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Past the Tolman–Oppenheimer–Volkoff limit, gravity overpowers neutron degeneracy pressure and neutron stars collapse, possibly to black holes. This essay by Graeme Heald suggests that a quark star could form under the event horizon of a black hole, with quark degeneracy pressure preventing the collapse to a singularity. (The Penrose singularity theorem article once claimed it doesn't apply to fermions, "It does not hold for matter described by a super-field, i.e., the Dirac field.")

Is such a quark star possible? (Or any other degeneracy-pressure-supported object under an event horizon?) If so, what's the minimum degeneracy pressure required to resist collapse for a given mass / Schwarzschild radius?

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Graeme Heald seems to be a kook with an engineering degree. Physics Essays is basically a vanity-publishing operation. E.g., it has published papers by kooks like Adrian Sfarti and Mike Fontenot.

The Penrose singularity theorem doesn't apply to fermions

This is not true. The only condition on the matter fields is the null energy condition. The WP article appears to have been edited by someone who didn't understand the topic. See p. 263 of Hawking and Ellis for a correct statement of the assumptions of the Penrose singularity theorem. I've corrected the WP article and added a comment on the article's talk page.

The Penrose singularity theorem applies to all realistic forms of matter (any form of matter that satisfies the null energy condition). Therefore if a trapped surface forms, you are going to get a singularity.

Modeling gravitational collapse is a specialized and highly technical field, and I'm not an expert on it. However, I believe that in realistic simulations, the formation of a trapped surface and an apparent horizon coincides closely with the formation of an event horizon (absolute horizon).

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  • $\begingroup$ Yes, it does make that claim: "It does not hold for matter described by a super-field, i.e. a Dirac field." $\endgroup$ – alexchandel Jan 13 '20 at 17:32
  • $\begingroup$ @alexchandel: I see. That sentence looks to me like nonsense inserted by someone who doesn't know the topic. I'll remove it and leave a comment on the article's talk page. $\endgroup$ – user4552 Jan 13 '20 at 17:39
  • $\begingroup$ Better to leave it, unless you can find a source applying the Penrose theorem in supergeometry. I've seen the claim elsewhere that it doesn't apply for Einstein-Yang-Mills-Dirac theory. $\endgroup$ – alexchandel Jan 13 '20 at 17:46
  • $\begingroup$ @alexchandel: See p. 263 of Hawking and Ellis for a correct statement of the assumptions of the Penrose singularity theorem. You may be misinterpreting something else you've read, or you may be reading kook papers such as the one by Heald. If you want to point me to your source of information, I would be happy to take a look. $\endgroup$ – user4552 Jan 13 '20 at 17:52
  • $\begingroup$ I do not know what OP's sources are, but here is an example of Dirac particles facilitating (Big Bang) singularity avoidance. $\endgroup$ – A.V.S. Jan 14 '20 at 18:08
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I think it is technically incorrect to say that gravity "overpowers" neutron degeneracy, from any given local frame on the neutronium bulk, pressure doesn't magically surge or drop when the event horizon is formed. Black holes do not form because of some failure in the ability of the exclusion principle to keep fermions from overlapping

In other words, it is not matter that is overpowered when a black hole form; is the causal connection between the local spacetime and the environment asymptotic spacetime that is breached

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  • $\begingroup$ No, it's correct. The event horizon is smaller than the remnant's radius, and there's a point beyond which gravitation overpowers electron / neuron degeneracy, and the white dwarf / neutron star collapses. That isn't related to an event horizon forming, and indeed some hypothesize stable (non-black) quark stars just above the TOV line. $\endgroup$ – alexchandel Jan 13 '20 at 17:30
  • $\begingroup$ This doesn't answer my question either. $\endgroup$ – alexchandel Jan 13 '20 at 17:33
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Well, not only black holes can have event horizont, but I guess you are interested in black holes.

In my opinion the best way to imagine it is the following : below the event horizont there is no matter only curvature. From the Einstein equations we know that curvature of the spacetime equals the stress energy tensor on the other side of the equation. The enormous gravitational pressure ripps apart everything inside from atoms to elementary particles. Until nothing stays but their energy in the form of curvature.

One interesting thing, that quarks appear in pairs, and separating them requires energy, but it will create more quarks due to confinement of the color field. Would it mean that these quarks are indestructible and kind of suck out the energy of the black hole from inside and filling it up, transforming it to a quark star? It might happen, but maybe quantum field theory forbids it.

The heat and gravity under the horizont is close to that of the early universe. What is sure, that there are unification scales at high energies, and at a certain energy the u(1) symmetry is restored and higgs won't give mass to the z and w bosons. They are crucial elements for the standard model to work. It means, that interaction between electrons not work properly because z and w decays to photon and back.. so electrons could get closer to each other with less repeling . But at this high energies the fine structure constant and actually the gravitational and cosmological constant becomes running parameters, altering everyday physics.

It's a place where possible theories can go wild without the fear of ever being falsified.

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    $\begingroup$ Well, not only black holes can have event horizont, but I guess you are interested in black holes. An event horizon is basically the standard definition of a black hole. Do you have some other definition in mind? The rest of this answer seems mostly wrong as well. E.g., you say, "The heat and gravity under the horizont is close to that of the early universe." The Schwarzschild spacetime is a vacuum solution, so there is no heat. $\endgroup$ – user4552 Jan 16 '20 at 0:09
  • $\begingroup$ @BenCrowell I meant the physical conditions. But yes there are other ways to create event horizont, I can give you two examples: Penrose wrote in one of his books, that if some neutron stars could get close to each other they could provide critical mass in a critical volume of space creating a horizont around them. May be a chaotic system, may be stable, but Penrose argued its possible. $\endgroup$ – Kregnach Jan 19 '20 at 23:42
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Can quark stars form under an event horizon? This is an excellent question that I have often thought about myself. Personally, I agree with Graeme Heald that quark stars can indeed form under an EH. Why do I agree? Well, quarks are fermions, and fermions must obey the Pauli Exclusion Principle. This results in quark degeneracy pressure at some high curvature of spacetime preventing the formation of the mythical BH singularity. But, I am not an expert. What I can offer is a "Gedankenexperiment" a lá Einstein which might lead to an easily performed "Realexperiment" to settle the issue. Assume it's true. What would likely happen? Suppose a massive star dies and starts its collapse. It crashes through electron DP, then it crashes through neutron DP, but it finally hits the wall of our assumed quark DP after an event horizon is formed. What would happen next? The matter and energy accelerating inward would be forced to bounce off the spherical quark degeneracy pressure. This is just like when a supernova starts its collapse and bounces off the spherical neutron degeneracy pressure, and then explodes outward nearly obliterating itself with stupendous energy. But, in the case of a quark star bouncing off quark DP, it is gravitationally contained within an EH. The quark star cannot obliterate itself outwardly like a supernova. So then what? The bounce off the quark DP boundary accelerates matter and energy outwards to counteract some of the accelerating matter still moving inwards. Thus, the EH jiggles, oscillates, or breathes. It's like a heartbeat. Once the bounce is finished most of the quark star is still within the EH, still gravitationally contained. But the infow-outflow bounce cycle keeps repeating. The only thing that escapes is some high-powered, low-frequency EM radiation. This is a result of the EH breathing in and out. Pretty dynamic stuff, mostly shielded from external view by the EH. But remember, stars and the universe itself are very, very dynamic! ... a singularity is not! So, this is the Gedanken. How do we prove it by a real experiment. Well, look for a cyclical EM signature, mimicking a heartbeat, emanating from the vicinity of a black hole. Or, more generally, look for a cyclical EM signature emanating from a white dwarf (electron DP), a neutron star (neutron DP), or a black hole (quark DP). There is likely some cyclical bounce involved with all these degeneracy pressures. In the case of neutron stars, we've already detected the phenomenon and call such pulsating stars pulsars. So, this long-winded answer may not be the expert answer you were searching for, but at this point in time there are no experts that really do know the answer. Besides, sometimes in physics it's good to think out-of-the-box like that patent clerk did with his ridiculous GR theory. Until presto, it was supported by the data from a solar eclipse experiment in 1919.

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