A star's collapse can be halted by the degeneracy pressure of electrons or neutrons due to the Pauli exclusion principle. In extreme relativistic conditions, a star will continue to collapse regardless of the degeneracy pressure to form a black hole. Does this violate the Pauli exclusion principle? If so, are theorists ok with that? And if it doesn't violate the Pauli exclusion principle, why not?
[Physics] Does black hole formation contradict the Pauli exclusion principle
black-holesfermionsgeneral-relativitypauli-exclusion-principle
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This answer expands on @Rex' s answer, so please read it to get the complete picture. It expands on the part elementary particles have in a black hole creation.
It is often said that when the gravitational force exceeds any outward forces or pressures, mainly the electron degeneracy pressure I'm thinking, the star collapses into a black hole. But how can this happen without the Pauli exclusion principle being violated? . . .
When the simple hydrogen equation does not hold because the potential has been distorted by the gravitational one, an electron can be captured by a proton. This makes a neutron and an electron neutrino. Neutrinos being weakly interacting escape and the neutrons make a neutron star which continues to collapse towards a black hole , if the mass is large enough. There is no problem with the Pauli exclusion or lepton number at this level. Neutrons are composed of quarks which are charged and also obey the Pauli exclusion principle. When the density due to the gravitational collapse becomes large then the whole will turn into a quark gluon plasma. That is as far as elementary particle interactions have taken us. Research is ongoing.
but I don't understand at what point we start our counting of states from the object being a star to a black hole. What happens in the middle? Is there a sharp change?
The point about a black hole is the total mass, such that it does not allow anything to escape from a certain radius. The quantum mechanical behavior from a certain point on is an effective theory joining quantum mechanics and gravitation, a process that is at the frontier of research. It depends on your definition of sharp. Supernovas are sharp.
Supernovae can be triggered in one of two ways: by the sudden reignition of nuclear fusion in a degenerate star; or by the gravitational collapse of the core of a massive star. In the first case, a degenerate white dwarf may accumulate sufficient material from a companion, either through accretion or via a merger, to raise its core temperature, ignite carbon fusion, and trigger runaway nuclear fusion, completely disrupting the star. In the second case, the core of a massive star may undergo sudden gravitational collapse, releasing gravitational potential energy that can create a supernova explosion.
It's a force like no other. It is fundamentally a quantum property and there is no classical way to think of it (at least to my knowledge). That's just how the universe is, and we haven't understood any deep reason "why" it should be that way. Mathematical consistency seems to dictate it.
It comes down to the observation that there can be some objects such that when you swap any two of them, the "wavefunction" (which describes the configuration, and can be squared to give the probability) picks up a negative sign. So when two of them sit on top of each other, the wavefunction of having A and then B is the negative of the wavefunction of having B and then A. The only mathematically consistent way to have this happen is for the wavefunction to be zero for any such configuration, which means that has zero probability of happening. We call such particles fermions.
Since electrons have spin $\frac{1}{2}$, the spin-statistics theorem tells us that they must behave like fermions.
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I don't have a very satisfactory description of the microscopic picture, but let me share my thoughts.
The Pauli exclusion doesn't quite say that fermions can't be squeezed together in space. It says that two fermions can't share the same quantum state (spin included). A black hole has an enormous amount of entropy (proportional to its area, from the famous Bekenstein-Hawking formula $S = \frac{A}{4}$) and hence, its state count is $\sim e^A$.
Now, this might not seem like a big deal since usual matter has entropy proportional to volume. However, volume of such collections is also proportional to the mass. This means that a counting of the number of states goes as $e^M$
For a black hole, it's Schwarzschild radius is proportional to the mass, hence $A \sim M^2$. So, the number of states scales as $e^{M^2}$ which is much much more than ordinary matter, especially if the mass is "not small". So there seem to be a lot of quantum states into which one can shove the fermions.
So it seems like the fermions should have an easier time in a black hole than in (say) a neutron star.