Exploring the Pauli Exclusion Principle and Black Holes

[Alfred Cann]

The event horizon of a black hole appears to be plastered with 'afterimages' of everything that ever fell into it. (Because gravitational time dilation makes every such object appear to stop at the event horizon.) Now, suppose an event horizon is 'full' as defined by the Pauli exclusion principle. If one more fermion falls into the black hole, would its afterimage violate the exclusion principle?
I think maybe, by the following reasoning. The event horizon radius is proportional to the black hole's mass. If an object of x% of the black hole's mass falls in, the event horizon radius increases by x%. The volume of an infinitesimally thin shell at the event horizon increases by x% cubed.
The volume of an object of a given mass depends on its density, which, for a proton is E18 kg/cubic meter. The average density of the volume inside the event horizon of a one solar mass black hole is 1.85 E19. Thus, in this case, it wouldn't fit, so would try to violate Pauli. What would happen? There are cases where Pauli gets violated, for example, the collapse of a white dwarf to a neutron star. For black holes exceeding about 20 solar masses there would be no problem. Electron density is not well defined, so I don't know what would happen in the case of an electron.
Your comments please.

 

[phinds]

I am at a loss to figure how you think the exclusion principle has anything to do with red shifted photons. What am I missing?

 

[Alfred Cann]

phinds,
To an outside observer it looks as if these objects are actually, physically, at the event horizon; not just images. They must, therefore, obey Pauli unless some great force forces them closer.

 

[Drakkith]

Now, suppose an event horizon is 'full' as defined by the Pauli exclusion principle.

That is not possible. Infalling objects see no barrier and free fall right past the horizon without even knowing.

here are cases where Pauli gets violated, for example, the collapse of a white dwarf to a neutron star.

The exclusion principle is never violated, even during the collapse of a white dwarf into a neutron star. In this process electrons, which are the dominant particles holding up the star, combine with protons to form neutrons and allow the star to collapse. At no point are particles ever in the exact same state.

 

[Chronos]

The perception of an infalling body 'freezing' at the event horizon for remote observers is purely an illusion. As Drakkith noted, the infaller sails right though unaware of any perceptual illusions suffered by remote observers. To do otherwise suggests a black hole can never actually form.

 

[sophiecentaur]

Now, suppose an event horizon is 'full' as defined by the Pauli exclusion principle.

If you are trying to look at the event horizon as a region with volume -which would be necessary for it to get "full" then how would you define this volume? Pauli is a QM concept and Black Holes are a GR concept so it is not surprising to find some contention when you try to bring them together.
I think this is a bit of an 'Angels on a pinhead' conversation.

 

[phinds]

phinds,
To an outside observer it looks as if these objects are actually, physically, at the event horizon; not just images. They must, therefore, obey Pauli unless some great force forces them closer.

As others have now pointed out, you misunderstand what is happening.

 

[Papa Doyle]

The mass of a black hole is proportional to the area (not diameter) of the event horizon. Every time anything falls into a black hole the event horizon gets bigger. Those two things should sort out the mystery. Also, it does not seem necessary to invoke Pauli to consider these questions except that, outside the event horizon, everything must obey the Pauli exclusion principle - I'm not sure anyone has a theory about PEP inside the horizon.

 

[sophiecentaur]

everything must obey the Pauli exclusion principle

But Pauli only says that Fermions can't share the same energy state. Under those conditions, surely the number of energy states would just go up and become a continuum. That's ignoring anything flashy like GR.

 

[Clever Penguin]

If one more fermion fell into the black hole, the area of the event horizon would increase.

 

[phinds]

If one more fermion fell into the black hole, the area of the event horizon would increase.

Yes, Papa already pointed this out in post #8, so what's your point?

 

[Alfred Cann]

That is not possible. Infalling objects see no barrier and free fall right past the horizon without even knowing.

I'm talking about what an outside observer sees: the object stops at the EH and stays there forever.

The exclusion principle is never violated, even during the collapse of a white dwarf into a neutron star. In this process electrons, which are the dominant particles holding up the star, combine with protons to form neutrons and allow the star to collapse. At no point are particles ever in the exact same state.

Sophie Centaur suggests a good solution: the generation of additional energy levels as needed.

The perception of an infalling body 'freezing' at the event horizon for remote observers is purely an illusion. As Drakkith noted, the infaller sails right though unaware of any perceptual illusions suffered by remote observers. To do otherwise suggests a black hole can never actually form.

Would you call the different observations of different observers in SR illusions? Then which one is true? No, this is not an illusion; it is reality for an outside observer. Of course, the infaller experiences no barrier.

If you are trying to look at the event horizon as a region with volume -which would be necessary for it to get "full" then how would you define this volume?

Good point. I suppose I was assuming a thickness like the diameter of the fermion. But if the thin shell representing the increase of the EH has a volume like that of the infalling fermion, it would obviously be much thinner. I don't understand the rest of your answer. Clearly, I don't know enough.

The mass of a black hole is proportional to the area (not diameter) of the event horizon.

Wrong. The Schwarzschild (ER) radius is proportional to the mass.

Every time anything falls into a black hole the event horizon gets bigger. Those two things should sort out the mystery.

Yes but how much bigger? My concern is whether it's big enough. For a big enough BH it obviously is, because of the low average density. But for a smaller BH it may not be.

 

[Drakkith]

Would you call the different observations of different observers in SR illusions? Then which one is true? No, this is not an illusion; it is reality for an outside observer. Of course, the infaller experiences no barrier.

You can call it what you'd like. The fact remains that the infalling particle hits nothing at the horizon and the exclusion principle doesn't apply. In fact, I don't think the observer ever actually sees the particle hit anything, so the exclusion principle wouldn't apply even from the observer's frame. The only way for the observer to need to use the PEP is for the particle to actually hit something. As the particle approaches the horizon, I expect that the observer sees it approach, but never hit, the matter that has fallen in before it. However, I admit that I'm not up to speed on the details of SR and GR, so I'd appreciate if someone could verify this for me.

 

[Khashishi]

Pauli exclusion principle is not violated when a white dwarf collapses into a neutron star. Rather, the electrons are pushed to higher energy levels. The extra energy comes from a drop in gravitational potential energy. This is only possible if the gravitational field is strong enough, so it puts a limit on the white dwarf mass. The lower energy levels do fill up, but there are always higher energy levels available. A neutron star is held up by neutron degeneracy pressure instead of electron degeneracy pressure, but similar story.

We definitely don't understand quantum gravity yet, so we really can't say anything definitive about the states within a black hole. I'm not sure if fermions exist inside a black hole, because at some point the degeneracy pressure for fermions will be so large that they will spontaneously convert to other forms of energy, such as bosons. Bosons do not have any problem piling up in the same state. However, a black hole has large entropy, so the internal state is probably very complex.

 

[sophiecentaur]

Rather, the electrons are pushed to higher energy levels.

But, under those densities, there would be more and more energy levels available. Pauli is the least of your worries in explaining what goes on under those conditions. It seems to me that the step from everyday QM to QM under strong relativistic conditions is equivalent to the step from Classical atomic theory to QM.