Black holes - a classical look
REFERENCE: None
Let m be a mass in the near vicinity of the surface of M (of planetary
size). Recall that the potential energy stored in the configuration of
the two masses m and M is given by
[Eq. 1]
U = -GMm/r.
Since the total mechanical energy E of the system is given by
E = K + U,
[Eq. 2]
E = K + U.
[Eq. 3]
we have
where K is the kinetic energy of the system.
then E = 0 so
K + U = 0.
If energy is conserved
[Eq. 4]
Of course, Eq. 4 is just a statement of the conservation of mechanical
energy. Suppose the small mass is given enough kinetic energy that it
is able to completely escape the gravitational pull of M. Taking into
account that the kinetic energy of m is K = mv2/2, Eq. 4 can be expanded
to read
[Eq. 5]
mv2/2 - mv02/2 + (-GMm/r) - (-GMm/R) = 0,
where R is the radius of the mass M (assumed to be a sphere) and
the initial velocity of m.
In order to escape completely from
must have its initial velocity at least great enough that m can
infinity and come to rest. We set the following conditions on Eq.
v0  vesc, r = , and v = 0.
v0 is
M, m
reach
5:
[Eq. 6]
Substitution of the conditions of Eq. 6 into Eq. 5 yields
vesc =
2GM
R
1/2
.
[Eq. 7]
During the Great War, a German artillery officer named Karl
Schwarzschild read Einstein's general theory of relativity.
The 1915
theory offered as yet unsolved equations describing a new theory of
gravitation.
Schwarzschild found a solution to what was called the
Einstein Equation.
He sent it to Einstein, and died shortly
thereafter.
Schwarzschild's solution predicted the existence of a black hole - an
object whose mass-to-radius ratio was such that even light could not
escape from it. We will derive this relationship simplistically here,
duplicating the effort of Laplace a century earlier.
We reason:
If the speed of light c is the limiting velocity in the
universe (Einstein's special theory of relativity, 1905), then if we
set
[Eq. 8]
vesc = c
in Eq. 7, we'll have that ratio. Doing so, we obtain what we will call
the Schwarzschild radius, the critical radius that a body of mass M
must reach in order to become a black hole (light is not able to
escape):
2GM
[Eq. 9]
RSch =
.
c2
We call that critical radius RSch the event horizon. The event horizon
is that distance from the center of a mass from which light cannot
escape. It is predicted that the actual radius of an object that is a
black hole collapses to 0.
Thus, when one speaks of the radius of a
black hole, one is really speaking of the radius of the event horizon.
You can use Eq. 9 to calculate the critical radius the sun or earth
would have to be squished to in order to become a black hole.
For
example, the sun would have to collapse to a radius of 2950 m (left as
an exercise).
You can see looking at Eq. 9 that even if the radius of an object does
not decrease, a black hole can be created by increasing the mass M (as
long as the radius doesn't increase too).
Stars in binary systems
often do just that - sucking matter from a smaller partner. If one of
the partners happens to be a neutron star (which we'll study later), it
turns out that its radius decreases with an increase in mass!
By using the formula for density,
 = 3M/4RSch3,
[Eq. 10]
we can show that
RSch =
3c 2
8G
1/2
.
[Eq. 11]
From Eq. 11, you see that you can get a black hole using material of
any density, so long as the radius is large enough.
As a final note on black holes, our derivation (like Laplace's in the
19th century) contains two errors which fortuitously cancel out. They
are...
(1)
Light does not satisfy K = mv2/2 because it has no m associated
with it.
(2)
Newton's law of gravitation does not hold when even light finds
it difficult to escape a massive body.
(3)
One last unrelated bit of information.
Schwarzschild did not
coin the term "black hole." John Wheeler did (in the 1960s).