Re: degeneracy pressure

[David Bowman <dbowman@TIGER.GEORGETOWNCOLLEGE.EDU>]

Regarding Justin's question:

> Maybe I can rephrase my question:
>
> Does "overcoming" degeneracy pressure violate the Pauli exclusion
> principle, and if the answer is no, why not?

The answer *is* no.

The Pauli Principle requires that a collection of identical
(otherwise noninteracting) point particle fermions have a degeneracy
pressure (i.e. ground state pressure or, more conveniently, the
limiting pressure when the temperature is tiny compared to the Fermi
temperature) that is a function of the particle density.  When the
Fermi energy is tiny compared to the rest energy of the particles
then the particles are all quite nonrelativistic in energy and then
the degeneracy pressure scales proportional to the 5/3 power of the
particle density.  When the density is so high that the Fermi energy
is huge compared to the rest energy of the particles then the partial
degeneracy pressure contributed by each species of particles present
scales as the 4/3 power of the density.  As a rarified degenerate
Fermion gas is compressed in the presence of an isothermal heat sink
the degeneracy pressure begins to grow as the 5/3 power of the gas
density until the fermions near the Fermi energy begin to have
relativistic single particle energies.  At this point the degeneracy
pressure continues to rise with increasing density at a rate which is
slower than the 5/3 power but faster than the 4/3 power.  As the
density is further increased as the system is further compressed so
that the vast majority of occupied states have single particle
energies much larger than the rest energy of the individual fermions
then the degeneracy pressure asymptotically continues to rise as the
4/3 power of the particle density.

Note that once the gas is so compressed that its ground state has a
significant fraction of its particles occupying relativistic states
then the gas will be begin to have spontaneous particle/antiparticle
pairs created and destroyed, and the gas will necessarily become a
mixture of many species of particles and their antiparticles as these
particle transmutations begin to occur spontaneously--even when the
temperature is effectively near zero temperature.  When this happens
the degeneracy pressure contributed by the original particle species
is merely the *partial* pressure contribution of that species.  The
total pressure of the whole mixture is then the sum of the partial
pressures of all the species present.

Since the particles (supposedly) point particles they can be, in
principle, compressed as close to each other as desired.  The only
penalty is that higher densities correspondingly, have higher
degeneracy pressures and require higher applied external squeezing
pressures.

Increasing the density of the fermion gas increases the energy of
the allowed occupied states because the decreasing volume of the
container increases the energy spacing (and the ground state
energy level) of the single particle particle-in-a-box energy
levels.  When the fermions are stacked into a set of spread out
levels they have more energy than when those levels are close
together.

> I would think it *does* violate the principle if particles which
> were previously prohibited from being close to each other are
> "forced" by gravity to be close to each other.

The single particle states are delocalized.  The concept of "close
together" in space doesn't really enter much.  The particles are
essentially separated by energy and momentum--not by space.

> Obviously I don't fully understand what is going on here.
>
> When we speak of "overcoming" gravity what we really mean is that a
> force is exerted on something which is greater than or equal to the
> force which gravity exerts and we can move the object against
> gravity.  Gravity is not really "overcome".

In the case of the gravitational collapse of a neutron star into a
black hole what happens is that once the mass of the star's nuclear
matter exceeds the Oppenheimer-Volkhoff limit then the pressure in
the center of the star due to increasing degeneracy pressure doesn't
increase fast enough to support the increasing gravitational weight
of the star's higher layers, and the whole thing contracts in a
runaway process.

In general relativity it is possible to show that the central
pressure required of a self gravitating mass of matter to support it
against further gravitational contraction diverges toward *infinity*
at a *finite* (but large) maximum central density of the core as the
object is shrunk.  We see that since the degeneracy pressure of any
collection of fermion matter only diverges at an infinite density and
is always finite at finite densities there must come a point where
once the mass of the star gets too large the star cannot be
supported and the gravitational contraction simply cannot be averted
by the degeneracy pressure.
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The reason why the needed supporting pressure diverges at a finite
density in general relativity is that in GR the sources of
gravitation (i.e. Ricci curvature of spacetime) include not only the
mass (and the energy) of the matter involved, but also the momentum,
the stress and the pressure of that matter.  Thus, when a star is
compressed and the matter is squeezed the gravitational forces
increase both because the matter is closer together a la the
Newtonian theory *and* because the internal pressure and stresses
are building up and that increased pressure increases the
gravitational field produced by the matter.  This nonlinear
positive feedback situation becomes so severe when the system's
mass is above the Oppenheimer-Volkoff limit that as the system
contracts and the internal pressure goes up the gravitational
forces go up much faster than would be the case with purely
Newtonian gravity (where the attractive force between point masses
is only an inverse square of their separation) and the increasing
gravitation produced by both the increasing mass density and the
increasing pressure makes the pressure required to support the
system against further collapse diverge at a *finite* central
density.

The best (worst depending on your point of view) case scenario in GR
is for a star whose density distribution is such that the star's
matter is so stiff it can be thought of as being a uniform density
star (the nuclear matter in a neutron star obeys this relatively
nicely up until close to the Oppenheimer-Volkhoff instability point).

In such a uniform density star it is possible to exactly solve
Einstein's equations of GR and the result is that in such a star the
central pressure diverges to infinity once the star's radius
(in Standard Coordinates) becomes as small as 9/8 of its
Schwarzschild radius.  If the star has a more realistic density
profile so it is denser on the inside than the outside, then
the radius where the central pressure diverges is *even larger* than
9/8 of the Schwarzschild radius.  Thus, the *theoretical minimum*
size of a stable spherical distribution of non-rotating self-
gravitating fixed amount of matter is a sphere whose radius (in
Standard Coordinates) is 9/8 of the Schwarzshild radius
corresponding to the amount of the mass of matter present.

> Since there is really
> no "force" keeping neutrons from collapsing into each other what is
> there to "overcome"?

The pressure due to the kinetic energy of the energetically stacked
particles is overcome.

> Thanks for your indulgence

I hope this explanation helps.

> Justin

David Bowman
David_Bowman@georgetowncollege.edu