Quark Stars(johncarlosbaez.wordpress.com)
johncarlosbaez.wordpress.com
Quark Stars
https://johncarlosbaez.wordpress.com/2024/08/02/quark-stars/
59 comments
As a layman, I find the idea of neutron and quark stars to be fascinating. What puzzles me though is what distinguishes a compressed mass of neutrons from unconfined quarks/gluons. I thought that the idea of a neutron having a "shell" with quarks inside it was just a visualisation tool, but the compressed neutrons must be providing some force to prevent them from collapsing further into unconfined quarks. That also raises the question of how the unconfined quarks/gluons provide a force to prevent collapse into black holes.
> what distinguishes a compressed mass of neutrons from unconfined quarks/gluons.
If the density and pressure get high enough, there is no longer a stable "neutron" state in which three valence quarks are bound together. You just have a soup of quarks. Calling the quarks "unconfined" is a bit of a misnomer since each of them is restricted to a very small "cell" in space. But they are "unconfined" in the sense of not being bound together by the strong interaction; within their small "cell" they move more or less like free particles.
> the compressed neutrons must be providing some force to prevent them from collapsing further into unconfined quarks.
"Collapse" isn't really a good word to describe this transition. If you add a small amount of mass to the object, it compresses a little more. If it compresses enough, there is something like a phase transition where the quarks stop being bound into neutrons; but the overall size of the object doesn't "collapse", it just gets a little smaller.
> That also raises the question of how the unconfined quarks/gluons provide a force to prevent collapse into black holes.
This is just the Pauli exclusion principle, as has already been said in response to you. It's more or less the same whether the quarks are bound into neutrons or not.
If the density and pressure get high enough, there is no longer a stable "neutron" state in which three valence quarks are bound together. You just have a soup of quarks. Calling the quarks "unconfined" is a bit of a misnomer since each of them is restricted to a very small "cell" in space. But they are "unconfined" in the sense of not being bound together by the strong interaction; within their small "cell" they move more or less like free particles.
> the compressed neutrons must be providing some force to prevent them from collapsing further into unconfined quarks.
"Collapse" isn't really a good word to describe this transition. If you add a small amount of mass to the object, it compresses a little more. If it compresses enough, there is something like a phase transition where the quarks stop being bound into neutrons; but the overall size of the object doesn't "collapse", it just gets a little smaller.
> That also raises the question of how the unconfined quarks/gluons provide a force to prevent collapse into black holes.
This is just the Pauli exclusion principle, as has already been said in response to you. It's more or less the same whether the quarks are bound into neutrons or not.
There is an outward force from the neutrons, called neutron degeneracy pressure. It's not really a case of the Pauli Exclusion Principle, as much as they are both different descriptions of what's going on with the wave function.
You're looking for the concept of a degeneracy pressure, specifically neutron and quark degeneracy pressures. The short version is that the Pauli exclusion principle forbids fermions from occupying the same state, and this manifests as an actual physical force that resists attempts to bring them close together.
https://en.wikipedia.org/wiki/Degenerate_matter
https://en.wikipedia.org/wiki/Degenerate_matter
> short version is that the Pauli exclusion principle forbids fermions
I'd like to point out (for others following along) that the Pauli exclusion principle isn't actually a separate rule (That is, something you'd have to apply after you do 'normal' physics). What is happening with the PEP is that if you start off with a wavefunction that has fermionic symmetry (that is, interchange of two particle swaps the sign of the wavefunction), the evolution via the Schroedinger equation will preserve that (much like it preserves the Integral(|psi|^2)=1 relation). Same for bosons.
So if you're writing a Quantum physics simulator, you don't need to put in a "Pauli Exclusion Rule" step.[1]
[1] Though depending on your representation you may toss one in for numerical stability.
I'd like to point out (for others following along) that the Pauli exclusion principle isn't actually a separate rule (That is, something you'd have to apply after you do 'normal' physics). What is happening with the PEP is that if you start off with a wavefunction that has fermionic symmetry (that is, interchange of two particle swaps the sign of the wavefunction), the evolution via the Schroedinger equation will preserve that (much like it preserves the Integral(|psi|^2)=1 relation). Same for bosons.
So if you're writing a Quantum physics simulator, you don't need to put in a "Pauli Exclusion Rule" step.[1]
[1] Though depending on your representation you may toss one in for numerical stability.
Thanks. It would appear that quark degeneracy pressure is the missing bit that I didn't know about and I don't really understand how that works, but then it does appear to be poorly understood in general.
Not an expert, but it could be that if the radius of a quark star grows slowly enough, then a very massive one would have a schwarzchild radius that is bigger than the star, making it a black hole but without a collapsed singularity.
If you are interested in this topic then i __highly__ recommend the PBS Spacetime channel on YouTube: https://youtu.be/1Ou1MckZHTA?si=enfHtWOSa9BwYRZ-
where they discuss this topic and so much more. it is truly a gold mine on this topic!
where they discuss this topic and so much more. it is truly a gold mine on this topic!
Love Spacetime, but I find there's a point about 2/3 of the way through every video where I realize I am totally lost in the explanation and I'm just letting the science talk spritz over me like a light summer rain. Vaguely hoping that I'm picking up some sort of osmotic education. I think it's like ASMR for wish-they-could-have-been physicists.
> letting the science talk spritz over me like a light summer rain
this made my day, thanks! Sometimes I watch an episode 2-3 times before it clicks, maybe you should try it too.
this made my day, thanks! Sometimes I watch an episode 2-3 times before it clicks, maybe you should try it too.
chiming in for a 'me too' post on hn
also notable, the q'n'a at the end smells staged since ever
also notable, the q'n'a at the end smells staged since ever
100% can attest I love this channel
I was on microphone duty as a grad student at a conference that had a crackpot who sat in the front row and asked questions about quark stars no matter the topic of the talk. Unfortunately the chairpeople of the plenaries kept calling on them and I kept having to hand them the mic...
> Right at the critical point, water looks weird. It look like a blur of droplets floating in gas. But if you look at any droplet you’ll see it’s full of bubbles of gas. And if you look in any of these bubbles you see it’s full of droplets. As you keep zooming in, you keep seeing basically the same thing…. droplets of liquid in gas, bubbles of gas in liquid…. until you get down to the scale of atoms. So we say this stuff has ‘conformal symmetry’.
This is akin to a rather derided religious position: "it's turtles all the way down".
The problem with all these claims is that one has to accept that we have established this in CERN or wherever. Have we? Can I check? Is it just the turtles story?
This is akin to a rather derided religious position: "it's turtles all the way down".
The problem with all these claims is that one has to accept that we have established this in CERN or wherever. Have we? Can I check? Is it just the turtles story?
Supercritical water is relatively easy to create in a laboratory. So yeah, it's been verified.
https://en.m.wikipedia.org/wiki/Supercritical_water_oxidatio...
https://en.m.wikipedia.org/wiki/Supercritical_water_oxidatio...
Great, how did you do it?
PS ... and what did you see?
PS ... and what did you see?
Quark stars might even be a stage on the way to collapse from neutron star to black hole. The collapse is fast, but not instantaneous, so what would those neutrons become once their degeneracy pressure is exceeded?
>what would those neutrons become once their degeneracy pressure is exceeded
That kind of depends on what the equation of state looks like for quark matter. If you look at black hole formation from a theoretical perspective (e.g. as a collapsing shell of matter), it is very much possible that neutrons transition directly into a black hole before entering a new state. Unfortunately, we have no idea about the equation of state and this stuff is very far beyond anything that could be studied experimentally in a collider. It's admittedly a fascinating topic, but there is very little rigorous science surrounding it.
That kind of depends on what the equation of state looks like for quark matter. If you look at black hole formation from a theoretical perspective (e.g. as a collapsing shell of matter), it is very much possible that neutrons transition directly into a black hole before entering a new state. Unfortunately, we have no idea about the equation of state and this stuff is very far beyond anything that could be studied experimentally in a collider. It's admittedly a fascinating topic, but there is very little rigorous science surrounding it.
> it is very much possible that neutrons transition directly into a black hole before entering a new state
At least the neutrons that start out being outside the Schwarzschild radius of the black-hole-to-be cannot transition directly into being in a black hole state because they have to first move from where they are to the event horizon, and that has to take some non-zero amount of time since they can't exceed the speed of light on the way to the event horizon.
It's possible that with the degeneracy pressure exceeded they just don't become something else, but instead simply move... through each other? That makes little sense. Since they are made up of quarks what makes sense instead is that they become quark soup on the way to the event horizon. Though there are other possibilities too. Maybe all the neutrons turn to photons going towards the incipient black hole and without becoming quark soup first, they they can travel through each other.
> Unfortunately, we have no idea about the equation of state and this stuff is very far beyond anything that could be studied experimentally in a collider. It's admittedly a fascinating topic, but there is very little rigorous science surrounding it.
The best we could do is check different theories for consistency, but we can't test them unless those theories make predictions about electromagnetic emissions or gravity wave emissions from a neutron star collapse that we might be able to observe and which could be used to test those predictions.
At least the neutrons that start out being outside the Schwarzschild radius of the black-hole-to-be cannot transition directly into being in a black hole state because they have to first move from where they are to the event horizon, and that has to take some non-zero amount of time since they can't exceed the speed of light on the way to the event horizon.
It's possible that with the degeneracy pressure exceeded they just don't become something else, but instead simply move... through each other? That makes little sense. Since they are made up of quarks what makes sense instead is that they become quark soup on the way to the event horizon. Though there are other possibilities too. Maybe all the neutrons turn to photons going towards the incipient black hole and without becoming quark soup first, they they can travel through each other.
> Unfortunately, we have no idea about the equation of state and this stuff is very far beyond anything that could be studied experimentally in a collider. It's admittedly a fascinating topic, but there is very little rigorous science surrounding it.
The best we could do is check different theories for consistency, but we can't test them unless those theories make predictions about electromagnetic emissions or gravity wave emissions from a neutron star collapse that we might be able to observe and which could be used to test those predictions.
>they have to first move from where they are to the event horizon
No. I mean, from their point of view it would seem they kind of have to do, but there are some serious open questions about what happens here exactly. Regardless, as an outside observer, you would never see them actually move behind the horizon, you would only see the black hole grow beyond the point where they used to be. For a detailed description of all this you can check out chapter 32 of MTW's Gravitation.
>Since they are made up of quarks what makes sense instead is that they become quark soup on the way to the event horizon
Again, this depends on the equation of state and the existance and location of transition points. We don't know any of those things - but we do know that any realistic model must include general relativity, because quantum effects alone are no longer sufficient to describe what happens here. At that level all bets are off.
>The best we could do is check different theories for consistency
That is one thing. But it is also kind of moot since this is where string theory has been stuck for half a century now.
No. I mean, from their point of view it would seem they kind of have to do, but there are some serious open questions about what happens here exactly. Regardless, as an outside observer, you would never see them actually move behind the horizon, you would only see the black hole grow beyond the point where they used to be. For a detailed description of all this you can check out chapter 32 of MTW's Gravitation.
>Since they are made up of quarks what makes sense instead is that they become quark soup on the way to the event horizon
Again, this depends on the equation of state and the existance and location of transition points. We don't know any of those things - but we do know that any realistic model must include general relativity, because quantum effects alone are no longer sufficient to describe what happens here. At that level all bets are off.
>The best we could do is check different theories for consistency
That is one thing. But it is also kind of moot since this is where string theory has been stuck for half a century now.
> It’s not every day we find quintillions of tonnes of a new state of matter
In terms of size, it’d still be quite unimpressive. About the volume of a million matchboxes. Would fit in a truck.
In terms of size, it’d still be quite unimpressive. About the volume of a million matchboxes. Would fit in a truck.
This also means how big a atom actually is compared to the elementary particles that it is made up of.
It could be said most of any thing is just empty space.
It could be said most of any thing is just empty space.
At this point we start wondering what’s the real difference between space and the stuff that occupies it. Or what “occupies” actually means.
I completely agree and although I am guessing that this probably won't be a very popular opinion, I see matter as being genuinely made-out-of-the-same-stuff-as-space but in a highly 'condensed' and dynamic form.
If you see matter in this way and imagine that the relationship between the space and the matter is probably-dynamic too, then it doesn't take a big leap to imagine that gravity could be thought-of as the outcome of the constant-consumption-of-space that must usually occur for the matter to continue its normal existence.. as if 'running' the matter uses up the 'execution cycles' of space (however I think space (even a vacuum) is probably quite structured itself and not mere-causation/computation, even at that level!)
If you see matter in this way and imagine that the relationship between the space and the matter is probably-dynamic too, then it doesn't take a big leap to imagine that gravity could be thought-of as the outcome of the constant-consumption-of-space that must usually occur for the matter to continue its normal existence.. as if 'running' the matter uses up the 'execution cycles' of space (however I think space (even a vacuum) is probably quite structured itself and not mere-causation/computation, even at that level!)
Wouldn't it be funny if black holes were actually made of quarks, with a core of some exotic higher form of quark we can't create in an accelerator.
nice! just like the strong interaction material from the three body problem!
Off topic, but the reference to critical point of water reminded me of this lovely video from ages ago: https://www.youtube.com/watch?v=2xyiqPgZVyw
The original link is now gone, and archive.org doesn’t have the other two videos, but I’m pretty sure I have them in my archive: https://web.archive.org/web/20080416125148/http://www.scienc...
The original link is now gone, and archive.org doesn’t have the other two videos, but I’m pretty sure I have them in my archive: https://web.archive.org/web/20080416125148/http://www.scienc...
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This is an exceptionally fun read for me since my name is Quark! The terms "quark star" and "quark matter" are completely new to me so I'll have to do some reading!
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ChatGPT just told me that
... the Schwarzschild radius for a quark-gluon plasma with a mass of 10^12 kg is approximately 1.485x10^-15 meters, which is extremely small.
Leaving aside the calculation (I don't know if it's correct), and the sense or not of relying on a fancy word-association machine for such calculations (I wouldn't), your Schwarzchild radius is even smaller than that and you're not a black hole - an object is only a black hole if it is entirely inside its Schwarzchild radius.
If the Schwarzchild radius is reached within a larger mass, wouldn't that radius become inescapable and eventually consume the remainder while growing in proportion?
Since I did a sanity check on the result which seems to have failed, why do you think that it's necessary to warn me not to rely on it?
The hostility on this site for using LLMs as a tool for exploration is confusing to me. I also learn a lot by asking questions of highly unreliable humans. LLMs don't seem to be much worse than that.
Since I did a sanity check on the result which seems to have failed, why do you think that it's necessary to warn me not to rely on it?
The hostility on this site for using LLMs as a tool for exploration is confusing to me. I also learn a lot by asking questions of highly unreliable humans. LLMs don't seem to be much worse than that.
I don't have a problem with your post, but the hostility I see is mostly out of a real concern that people posting LLM comments could degrade discussion. It's probably also association, because a lot of people have started posting comments that just say "ChatGPT says..." and nearly the entirety of the body is just LLM output.
Honestly, a lot of people are just sick of seeing anything about LLMs and don't like seeing it at all in threads that aren't already about it.
Honestly, a lot of people are just sick of seeing anything about LLMs and don't like seeing it at all in threads that aren't already about it.
Exactly. It's not appropriate for this type of question. LLMs can summarize the global internet sentiment about something pretty well, but they aren't the right architecture for answering precise scientific questions, nor will they ever be. It's like asking an old, experienced physicist to guess answers to things instead of solving equations.
They're already better than I'd have expected possible, so I wouldn't say "never" — but also yes I agree with you, currently I'd compare their output to an off-the-cuff answer from an experienced professional (or an undergraduate trying hard).
As a physicist I can say that unless ChatGPT is asked a definitional question, or a very common question (whose solution is likely described many places on the internet anyway), it is very likely to be wrong. Personally more than 90% of the questions I've asked it have been flat-out wrong so I stopped using it entirely.
This is maybe cause ppl don’t want to rely on some machine hallucinating wrong answer when trying to answer fundamental questions but rather use real wisdom instead.
As others have said, that's not what a Schwarzschild radius means. You would only get a black hole if all the matter is inside the radius.
The Schwarzschild radius comes from the free space solution to Einstein's equation. It only holds in free space. So if you want to use it here, all the matter would need to be within Schwarzschild radius, and then you could calculate what happens outside that radius (again, in the matter free space outside). If you want to know how gravity behaves inside a star you have to use a different tool.
The Schwarzschild radius comes from the free space solution to Einstein's equation. It only holds in free space. So if you want to use it here, all the matter would need to be within Schwarzschild radius, and then you could calculate what happens outside that radius (again, in the matter free space outside). If you want to know how gravity behaves inside a star you have to use a different tool.
Lots of other replies so hopefully it makes sense but just to be clear: 'the' Schwarzchild radius isn't one specific number, it's based on the mass of an object. Every massive object has a different SR (also called its Event Horizon), and if you compress the object entirely to within that radius, it will become a black hole. It's just a measure of how small the object needs to be before light cannot escape its surface.
> If the Schwarzchild radius is reached within a larger mass, wouldn't that radius become inescapable and eventually consume the remainder while growing in proportion?
If I understand your question correctly, no. If I have a mass M that corresponds to a Schwarzchild radius r, but the mass occupies a larger radius R, no, it will not create a black hole, since the mass m within r is less than the critical mass M. The portion of M that is within r can still escape that radius.
If I understand your question correctly, no. If I have a mass M that corresponds to a Schwarzchild radius r, but the mass occupies a larger radius R, no, it will not create a black hole, since the mass m within r is less than the critical mass M. The portion of M that is within r can still escape that radius.
It's fairly well-accepted that tiny black holes are quite unstable:
https://en.wikipedia.org/wiki/Hawking_radiation
On a larger scale, kinda yes - a small-ish black hole would generally "feed" from any near-enough object. Though that feeding will generally be extremely energetic and messy, and fling most of the victim away at high velocity.
https://en.wikipedia.org/wiki/Hawking_radiation
On a larger scale, kinda yes - a small-ish black hole would generally "feed" from any near-enough object. Though that feeding will generally be extremely energetic and messy, and fling most of the victim away at high velocity.
You only have a black hole and an event horizon if the mass whose Schwarzschild radius you are calculating is completely contained within said radius.
ant6n(1)
Neutron stars weigh in at about 1.4 solar masses, or 10^30kg, which gives a Schwarzschild radius of about 3mi.
A neutron star can be viewed as a mass-just-shy-of-a-black-hole, so I'd expect it to be relatively similar in scale.
A neutron star can be viewed as a mass-just-shy-of-a-black-hole, so I'd expect it to be relatively similar in scale.
> Neutron stars weigh in at about 1.4 solar masses
Observed neutron stars actually have a range of masses, from the 1.4 solar mass point (which is the maximum mass for a white dwarf, so one would expect to see neutron stars of about that mass that were just over the white dwarf limit) up to, IIRC, almost 3 solar masses for the largest one that has been observed.
> A neutron star can be viewed as a mass-just-shy-of-a-black-hole
Not in general, no. A neutron star just under the maximum mass limit for neutron stars, which is believed to be about 3 solar masses, could be sort of viewed this way, but even then it's not really correct, since there is nothing that forbids a black hole with a mass smaller than that from existing. It's just extremely unlikely that such a black hole could be formed by the collapse of a star, since such a collapse would be expected to stop at the neutron star stage (or at the white dwarf stage if the star was less than 1.4 solar masses).
Observed neutron stars actually have a range of masses, from the 1.4 solar mass point (which is the maximum mass for a white dwarf, so one would expect to see neutron stars of about that mass that were just over the white dwarf limit) up to, IIRC, almost 3 solar masses for the largest one that has been observed.
> A neutron star can be viewed as a mass-just-shy-of-a-black-hole
Not in general, no. A neutron star just under the maximum mass limit for neutron stars, which is believed to be about 3 solar masses, could be sort of viewed this way, but even then it's not really correct, since there is nothing that forbids a black hole with a mass smaller than that from existing. It's just extremely unlikely that such a black hole could be formed by the collapse of a star, since such a collapse would be expected to stop at the neutron star stage (or at the white dwarf stage if the star was less than 1.4 solar masses).
The neutron star mass limit has a formal name.
https://en.m.wikipedia.org/wiki/Tolman–Oppenheimer–Volkoff_l...
https://en.m.wikipedia.org/wiki/Tolman–Oppenheimer–Volkoff_l...
On the one hand that's interesting, on the other hand eponyms are generally a bad idea. If you mention "Tolman-Oppenheimer-Volkoff limit" the chances of being understood outside of a very specialized audience are far lower than if you mention "neutron star mass limit".
On the other hand, the Chandrasekhar limit is somewhat more well known.
Excellent points! Galaxies have more mass than either, but behave very differently. I would correct my statement by saying it's the concentration of mass (density) that defines neutron stars and black holes.
> it's the concentration of mass (density) that defines neutron stars and black holes.
Not really, no. What defines a neutron star is that it is in hydrostatic equilibrium supported by neutron degeneracy pressure. If we include the quark-gluon plasma phase, we can just amend that to being supported by quark degeneracy pressure (and noting that "neutron" is just a special case of "quark" for this purpose). Whereas a normal star is in hydrostatic equilibrium supported by thermal pressure (with fusion reactions providing the heat source). And a galaxy is not in hydrostatic equilibrium at all, it's composed of stars in free-fall orbits. The average densities of the objects in all three of these cases are consequences of the above.
A black hole is defined by having an event horizon and being vacuum. There is no well-defined concept of "density" for a black hole, nor is there "concentration of mass" in the sense of the hole being made of matter; it's not, it's vacuum. The "mass" of a black hole is a global feature of its spacetime geometry.
Not really, no. What defines a neutron star is that it is in hydrostatic equilibrium supported by neutron degeneracy pressure. If we include the quark-gluon plasma phase, we can just amend that to being supported by quark degeneracy pressure (and noting that "neutron" is just a special case of "quark" for this purpose). Whereas a normal star is in hydrostatic equilibrium supported by thermal pressure (with fusion reactions providing the heat source). And a galaxy is not in hydrostatic equilibrium at all, it's composed of stars in free-fall orbits. The average densities of the objects in all three of these cases are consequences of the above.
A black hole is defined by having an event horizon and being vacuum. There is no well-defined concept of "density" for a black hole, nor is there "concentration of mass" in the sense of the hole being made of matter; it's not, it's vacuum. The "mass" of a black hole is a global feature of its spacetime geometry.
Right, but are those phenomenon not also consequences of density? If we compressed stellar or even terrestrial material down the density of a neutron star, would we not get neutronium as a result?
I agree that the density of a black hole will depend on choice of observer, but nothing stops us from picking a reasonable one, such as a Schwarzschild observer. Mass (as you've mentioned) and volume (Schwarzschild radius) are both we enough for us to define a global average density. It does not behave intuitively though, increasing mass actually decreases density, which does seems to contradict my earlier point...
I agree that the density of a black hole will depend on choice of observer, but nothing stops us from picking a reasonable one, such as a Schwarzschild observer. Mass (as you've mentioned) and volume (Schwarzschild radius) are both we enough for us to define a global average density. It does not behave intuitively though, increasing mass actually decreases density, which does seems to contradict my earlier point...
> are those phenomenon not also consequences of density?
No, density is a consequence of the object's structure. Hydrostatic equilibrium supported by degeneracy pressure leads to higher density than hydrostatic equilibrium supported by thermal pressure.
> I agree that the density of a black hole will depend on choice of observer
That's not what I said. I said there is no well-defined density of a black hole at all. That's because the hole is vacuum and has no well-defined spatial volume.
> nothing stops us from picking a reasonable one, such as a Schwarzschild observer
There are no Schwarzschild observers inside a black hole. Schwarzschild coordinates in the interior of a black hole give an infinite answer for the "spatial volume", which, of course, is not well-defined.
> volume (Schwarzschild radius)
The Schwarzschild radius is derived from the surface area of the hole's horizon. It does not mean the hole has a well-defined volume. It doesn't.
No, density is a consequence of the object's structure. Hydrostatic equilibrium supported by degeneracy pressure leads to higher density than hydrostatic equilibrium supported by thermal pressure.
> I agree that the density of a black hole will depend on choice of observer
That's not what I said. I said there is no well-defined density of a black hole at all. That's because the hole is vacuum and has no well-defined spatial volume.
> nothing stops us from picking a reasonable one, such as a Schwarzschild observer
There are no Schwarzschild observers inside a black hole. Schwarzschild coordinates in the interior of a black hole give an infinite answer for the "spatial volume", which, of course, is not well-defined.
> volume (Schwarzschild radius)
The Schwarzschild radius is derived from the surface area of the hole's horizon. It does not mean the hole has a well-defined volume. It doesn't.
> No, density is a consequence of the object's structure.
Which is a consequence of the gravitational pressure, no? Quark degenerate matter without a confining potential would not be at equilibrium.
> There are no Schwarzschild observers inside a black hole.
Of course not, but the definition of the Schwarzschild radius does not require a black hole, just mass. For example, we define the Schwarzschild radius of the Sun to be about 3km. Outside of black hole dynamics, this defines a volume of space, and we call an object whose radius is smaller than its Schwarzschild radius a black hole. I understand that black holes are fundamentally dynamic processes, but can we not define the behavior at the limit?
Which is a consequence of the gravitational pressure, no? Quark degenerate matter without a confining potential would not be at equilibrium.
> There are no Schwarzschild observers inside a black hole.
Of course not, but the definition of the Schwarzschild radius does not require a black hole, just mass. For example, we define the Schwarzschild radius of the Sun to be about 3km. Outside of black hole dynamics, this defines a volume of space, and we call an object whose radius is smaller than its Schwarzschild radius a black hole. I understand that black holes are fundamentally dynamic processes, but can we not define the behavior at the limit?
> Which is a consequence of the gravitational pressure, no?
Not just gravity alone, no. (And "gravitational pressure" is not a good description of gravity's effect.) Gravity plus the fact that the object does not have a heat source inside it (fusion) to provide thermal pressure. With thermal pressure an object with the same number of particles in it would be much larger--an ordinary star--and its average density would be much smaller.
> the definition of the Schwarzschild radius does not require a black hole
But if the object is not a black hole the Schwarzschild radius is physically meaningless. It only has physical meaning for a black hole, and its physical meaning, as I said, has to do with the area of the hole's horizon, and only that.
> this defines a volume of space, and we call an object whose radius is smaller than its Schwarzschild radius a black hole
No, this is not correct. We call an object a black hole if it has an event horizon. A black hole doesn't have a well-defined "radius" in space any more than it has a well-defined volume. A black hole is certainly not just an object "made of stuff", like a neutron star, that just happens to have collapsed further. It is a different kind of thing from any ordinary object.
> I understand that black holes are fundamentally dynamic processes, but can we not define the behavior at the limit?
What you are describing is not a "limit" of anything. It's simply a wrong description of the physics of a black hole. If by "dynamic" you are talking about the process of forming a black hole from the collapse of a massive object, you're still not correctly describing the end point of that process.
Not just gravity alone, no. (And "gravitational pressure" is not a good description of gravity's effect.) Gravity plus the fact that the object does not have a heat source inside it (fusion) to provide thermal pressure. With thermal pressure an object with the same number of particles in it would be much larger--an ordinary star--and its average density would be much smaller.
> the definition of the Schwarzschild radius does not require a black hole
But if the object is not a black hole the Schwarzschild radius is physically meaningless. It only has physical meaning for a black hole, and its physical meaning, as I said, has to do with the area of the hole's horizon, and only that.
> this defines a volume of space, and we call an object whose radius is smaller than its Schwarzschild radius a black hole
No, this is not correct. We call an object a black hole if it has an event horizon. A black hole doesn't have a well-defined "radius" in space any more than it has a well-defined volume. A black hole is certainly not just an object "made of stuff", like a neutron star, that just happens to have collapsed further. It is a different kind of thing from any ordinary object.
> I understand that black holes are fundamentally dynamic processes, but can we not define the behavior at the limit?
What you are describing is not a "limit" of anything. It's simply a wrong description of the physics of a black hole. If by "dynamic" you are talking about the process of forming a black hole from the collapse of a massive object, you're still not correctly describing the end point of that process.
The result is correct. But I don't understand your conclusion. Where does that number 10^12 kg come from?