Black hole - contradiction to the Pauli exclusion principle?

[raul_l]

Hello.

I was thinking. The collapse of a stellar nucleus into a black hole is an apparent contradiction to the Pauli exclusion principle, right? So which one of those theories fails at that point - quantum mechanics or the theory of relativity? I used to think that it's the theory of relativity because it's accurate in predicting the existence of black holes but not very good at describing what's inside them. Also, to me the exclusion principle seems too fundamental to fail.
However, recently I took a course is astrophysics and if I understood correctly then during the collapse the density gets higher than that allowed by the exclusion principle (the degeneracy pressure). And that made me confused.

Can anyone shed some light on this?

 

[HallsofIvy]

How is it a contradiction of the Pauli exclusion principle?

 

[raul_l]

Well, the Pauli principle states that two quantum states can't occupy the same quantum state. Now, taking the uncertainty principle also into account it means that every particle has to occupy a finite volume of space. So there's a distance $$\delta x$$ between two particles and you shouldn't be able to push those two particles closer than that without violating the Pauli principle. Right? However, in order to form a black hole you have to push matter further together and exceed the limits set by the Pauli and Heisenberg principles until a gravitational collapse occurs.

Of course, I might be wrong about these things. I've had some courses in quantum mechanics and some courses in astronomy but when it comes to black holes (and especially their formation) I'm unable to connect the two.

 

[DaveC426913]

We don't know what the laws are that dictate what happens at the centre of the BH, but matter stops being matter at some point in the compaction process.

Consider, by comparsion, what happens to matter as it is cooled to near absolute zero. The atoms smear out into what's called a Bose-Einstein condensate.

I'm not suggesting this is what happens at the centre of a BH, I'm merely pointing out that the well-behaved atomic state of matter has boundaries.

 

[humanino]

So there's a distance $$\delta x$$ between two particles and you shouldn't be able to push those two particles closer than that without violating the Pauli principle. Right?

No. A state is not only a spatial position. It is also a momentum, a spin... So you can have two particles arbitrarily close to each other provided they have different momenta. You can also have one single particle located arbitrarily well provided you have enough uncertainty on its momentum. It is not unconceivable that, as the particle gains momentum while approaching the singularity, the uncertainty of the momentum increases, compensating for the uncertainty in the position decreasing. As was mentioned already, we do not know the full quantum theory of gravity. However, as such, it seems unlikely that the arguments presented should be a problem.

 

[Naty1]

Good question...as noted above nobody knows what goes on at the singularity but likely our understanding of time and matter and energy will be profoundly changed when we do.

 

[raul_l]

Ok, so the exclusion principle still holds. But now I'm wondering about another thing.
The degeneracy pressure comes from the fact that as particles are pushed tightly next to each other their momentum increases and momentum is what constitutes pressure.
$$\Delta x \Delta p \ge \frac{\hbar}{2}$$
So the denser the matter the greater its degeneracy pressure. By that logic we should never be able pack matter to tightly that it would form a black hole because the degeneracy pressure would reach infinity. But as we know black holes still form at some point. My guess is that as the pressure from gravity is increased the degeneracy pressure is also increased but not as fast, i.e. gravity becomes more dominant as the mass is increased. The two become equal at the Chandrasekhar limit at which point gravity takes over and nothing can stop the gravitational collapse.

 

[ZikZak]

So the denser the matter the greater its degeneracy pressure. By that logic we should never be able pack matter to tightly that it would form a black hole because the degeneracy pressure would reach infinity. But as we know black holes still form at some point.

No, by that logic, matter should collapse more quickly, not less quickly than without the pressure. Pressure is contained energy. In the relativistic regime, it therefore gravitates. Pressure only acts outwards in a nonrelativistic star. Add enough matter, enough pressure, to a star, then the momenta become relativistic and the pressure acts to accelerate, not oppose, the collapse. This is what happens in a supernova, for instance.

 

[_PJ_]

As others have stated the nature of a singularity and its effects are known to break many 'laws' of physics as we know them. However, I would suggest that the energetic temperatures implied at the core of a black hole would likely restore symmetry to any particles before they reach a singularity, so the asymmetrical characteristics of these particles would be lost, therefore invalidating the exclusion principle.

 

[xantox]

So the denser the matter the greater its degeneracy pressure. By that logic we should never be able pack matter to tightly that it would form a black hole because the degeneracy pressure would reach infinity. But as we know black holes still form at some point.

A black hole forms when its absolute horizon forms – that can happen even at water density or less, for a sufficiently large hole. While the details of the "center" can only be described by a theory of quantum gravity (which could possibly remove the singularity) this should not affect the possibility for a black hole to form in the first place.