starship-design: Blackhole

[Steve VanDevender <stevev@efn.org>]

Timothy van der Linden writes:
 > >I believe Isaac is right about this.  Very small black holes evaporate
 > >rapidly, to the point of being violently explosive.  The evaporation
 > >rate is a function of the gravity gradient at the event horizon, which
 > >goes asymptotic as the mass of the hole approaches zero.  Imagine the
 > >energy release from a few hundred tons of mass turning into a spray of
 > >high-energy subatomic particles within a tiny fraction of a second.
 > 
 > I already wrote that it is said that smaller black holes evaporate faster,
 > and I also assumed the radiation per surface area was bigger.
 > The total surface however is getting smaller too. My suggestion was that the
 > total surface got smaller faster than the emission rate per surface area
 > gets bigger.
 > So you didn't really address the point I made.
 > After having thought things over, I conclude that the radius decreases
 > linearly with the mass (R=2G*M/c^2). So the surface decreases quadratically
 > while the mass decreases linearly.
 > The question is: Does the "gravity gradient" increase linearly,
 > quadratically or even faster with the radius decrease? 

My understanding (which is blissfully free of any messy equations :-) is
that the rate of virtual particle creation is proportional to the rate
of change of gravitational force near the black hole's event horizon.
Since the force of gravity at r is proportional to 1/r^2, then the
derivative of that is proportional to 1/r^3.  So although the surface
area of the black hole is proportional to r^2, the rate of evaporation
is proportional to r^2/r^3 or 1/r.  Furthermore the Schwarzchild radius
for a "classical" black hole is directly proportional to the black hole
mass.  So despite the shrinking surface area the rate of evaporation
increases asymptotically as the black hole shrinks and the black hole
finishes evaporating with an impressive bang.