Re: black hole and special relativity

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Regarding Brian's comment:

> Chandra comes up with a similar number (epsilon = 0.94) when considering
> the limiting radius of degeneracy for a 'white dwarf' in
> Stellar Structure
> Chapter XI 'Degenerate Stellar Configurations'
> S Chandrasekhar
> UChicago, 1939.

Why do you call this a similar number?  The radius of a white dwarf is
between 3 and 4 orders of magnitude greater than the Schwarzschild radius
corresponding to their mass. The Chandrasekhar limit on white dwarf mass
has *nothing* to do with general relativity and black holes. It has to do
with the relationship between the density dependence of the pressure
developed by the high density of the degenerate electrons (i.e. the
equation of state for degenerate electrons) and the mass dependence of
gravitationally produced pressure (in a fully Newtonian regime) at the
center of the dwarf caused by the gravitational weight of the overlying
layers of the dwarf, where the density of each layer is self-consistently
determined by the local equation of state and the cumulative weight of
the layers farther out from the center.  The Chandrasekhar limit occurs
when the mass of the whole dwarf gets so high that there is no longer a
self consistent solution for the radial density/pressure profile because
the gravitational pressure from the increase in weight by further
compression of the degenerate medium increases with mass faster than the
needed supporting degeneracy pressure does.

I suppose one could call the upper bound on a white dwarf mass 'similar'
to the GR bound on the minimal radius bound a dense object must have to
be able to prevent its collapse to a black hole in that in both cases a
threshold bound exists which, when violated, makes further collapse
inevitable.  But the GR bound is a bound on the maximum density (or
equivalently, a minimum radius) allowed for a *given* mass of any
amount that could conceivably be capable of resisting the object's
collapse to a black hole, whereas the Chandrasekhar limit is a upper
bound on the *total mass* for an electron degenerate system.  One is a
bound on the degree of compression allowed for a given mass, and the
other is a bound on the total mass allowed.

So even though these two bounds may have at first blush a somewhat
superficial resemblance, they really are quite different in character
(not to mention in the physics involved as well).

Come to think of it, as I recall, even at the Oppenheimer-Volkhoff limit
for the maximum mass that a neutron star can have before it must collapse
to a black hole, its radius is, I think, something like a half of an
order of magnitude greater than its Schwarzschild radius would be for
a Schwarzschild black hole of the same mass.  So even in this more
relevant case the GR bound of 9/8 < R/r_s is not very closely approached.

David Bowman
David_Bowman@georgetowncollege.edu