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My understanding is that the Pauli exclusion principle is always valid for fermions, as long as these fermions exist.

For example, when a star collapses under gravity, at first the Pauli exclusion applies to electrons until they combine with protons, at which point leptons leave as neutrinos and a neutron star forms.

From this point, the Pauli exclusion applies to neutrons until they break under gravity into quarks potentially forming a Quark Star:

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At this point the Pauli exclusion applies to quarks. The Up and Down quark are stable elementary particles that cannot break down into anything smaller like atoms or neutrons. Therefore, it would seem that the Pauli exclusion would still apply to quarks and prevent the collapse no matter how strong gravity is.

What is the current reasoning of how gravity overcomes the Pauli exclusion principle for quarks to form a black hole?

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You may be misunderstanding the Pauli Exclusion Principle (PEP).The PEP states that no two fermions can occupy the same quantum state, not that they cannot occupy the same space or be compressed to whatever density you like. The quantum states here consist of two spin states for every possible momentum state. In a degenerate gas, all these states are filled up to the Fermi energy. All that happens when the neutron star (or quark star) gets smaller (or collapses), is that the Fermi energy just keeps increasing as the neutron (quark) density climbs, and the neutron (quark) degeneracy pressure just keeps increasing as a consequence.

However, in General Relativity, pressure (like mass/energy) is a source of gravitational curvature and actually increases the required pressure gradient needed to support the star. At a certain threshold radius - a small factor larger than the Schwarzschild radius, a point of instability is reached where increasing the pressure is actually counter-productive. Beyond this, you can make the pressure as large as you like and it will not prevent the formation of a black hole.

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  • $\begingroup$ So the momentum (pressure) increases and the kinetic energy contributes to gravity. I get it. The question remains, to what limit? It would seem that closer to the singularity the kinetic energy would increase without a limit, but this doesn't seen possible, as it would also make the total mass larger than it was. I asked this before, but the answer wasn't very clear to me: physics.stackexchange.com/questions/356029/… I wonder if you have a better answer to that question. Thanks! $\endgroup$ – safesphere Nov 1 '17 at 23:55
  • $\begingroup$ It was the only one and I accepted it, as I was sure it wasn't incorrect, but you can see a long discussion there indicating the answer wasn't too clear. This doesn't prevent anyone from posting a better answer though, does it? In fact, I was surprised that the question didn't generate more interest, as I am sure most people have a little idea of what the relevant physics is. $\endgroup$ – safesphere Nov 2 '17 at 0:10
  • $\begingroup$ @safesphere To talk about anything near the singularity, you need a theory of quantum gravity. It's entirely natural that things seem non-physical if we try to apply theories without quantum gravity near a singularity. $\endgroup$ – Chris Nov 2 '17 at 0:22
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    $\begingroup$ The "gravitational mass" of a compact object is less than the sum of its baryonic components because it is bound and so the sum of its internal (kinetic) energy and potential energy is negative. @safesphere $\endgroup$ – Rob Jeffries Nov 2 '17 at 0:23
  • $\begingroup$ @Chris How about a half way to the singularity or 90%... :) $\endgroup$ – safesphere Nov 2 '17 at 0:29

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