# How could a neutron star collapse into a black hole?

White dwarfs usually do not collapse, as they have electron degeneracy pressure due to the Pauli exclusion principle. However, if one accretes mass beyond the Chandrasekhar limit, it is energetically favorable for the electrons to combine with protons and form neutrons. This gives us a neutron star.

However, neutron stars usually do not collapse into black holes due to neutron degeneracy pressure. How is it possible that beyond the LOV limit, the Pauli exclusion principle no longer prevents the collapse? Shouldn't it still prevent neutrons, which are fermions, from being compressed together any further?

I've seen answers involving quark stars, but those are purely hypothetical. What is the most accepted explanation for this?

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Are you talking about an accreting neutron star? – Dean Mar 16 at 19:54
@Dean Yes, if it accretes enough mass to collapse into a black hole. – Sir Cumference Mar 16 at 19:56
Possible duplicate of How does neutron star collapse into black hole? – James Kilfiger Mar 25 at 21:11
@JamesKilfiger I was specifically wondering how it could overcome the Pauli exclusion principle, which was never mentioned in that question. I also clearly stated I didn't want any answers involving hypothetical quark stars. I wanted the most accepted explanation for this. – Sir Cumference Mar 26 at 3:37

The scenario you describe may occur. On the other hand it may actually be that neutronisation in a white dwarf is the trigger for a thermonuclear type Ia supernova.

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 gets smaller (or collapses), is that the Fermi energy just keeps increasing as the neutron density climbs, and the neutron 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.

Even inside the BH there is not necessarily a problem with the PEP. You can compress fermions to infinite density so long as they can have infinite momentum.

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Great answer! But as long as the neutron star gains mass, will the Fermi energy continue rising? – Sir Cumference Mar 16 at 23:32
@SirCumference Yes it will. roughly proportional to $\rho^{1/3}$. But as I explained, in GR this cannot halt the collapse - it is self-defeating. – Rob Jeffries Mar 16 at 23:37
And when the Fermi energy increases, not all of the states will still be filled up to the Fermi energy, right? – Sir Cumference Mar 17 at 1:19
@SirCumference Yes they will. That is what happens in a degenerate gas. As you increase the number of fermions per unit volume, they completely fill states up to a higher momentum (and hence energy) and thus exert a greater pressure. – Rob Jeffries Mar 17 at 7:08

In Layman's terms, the Pauli exclusion principal wouldn't need to be overcome to form the black hole. A Neutron star of a certain size will shrink below it's Schwarzschild radius naturally. That's not hard to see. In fact, like white dwarfs, Neutron stars grow smaller in radius as they gain mass. The maximum mass wouldn't be much more than 2.5 or so solar masses past which the Neutron Star couldn't avoid becoming a black hole.

The relativistic effects get complicated, such as what precisely happens at the 100% time dilation and beyond.

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Now, as to what happens inside the black hole, there's two general points I can make. One is, as the neutrons (quark matter, whatever it is), grows more compact the weight and force to compact it further keeps increasing. That's fairly obvious. It almost becomes the unstoppable force (weight and gravitation) vs the immovable object (Pauli exclusion) question. The problem with knowing exactly what happens is essentially the singularity problem. The math breaks down. I don't think anyone knows.

Another way that I like to look at it, is Gluons, like photons, move at the speed of light. Inside a black hole, Gluons, like Photons would be drawn towards the center, not able to fly outwards and that property might greatly shrink the size of a Proton or Neutron down to the size of . . . maybe an Electron?? but again, who knows? Maybe some kind of quantum tunneling keeps the size of the Neutrons somewhat consistent but the gravitational escape velocity exceeding the speed of light could greatly reduce the more standard/observed size of the Neutrons. (I think).

I know you asked for the most accepted explanation and I've only touched on this from a layman's POV, so, hopefully someone with a bigger brain than me will answer this one more precisely to your specific question.

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Let me upvote and make a positive comment. This summary above is quite accurate, but you could make the point in the first paragraph that as you add mass to a neutron star, it gets smaller. Secondly, once collapse to a black hole is established it is not a question of forces; there is (in GR) no possible avoidance of collapse to a singularity in the same way that outside a black hole it is not possible to stop moving forward in time. – Rob Jeffries Mar 25 at 10:15

I can see why your drawing the comparison with white dwarfs, but in reality we don't yet understand the effects of gravity and pressure well enough at those densities to be able to know for sure if an accreting Neutron Star will collapse directly into a black hole or go into some intermediate phase like the quark stars you mentioned.

As far as I've always been taught the accepted view is that the Pauli exclusion principle only works so far, and once that is overcome by gravity the star collapses into a singularity and forms a black hole. In my time studying on my undergraduate course I heard no mention of quark stars or any other intermediate step (not that I was looking though).

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My question is why the Pauli exclusion principle can be overcome. Isn't impossible for two fermions to occupy the same quantum state? – Sir Cumference Mar 16 at 20:07
Ah right, well i found this question on the physics page which might or might not answer your question. physics.stackexchange.com/questions/93988/… – Dean Mar 16 at 20:18
There is no question of the PEP being "overcome by gravity". It works just fine at any density you like. – Rob Jeffries Mar 16 at 23:06
Ben Crowell's answer is the correct one, though he doesn't explain why the PEP can't balance gravity at higher densities (see my answer). – Rob Jeffries Mar 16 at 23:10

Yes, if there is an astronomical body near it, and the star's gravity feeds itself from the nearby body.

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Your answer is very short and could be improved by extending and explaining further – James Kilfiger Mar 25 at 16:53