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...
One really weird thing about the above formula for the entropy is that
it requires that black holes have a negative heat capacity. This means
that any equilibrium they might have with an external heat bath is
unstable and violates LeChatier's principle. The heat capacity of an
object is related to the negative of the second partial derivative of
the object's entropy w.r.t. its internal energy (i.e.
d^2 S/dE^2 = -1/(C_v*T^2) . A normal system has a stable equilibration
behavior w.r.t. heat transport when its heat capacity is positive, or
equivalently, when it's entropy is a concave down function of its
internal energy. Now black holes have an entropy that is proportional
to their horizon area and that area tends to be proportional to the
horizon's radius, and the (gravitational) radius tends to be
proportional the hole's mass, and the mass is just the internal energy
as measured in kg rather than J. Thus the entropy tends to be
quadratic in the internal energy. This is a parabolic function that
curves upward with a positive second derivative. Another way to see this
is to realize that the temperature is proportional to the surface
gravity (at the horizon) and the surface gravity is inversely
proportional to the horizon radius and the radius is proportional to
the mass/energy. Thus, as the mass/energy goes up the surface gravity
and the temperature go down. This opposite tendency for changes in
energy and in temperature is a manifestation of the hole's negative heat
capacity.
To see what this unstable negative heat capacity implies, let's consider
a black hole in equilibrium with a surrounding heat bath at the same
temperature. In this equilibrium the hole radiates as much energy out
into the heat bath as it gets back from it in the form of radiation
descending through the horizon. Now suppose a thermal fluctuation occurs
that slightly gives the hole a little more energy from the bath. This
fluctuation increases the hole's mass which *decreases* its temperature
and this establishs a temperature difference between the hole and the
bath. Heat transport will occur across this temperature difference with
the energy going from the hotter bath to the colder hole. This will cool
the hole's temperature further. This will cause the hole to eat ever
more and more of the energy of the bath until it gobbles up the whole
bath. OTOH, suppose the initial fluctuation transfered a little energy
from the hole to the bath. Then the hole's temperature would increase.
This would make the hole hotter than the bath and there would then be a
net flow of energy from the hot hole to the cold bath. As the hole
radiates away its mass its temperature gets ever hotter thus increasing
the radiation rate until the evaporating hole disappears in a final last
explosion of radiation.
We can think of the 2.76 K Cosmic Background as a Heat bath for black
holes that are not near other energy sources (i.e. stars, etc.). If such
a hole has a mass that is greater than about 4.45 x 10^22 kg (about 60%
of the mass of the Moon) it will not evaporate, but will grow ever
larger. But if the hole has a mass less than this amount it would
eventually evaporate (assuming it is not fed by some other nearby
radiation or matter).