How long do black holes last?
1) The Schwarzchild radius of a Black Hole is the radius of the 'event horizon', past which even light cannot escape.
Find an expression for the Schwarzchild radius of a black hole using the notion of escape velocity (non-rigorous derivation :-o)
2) Over time, black holes lose energy and hence mass via "
Hawking Radiation". The power emitted by a Black hole is:
, where
, and
is
Planck's constant,
is
Boltzmann's constant, and
is the
speed of light,
is the surface area,
is emissivity, and
is temperature.
Since we may assume a black hole to be a perfect blackbody (perfectly absorbs and emits radiation),
.
Given that the temperature of a black hole is:
, where
is the
Gravitational constant, and the energy of a black hole is
, form a differential equation to find the lifetime of a black hole in terms of its initial mass.
3) How long would a black hole the mass of the sun last (
)? How about the mass of a proton (
)?