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Black Holes & Fermi exclusion principle

  1. Aug 14, 2003 #1
    After reading "Hyperspace" by Kaku, I was slightly puzzled after his chapter on collapsing stars.

    He states that white dwarfs and neutron stars remain stable, because the exclusion principle counter-acts the gravitational collapse.

    He then sais the stars who have reached the end of the fusion cycle and who are composed of mostly iron have no forces balancing gravity anymore, and hence these collapse to a point.

    My question is, where does the exclusion principle go here? Why would it suddenly be possible to precisely know the quantum state of the matter inside the singularity? Obviously I'm missing something here, but some insight you might would be greatly appreciated.

  2. jcsd
  3. Aug 14, 2003 #2
    White dwarfs remain stable due to electron degeneracy pressure, which comes from the Pauli Exlusion principle. Basically, this says that no two electrons can share the exact same quantum state (especially when we compress them very close together, as in a white dwarf). As the star collapses, the electrons gets pushed very close together, allowing us to know their position better. According to the Heisenburg Uncertainty Principle, this means the uncertainty in their momentum must increase, resulting in a larger "momentum space" if you like. Ultimately it is this spread in momentum that generates our outward pressure.

    For large stars, gravity is strong enough to overwhelm this force (which is very strong) and the matter continue to gets mashed together. Electrons combine with protons to form neutrons, which are fermions as well. When they get close enough together, we get neutron degeneracy pressure, which is what holds a neutron star together against gravity.

    Ultimately, very large stars have strong enough gravity to overcome even this. These will form black holes.

    As far as the uncertainty principle (or exclusion principle, whatever you like) within the singularity, all I can say is that the normal laws of physics do not apply there. It is such an exotic state of matter, that no one really knows what is going on inside. However, it is safe to say that the Pauli exclusion principle as we know it no longer applies.
  4. Aug 14, 2003 #3


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    Frank Shu's astrophysics text "the Physical Universe" provides the formulas you need to calculate the size of a neutron star of any given mass

    this is controlled by the degeneracy pressure, or fermion pressure, of neutrons

    at a central density rho there is only so much fermion pressure and unless it is enough to balance the central pressure of a self gravitating sphere of given mass and size, the star will collapse further----until (and if) the central fermion pressure rises to where it equals the hydrostatic pressure at the center.

    so one calculates the size of a neutron star of mass M by setting two formulas equal and solving for R.

    It will turn out that MR3 is a constant (I will tell you later this constant).

    So if you could increase the mass of a neutron star by 8-fold, the volume would shrink 8-fold and the radius would shrink by half.

    the more massive neutron stars are the smaller ones.

    At some mass (perhaps 3 solar masses) the radius is less than the Schwarzschild radius----PRESTO: the neutron star is a black hole. It is all inside its event horizon. What things are like inside there we cannot yet say.

    Do not worry about "singularity". this word refers to fact that GR breaks down at very high density. The GR theory reaches the limits of its applicability and fails to compute. When there is a quantum theory of general relativity then perhaps we humans will understand better about inside the Schwarzschild radius. But for now we can say "it is a black hole, now the star is all inside the event horizon."

    THE HEISENBERG does not prevent a neutron star from being compressed to size of one cubic meter! All Heisenberg says is that when the Δx of the pointparticle is very small its momentum uncertainty Δp is very large!!!

    If you have many neutrons or quarks in a box (a kind of fermion gas) and you compress it this increases the kinetic energy of the gas inside. So you are doing work! So there is pressure!!!

    The Heisenberg just says how much pressure.

    You can still keep squeezing the box, if you are strong enough.

    They are only points. And Δx can be made very small.

    I will get the approximate formula for the fermion pressure (in terms of the number density of neutrons----or equivalently in terms of mass density) and post it later.

    The analysis is very similar to the way the Chandrasekhar mass formula is derived---but that concerns the fermi pressure of electrons being overcome, allowing collapse to neutron matter.

    We next must consider the fermi pressure of the neutrons being overcome to the point that a black hole forms.

    WHOAH! I had to be away from computer and I see now futz has answered this. My post is not really needed. If you want the formulas for fermion pressure and size of neutron star, just say. Otherwise I will forget it.

    Last edited: Aug 14, 2003
  5. Aug 15, 2003 #4
    Thanks for that. I was indeed worried about the thing being an actual mathematical point (whatever that may be :smile: ). I suspected it had something to do about us not being able to observe the thing once it had collapsed to a black hole. That and the fact that at those distances GR does indeed break down.

    I had forgotten completely about the Schwarzschild radius. So I guess I wasn't too confused after all. Yay!
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