I
will actually have an intelligent answer for this.
Really! So
keep reading!
As
far back at 1798, the French mathematician PierreSimon Laplace (1749  1827)
postulated the existence black
holes, using Newton’s theory of gravitation.
He calculated that if a star were massive enough and if its
radius were small enough, the force of gravity near the star would
be so strong that not even light could escape from it.
Fast forward to the 20^{th} century.
In 1939, Robert Oppenheimer (1904 – 1967), “the father of
the atomic bomb,” and one of his students Hartland Snyder (1913
– 1962) made similar calculations using the more refined formulas
of relativity. A star of
sufficient mass would undergo a final catastrophic collapse to a
tremendously high density and to a size less than or equal to the
“Schwarzchild radius.” (That
is 2GM/c^{2}, where G
is the gravitational constant, M
the mass of the star, and c
is the speed of light in a vacuum.)
The result would be a black hole, which any light emitted
from the star would be dragged back by the gravitational field.
Hmmmm. Didn’t
Laplace come to a similar conclusion about two centuries
earlier?
According
to Einstein’s theory of special relativity, nothing in the
physical universe can travel faster than the speed of light.
So if light is not fast enough to escape a black hole, how
can anything else escape? It’s
been said, there are either holes like this in the universe, or
there are holes in the theory of relativity.
So the search began in the heavens for black holes, yielding
many likely candidates.
In
the constellation Cygnus, Cygnus X1 was discovered in 1964 as one
of the strongest Xray sources seen from Earth.
As the invisible member of a binary star, Cygnus X1 sucked
material gravitationally from its visible companion.
This material formed a rotating disk, which astronomers
detected by its Xrays. In
1990, after losing a bet to physicist Kip Thorne, Professor Stephen
Hawking conceded the observational evidence made a strong case for a
gravitational singularity at Cygnus X1, that is, a black hole.
Cygnus
X1 was one of the first evidences to show that black holes were not
just creations in mathematics or science fiction, but actual
physical entities. Since
then, the evidence for black holes has become so overwhelming that
astronomers now treat them as normal fixtures in the universe.
Millions of ordinary black holes are believed to be peppering
each of the estimated 50 billion to 100 billion galaxies in the
universe. Giant black
holes are believed to punctuate the centers of most galaxies, even
our own.
So
what does it look like inside a black hole?
To answer this, I will focus on one of its classical
definitions  its escape velocity is the speed of light or greater.
The
total Energy E_{total}
of a particle of mass m
interacting with a planet of mass M
is the sum of its Kinetic
Energy and its Potential
Energy. Conservation
of energy dictates that this sum is a constant.
We have,
E_{total
= }Kinetic
Energy + Potential
Energy = constant
In
Newtonian Mechanics, particle m’s
Kinetic Energy can be defined as ½mv^{2},
where m is its mass, and v its velocity. The
Potential Energy of particle m
and planet M is GMm/R. (G
is the gravitational constant, M
the mass of the planet, and m
the mass of the particle. R
is the distance between the center mass of planet M and the center of mass of particle m.) Hence,
E_{total}
= ½mv^{2
}^{ }GMm/R =
constant.
If
E_{total} <
0, the particle m is bound to the planet M
(such as the moon is bound in its orbit about the earth and most of
us are stuck on this earth). The
particle m escapes planet M’s
orbit (such as the Mars Phoenix space probe escaped Earth), if E_{total}
≥ 0.
The minimum velocity, the escape
velocity v_{escape}, for particle m to break from the bonds of planet M is when E_{total}
= 0.
E_{total}

Comments

<
0
(less than zero)

Particle
m is bound
by planet M

≥
0
(greater than or equal to zero)

Particle
m is not bound
by planet M

=
0
(equal to zero)

Minimum
velocity in which particle m
escapes planet M

Hence,
E_{total}
= 0 = ½mv_{escape}^{2
}  GMm/R.
If
we solve for v_{escape}, we get
v_{escape}
= √2GM/R.
The
escape velocity does not depend at all on the mass of the particle m.
If we set the escape velocity for the speed of light c,
(which is about
300,000 kilometers per second)
then
v_{escape}
= c =√2GM/R.
Solving
for M/R, we see this ratio is a constant:
M/R
= c^{2}/2G
So
this classical condition of a black hole is solely dependent on its M/R
ratio, the mass of the planet M
and the distance R between
the centers of mass between particle m
and planet M.
In MKS units (meter (m) –kilogram (kg) –second (s)),
c^{2}/2G = 6.75
x 10 ^{26} kg/m.
If
we solve for R, we have the “Schwarzchild
radius”, R_{Schwarzchild}, as mentioned earlier in this
article:
R_{Schwarzchild}
= 2GM/c^{2
}
Note: For nonrotating black
holes, the Schwarzchild radius defines the area of its event horizon, the boundary in which events inside the black hole
cannot affect an observer outside the black hole.
Black
holes can be very, very dense. The
tidal forces (the difference in gravitational forces across a body)
can rip an object apart as it approaches the event horizon.
Yet, the M/R ratio shows that black holes need not be
extremely dense, only sufficiently massive.
The more massive the black hole, the less dense it has to be.
Below are some simple
calculations demonstrating that.
Let
M be the mass of the black hole, R its Schwarzchild radius.
Its mean density Mass/Volume is proportional to M/R^{3}.
For the sake of clearing up the clutter, let the constant K =
c^{2}/2G. Therefore,
the M/R ratio = K and the minimum mass required for a black hole of
radius R is M = KR.
The
table below shows the relative density based on our black hole’s
mass for different radii.
R

M
= KR

≈
M/R^{3 }(density)

1

K

K

2

2K

2K/8
= K/4

3

3K

3K/27
= K/9

4

4K

4K/64
= K/16

5

5K

5K/125
= K/25

…

…

…

n

nK

nK/n^{3}
= K/n^{2
}

As
we see in the table, as the mass of the black hole increases by a
factor of n, its mean density decreases by a factor of 1/n^{2}. Hence,
the more massive the black hole, the less its mean density needs to
be.
Now,
let’s crunch some numbers for familiar objects whose radii have
collapsed to the Schwarzchild
radius R.
Object

Mass

R_{Schwarzchild}

M/(4/3πR^{3})
(density)

Speck
of dust

10^{5}
g

1.5
x 10^{33} cm

7.3
x 10^{92} g/cm^{3
}

Earth

6
x 10^{24} kg

0.89
cm

2.0
x 10^{27} g/cm^{3}

Sun

1
M_{☼}

3
km

1.8
x 10^{16} g/cm^{3}

Star

2
M_{☼}

6
km

4.6
x 10^{15} g/cm^{3}

Galaxy

10^{11}
M_{☼}

3
x 10^{11 }km

1.8
x 10^{6}
g/cm^{3}

100
billion galaxies
(Universe??)

10^{22} M_{☼}

3.1 x 10^{9 }light years

1.8
x10^{28} g/cm^{3}

Note:
g is grams; kg
kilograms. M_{☼}_{
}is_{ }a
solar mass, the mass of our sun,
approximately 2 x 10^{30} kg.
cm is centimeters, km
kilometers. A light
year is approximately 9.5 x 10^{12} km.
As a reference, water has a density of one g/cm^{3}.
Looking
at the table above, if we collapse the matter of a speck of dust to
the Schwarzchild
radius, its mean density is about 10^{93}
g/cm^{3}. If
matter that dense were the size of the nucleus of an atom, it
approaches containing all the mass of the known universe.
(Now, that is very, very, very dense!)
For a mass the size of our galaxy, its Schwarzchild radius is 3 x 10^{11 }km,
which about 50 times the radius of Pluto’s orbit about the sun.
Its mean density is about 10^{6} g/cm^{3},
which is only that of a light gas.
Now,
consider 100 billion galaxies. Its
Schwarzchild
radius is 3.1 x 10^{9 }light
years, approaching the radius of our known universe, which seems to
have the upper limit of 4.6 x 10^{10 }light
years. The mean density
is about 10^{28} g/cm^{3}, which is an exceedingly
thin gas, an effective vacuum! If
the mass of the known universe were about 10 times more massive as
some astronomers have estimated (at 100 billion galaxies), then the
condition for M/R ratio would be satisfied for a black hole.
Astronomers
have used a formula based the massluminosity to estimate the mass
of galaxies and clusters. The
Virial Theorem (which allows
the average total kinetic energy to be calculated for complicated
systems) and
Kepler’s Laws (which
describe the motion of planets in a Solar System) predict
missing mass within clusters and galaxies that the massluminosity
relation does not account for. Theories
say dark matter makes up this missing mass, such as nonluminous
stars, neutrinos, ultradense black holes, and hot, gas like
material distributed throughout space.
And I’m sure I haven’t even scratched the surface on this
topic. That is for
another time, perhaps.
What
does it look like inside a black hole?
Just open your eyes and look around.
We may be in one!
There
are other factors than the M/R ratio to consider when defining a
black hole. In 1979, I
had the privilege of talking in person to physicist Kip Thorne.
(He autographed his article, “The Search for Black Holes”
in my “Readings from Scientific American – New Frontiers in
Astronomy.”) We
discussed the possibility of the universe being a giant black hole.
Dr. Thorne made a case that the universe does not meet the
boundary conditions for one. (1)
The body must be contracting.
Evidence shows the universe is expanding (and at a faster
rate than expected.) (2)
There must exist a vacuum between the black hole and its event
horizon. Does the
universe have an event horizon?
God only knows – I say this reverently.
Also, it does not seem
possible to send a satellite in orbit around the universe to
determine its total mass – as we would do about a planet or binary
star.
Some
Conclusions:
A
black hole does not need to be super dense and rip us apart, crush
us to pieces, and squeeze us out of existence with its tidal forces.
A solar system full of water would qualify as a black hole.
It is possible to be inside one and everything seems quite
normal. As for the
universe qualifying as a black, that is a matter of definitions,
philosophy, religion, and new science that goes beyond the dark ages
when I was in school.
