature of the distant future of the universe. The current thinking seems to imply that once a black hole is formed, even when its mass drops below its initial mass, it remains a black hole forever (causing the "black hole age" when heat-death approaches). I was wondering why this is?
We don't know. I think this depends on whether there is a singularity or not. If there IS, then the black holes stays a black hole even as it loses mass because it's density is infinite at the singularity. It will on disappear once all of it's mass has been radiated away. If there is NOT a singularity, then it depends on the structure of matter in the extreme environment of a black hole.

What do you mean "space collapsing"? Space does no such thing. It experiences curvature that increases as you add mass and energy into an area of space. There is no collapse, there is simply no event horizon one second, and then there is one the next second. Meaning that one second the curvature is very very high, but not quite high enough, and then you add mass/energy and the curvature increases to the point where there are no paths in spacetime that will lead out of a black hole. It is a smooth curve from an area with no mass, to an area with black hole mass and density.
I fully understand that, I'm just clarifying that there should be no physical change to the matter itself as the black hole forms and using the word "collapse" as a turn of phrase. I'm emphasising that the change in the curvature of the spacetime is what causes the "collapse" from neutron star to black hole, rather than a change in the matter that makes up the candidate object.

Because that's what our math tells us and we lack sufficient understanding to say either way. The math says that all matter is pulled towards the center of a black hole and no known force can withstand the increasing force provided by gravity.
Unless I've missed something, I'm afraid it's not the mathematics stating that a black hole has infinite density, rather the thinking that apparent zero volume is the same as actual zero volume. Call me a pedant, but the term "infinite" density is not only wrong but misleading in this context.

We don't know. I think this depends on whether there is a singularity or not. If there IS, then the black holes stays a black hole even as it loses mass because it's density is infinite at the singularity. It will on disappear once all of it's mass has been radiated away. If there is NOT a singularity, then it depends on the structure of matter in the extreme environment of a black hole.
I agree, although personally I'm rather unconvinced a true singularity can exist even within a black hole, rather that they are simply a large (most likely neutron degenerate although other exotic forms of matter may exist) masses that are hidden behind an event horizon, forming the illusion of a singularity.

Assuming that true (ie non-illusory) singularities do not form, a follow-on related question is what happens to neutron degenerate matter when it compressed under 10 billion solar masses in a supermassive black hole! I'm sure the answer to that is "nobody knows" but it's a fascinating thought.

Drakkith
Staff Emeritus
Unless I've missed something, I'm afraid it's not the mathematics stating that a black hole has infinite density, rather the thinking that apparent zero volume is the same as actual zero volume. Call me a pedant, but the term "infinite" density is not only wrong but misleading in this context.
How so?

How so?
The term infinite density is applied due to the calculation: Density = mass / volume with the volume of a black hole => zero

However, it seems clear that a black hole does not have zero volume, simply that the "internal" volume of the black hole is hidden by the event horizon. This gives black holes large, but finite, true densities, despite perhaps apparent infinite density.

Since this apparent infinite density serves no mathematical usefulness, as you so eloquently put, and is merely an illusion caused by the event horizon, referring to a black hole as "an object with infinite density" which I read and hear a lot on many levels of discussion over black holes, is misleading.

Drakkith
Staff Emeritus
I agree that it is misused in normal conversation, and in many popular science shows, but so is everything in physics.

bcrowell
Staff Emeritus
Gold Member
Then what form would it take? Neutron degenerate matter is what goes in, and there is effectively nothing more dense...
Not true.

Imagine you had an isolated neutron star with almost exactly the mass required to collapse into a black hole. Then, a picogram of material (or whatever arbitrary mass) is added to the neutron star, causing spacetime to distort, hiding the neutron star behind an event horizon. ie a black hole forms.

I'm avoiding saying that the neutron start "collapses" into a black hole because it is not the neutron star that is collapsing, it is the surrounding spacetime.
Both happen. The neutron star collapses and changes its structure, and there is also a dynamical evolution of the surrounding spacetime.

For example - if you were passing into the event horizon of a black hole, you would not notice anything special happening at that moment (apart from the intense gravity and the tidal stresses that would depend on the mass of the black hole you were falling into, but that would be present from far outside the event horizon).
The tidal stresses become infinite at the singularity. After it hits the singularity, infalling matter no longer exists in spacetime. Its mass has been contributed to the singularity, which GR doesn't even describe as a point in spacetime.

Not true.
Care to expand?
I know quark matter, which remains hypothetical, has been suggested but you seem to be being very definitive here.

Both happen. The neutron star collapses and changes its structure, and there is also a dynamical evolution of the surrounding spacetime.
In what way does its structure change?
In the thought experiment I've proposed, the stellar material of the neutron star that's on the edge of becoming a black hole only experiences a truly miniscule increase in density. I find it hard to believe that such a change will have any effect on the stellar material in its own reference frame.
I would suggest that the change that occurs is only apparent from someone outside of the newly-formed event horizon, the tiny, but crucial, increase in spacetime curvature causes the star to appear to collapse inwards on itself.

The questions I'm posing are probing whether this is an illusory change or a real change.

If the answer is contrary to what I'm suggesting, I would like explanation of why this is not the case, and by what mechanism the original matter acquires this infinite density characteristic. The fundamental question is, do singularities actually exist or are they an illusion?

It's obviously a discussion for which we don't have a solution, and I'm in the "they don't" camp, however to me, it's an interesting subject also on a subatomic scale, if a new infinite-density point, a singularity, is created from neutron degenerate matter (and whatever else falls into it) - is charge (I know the answer to that one), parity, flavour, number etc (depending on the nature of the incoming matter). conserved? If so, how?

I'm wanting to hear the contrary view - I am a scientist afterall. (Chemistry, Physics is a hobby).

Since all matter that falls into the singularity would be thusly converted, the process must be ongoing throughout the singularity's life. This leads to the conclusion that the process is not driven by the mass of the singularity as that can be arbitrarily small as the singularity evaporates, or we could end up in a strange situation where a singularity is "coated" in matter if this process shuts down once the singularity drops to some low mass.

Chronos
Gold Member
You do not need infinite density for an event horizon to form. Any object with a radius under the schwarzschild limit will form an event horizon, thus becoming a black hole. For the sun, the critical limit is just under 3 kilometers. Density is not critical in determining the schwarzschild radius. The Milky Way, for example, would form an event horizon at a density of around 3.7 x 10^-8 gm/cc. The formula for calculating critical radius is V ~ p^[3/2] where V is the volume of the object and p is its density. At the density of a neutron star [~7 x 10^14 gm/cc], a little mathemagic indicates it would form an event horizon at a radius of a little over 16 km. The typical neutron star has a radius around 12 km, so you can see it is fairly close to the critical size. Disclaimer - the estimated critical radius is a naive calculation without any corrections.

bcrowell
Staff Emeritus
Gold Member
You do not need infinite density for an event horizon to form. Any object with a radius under the schwarzschild limit will form an event horizon, thus becoming a black hole.
The Penrose singularity theorem says that if a spacetime has a trapped null surface, it's guaranteed to have at least one black-hole-type singularity. Basically what that means is that once an event horizon forms, you're guaranteed to get a singularity inside it -- and this is going to happen within a very short proper time.

Then what form would it take? Neutron degenerate matter is what goes in, and there is effectively nothing more dense...
Not true.
Care to expand?
I know quark matter, which remains hypothetical, has been suggested but you seem to be being very definitive here.
You're the one making the claim, so it's really up to you to support it. But you've already alluded to the Tolman-Oppenheimer-Volkoff limit, whose existence disproves your claim. Past that limit, whatever it is, it's guaranteed not to be degenerate neutron matter.

None of what I'm saying is at all controversial. You can find a discussion of this in, for example, the type of textbook used in gen ed college astronomy courses. For someone with your background as a chemist, a good book on this topic is Exploring Black Holes: Introduction to General Relativity, by Taylor and Wheeler.

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I've always had an issue with the term "infinite density" being applied to a black hole.
It's a physical consequence of general relativity. The only way out is to assume some new physics, probably quantum gravity.

All well and good, but isn't it actually the case that black holes only have apparent zero volume due to spacetime curvature making its internal volume hidden rather than there being actual zero volume?
No. Assuming GR is correct once the thing starts collapsing into a black hole then it's going to keep going until you have a finite amount of matter in zero volume. Put another way, the trouble with black holes is that you have a real area of infinite curvature.

To wit: If some omnipotent being (regardless of impossibility) were to pass within the event horizon of the black hole, reach to its centre of mass and scoop up a handful and return, wouldn't this material simply be neutron degenerate matter?
Nope. There would be neutron matter on the way in, but what happens is that as density increases, the degeneracy lifts and so the star starts collapsing again. Dividing neutrons into smaller particles makes the problem *worse* because with smaller particles you end up with less degeneracy.

The only way you can avoid more collapse was if you had neutrons combine into *larger* particles. There are good quantum mechanical reasons to think that this won't happen. Large particles tend to be more unstable because there is nothing to keep them from decaying into smaller particles, and the number of possible particles is limited by various nuclear experiments.

Or to take a more possible example: if the black hole loses sufficient mass via evaporation/hawking radiation etc, could the original stellar remnant re-emerge as the total (but not apparent) density of the black hole drops to the point where spacetime can resume its "normal" curvature?
No. Since the matter gets converted into the radiation.

The conventional wisdom seems to suggest that once you have a black hole, you always have a black hole until it evaporates and spontaneous disappears (leaving some kind of topological defect?), even if its mass drops below the point where the original black hole formed - spacetime being plastic rather than elastic. I was wondering why this is - does spacetime have an elastic limit, and if so, what equation or experiment has determined this?
There's no way in classical GR of having a black hole mass drop. If you add in quantum effects, then you get evaporation. It's *possible* that something very strange will happen with quantum mechanics, but that's new physics.

Perhaps I simply take the "rubber sheet" analogue too far, but I can't shake the idea that there is an internal volume to a black hole - constructed out of spacetime hidden by the event horizon.
This is prohibited by classical GR. Now if you do quantum gravity, your guess is as good as mine, and the black hole might do something bizarre like create a new universe.

I fully understand that, I'm just clarifying that there should be no physical change to the matter itself as the black hole forms and using the word "collapse" as a turn of phrase. I'm emphasising that the change in the curvature of the spacetime is what causes the "collapse" from neutron star to black hole, rather than a change in the matter that makes up the candidate object.
But that's not true. Black holes form when the density of matter exceed a critical density. Degenerate matter is degenerate because you've filled up all of the energy levels. As you increase the density, special relativity insures that there will be more empty energy levels. Once you go past a critical value, the degeneracy disappears, you lose pressure support, and the star really collapses.

So in the case of neutron stars, there is an actual change in the behavior of the material that leads to black holes. What exactly the critical density is at turns out to be a hugely complex calculation. However, relativity insures that there is a critical density.

One other way of looking at is that nothing can go faster than the speed of light including sound. speed of sound is proportional to pressure / density. Since the speed of sound is finite, if you increase the density, then pressure has to be limited. Since force is d P /d r, as the pressure flattens, the force that prevents collapse goes to zero.

Unless I've missed something, I'm afraid it's not the mathematics stating that a black hole has infinite density
Nope. Black hole has infinite density at the singularity (ignore quantum effects).

I agree, although personally I'm rather unconvinced a true singularity can exist even within a black hole, rather that they are simply a large (most likely neutron degenerate although other exotic forms of matter may exist) masses that are hidden behind an event horizon, forming the illusion of a singularity.
This won't work. Any exotic matter makes the object *less* resistant to collapse. For you to have matter that prevents total collapse, it has to be matter for which relativity does not apply.

It's a general result from relativity that degenerate matter collapses once you hit a critical density. It has to do with the fact that sound can't travel faster than the speed of light.

Assuming that true (ie non-illusory) singularities do not form, a follow-on related question is what happens to neutron degenerate matter when it compressed under 10 billion solar masses in a supermassive black hole! I'm sure the answer to that is "nobody knows" but it's a fascinating thought.
It lose pressure support and the collapses. The details are horrendously complex, but the basic picture isn't. You can use the relativistic equations and show that *any* matter will collapse if the density is high enough. One other way of thinking about it is that nothing can travel faster than light so if you have something traveling close to the speed of light, it gets "floppy."

In what way does its structure change?
Degenerate matter becomes non-degenerate. The neutrons (or whatever) become relativistic, and once that happens you lose pressure support.

In the thought experiment I've proposed, the stellar material of the neutron star that's on the edge of becoming a black hole only experiences a truly miniscule increase in density. I find it hard to believe that such a change will have any effect on the stellar material in its own reference frame.
You drop a tiny bit of matter onto a white dwarf at Chandrasekar mass and it collapses.

A degenerate object that is at the edge of collapse is dynamically unstable. Once you increase the density, you open up unfilled energy states. This kills pressure support. This increases density, which opens up even more unfilled energy states, and it collapse even faster.

In a white dwarf, you end up with a supernova + maybe a neutron star. For a neutron star, you end up with a black hole.

The questions I'm posing are probing whether this is an illusory change or a real change.
It's real.

If the answer is contrary to what I'm suggesting, I would like explanation of why this is not the case, and by what mechanism the original matter acquires this infinite density characteristic. The fundamental question is, do singularities actually exist or are they an illusion?
That's another question. In classical GR, singularities form. That's just what the equations say. It's possible even likely that before you hit the singularity something weird happens.

But that's a different question than what happens to neutron degenerate matter.

It's obviously a discussion for which we don't have a solution, and I'm in the "they don't" camp, however to me, it's an interesting subject also on a subatomic scale, if a new infinite-density point, a singularity, is created from neutron degenerate matter (and whatever else falls into it) - is charge (I know the answer to that one), parity, flavour, number etc (depending on the nature of the incoming matter). conserved? If so, how?
There a lot of stuff in between "neutron degenerate matter collapses" and "singularity forms." We can say a lot about neutron degenerate matter since we can do experiments with neutrons and degenerate matter. What happens at the singularity is theoretical. You take GR and you mathematically get a singularity. This might mean that GR is incomplete.

But the stuff around the singularity can't be degenerate matter unless relativity is totally wrong.

Also at a more technical level, if you are familiar with the themodynamics of Fermi-Dirac gases, most of this follows from that. What you do is to replace the energy expression with the relativistic one and that gives you a critical density beyond which there is no degeneracy.

In the thought experiment I've proposed, the stellar material of the neutron star that's on the edge of becoming a black hole only experiences a truly miniscule increase in density. I find it hard to believe that such a change will have any effect on the stellar material in its own reference frame.
Well, this "miniscule" change will make local escape velocity go over c. After this, NOTHING can stay motionless, not even "neutronium" or hypothetical "quarkonium" matter. Everything will start sliding down to the center.

It's fun to solve Einstein equations in ordinary spherical coordinates for the black hole interior: you will find that ds^2 (in +--- notation) is maximized *along the direction to the center of the hole*. Which means that this direction is the *time axis* now, it's not a spatial direction anymore.

IOW: all matter falls into the center because the center is now in it's *future*, and you cannot avoid arriving into your future :)