Given their bizarre properties we might question
whether a black hole could even exist.
This effectively comes down to the question:
Is it possible for an object to undergo gravitational
collapse?
In the early days of relativity theory most
physicists, including Einstein, intuitively
felt this was impossible, that some physical
process would prevent the density of matter
from growing large enough to create an event
horizon.
Imagine a star as a spherically symmetric
ball of gas.
At any spherical boundary within the star,
there will be a gravitational force pressing
inward, representing the weight of all the
stellar material outside the boundary.
We�ll quantify the effect of gravity by
the amount of energy required to pull the
star apart.
Viewing the star in cross section, imagine
we somehow take hold of the top layer of material.
And working against gravity, pull it away
to infinity.
We continue to pull the star apart.
Layer by layer.
Until nothing is left.
Calculating the total work required to disassemble
the star we find that it varies as the star
mass M squared over the star radius R multiplied
by a constant which we don�t bother with
here.
We define the gravitational energy of the
star to be the negative of this work of disassembly.
Physical systems tend toward the state of
lowest energy.
If gravity was the only force acting, this
would be an energy of minus infinity at R
equals zero.
Gravity wants the star to collapse.
For the star to resist collapse and remain
in equilibrium, the gravitational force must
be offset by an outward pressure of equal
magnitude.
The primary sources of stellar pressure are:
First, gas pressure, analogous to Earth's
atmospheric pressure although under much more
extreme conditions.
Gas pressure varies with temperature.
At higher temperatures, gas particles are
on average moving faster, and so transfer
more momentum when they collide.
Second is radiation pressure, due to transfer
of momentum by the thermal emission, absorption
and scattering of electromagnetic waves inside
the star.
This depends on temperature even more strongly
than gas pressure.
Now, both of these sources of pressure vary
with temperature.
At non-zero temperature a star will radiate
power into space.
This must be made up by an internal power
source.
Otherwise the star will cool.
Nuclear fusion is the primary source of this
power.
As light nuclei fuse to form heavier nuclei,
energy is released, but only up to a point.
For nuclei heavier than iron and nickel, fusion
requires a net input of energy, such as can
occur in a super nova explosion.
So, even though it may take billions of years,
a star is ultimately destined to exhaust all
internal sources of energy and grow cold,
bringing gas and radiation pressure to an
end.
What remains is the phenomenon sometimes referred
to as the �stability of matter,� the fundamental
explanation of why solid matter is solid,
the reason why we don�t fall through the
floor.
To be completely rigorous, we would need to
solve for the quantum mechanical state of
all nuclei and electrons in the frozen star
and the resulting forces opposing gravity.
Instead, what we will consider is a very rough,
back of the envelope type calculation of what
is expected to be the dominant effect at extreme
densities, so-called electron degeneracy pressure.
We first consider a non-relativistic analysis.
Let�s denote the star�s radius and mass
by big R and big M.
Assume that each electron, with mass little
m, is constrained to a region of radius little
r
and that there are n sub e electrons per unit
mass.
In most elements there is approximately one
electron per proton-neutron pair, so this
is more-or-less a fixed value.
The total number of electrons in the star
is then n sub e times the star mass.
This is also the volume of the star divided
by the volume per electron, which varies as
the cube of big r over little r .
Solving this expression for little r we find
that it varies as big r over the cube root
of big m . Here and in what follows we drop
physical constants and focus on the effects
of the star�s radius and mass.
The uncertainty principle of quantum mechanics
implies that if an electron is constrained
to a region of dimension little r it must
have a momentum p on the order of Planck�s
constant h over r . Using the above expression,
this varies as the cube root of the star�s
mass over its radius.
The non-relativistic expression for the kinetic
energy of an electron is its momentum squared
over twice its mass.
This varies as the two thirds power of star
mass over the square of star radius.
This kinetic energy is not dependent on the
star�s temperature.
It�s an intrinsic quantum mechanical property
even at absolute zero.
Multiplying by the number of electrons we
get the total kinetic energy.
Since the number of electrons varies as the
star�s mass we obtain a total kinetic energy
of big m to the five thirds power over big
r squared.
We�ve already seen that the gravitational
energy varies as the star�s mass squared
over its radius.
Combining these two terms and removing a common
factor
we see that, for any mass, the total energy
grows arbitrarily large as the radius grows
arbitrarily small.
From this we conclude that gravitational collapse
is impossible.
Now we look at the relativistic analysis.
The steps are the same until we get to the
electron kinetic energy.
The relativistic expression for kinetic energy
is square root of p squared plus m squared,
minus m . For small p this agrees with the
classical expression, but for very large p
, which corresponds to very small star radius,
it behaves like p � not like p squared.
The result is that the total kinetic energy
varies as one over the star radius � not
one over the radius squared.
Combining this with the gravitational energy
We get a total energy that varies as a constant
over the star radius.
We see that if the star�s mass is small
the constant will be positive, so the energy
would again approach positive infinity as
the radius approaches zero.
However, if the mass is large enough, the
constant will be negative, and the energy
will approach negative infinity as the radius
approaches zero.
That is, the star will undergo gravitational
collapse.
This phenomenon is called the Chandrasekhar
limit
after Subrahmanyan Chandrasekhar
who derived it, rigorously, in a series of
papers starting in 1931.
He was able to solve for the ultimate radius
of a star as a function of its mass
and showed that this radius is zero if the
mass exceeds a certain limit.
Current calculations set this limit at about
1 point 4 times the mass of our sun.
Now, the process of stellar evolution is typically
dynamic and often violent, resulting in some
of the star�s mass being ejected as a planetary
nebula
supernova remnant, or in some other phenomenon.
However, if the mass of the remaining core
is above the Chandrasekhar limit then it will
not be able to resist gravitational collapse.
But, what does it collapse to?
The answer is, not necessarily a black hole.
By the 1930s a theoretical picture of nuclear
physics was emerging.
Nuclei were known to contain protons and neutrons.
It was thought possible that under certain
conditions an electron could be captured by
a nucleus, converting one of the protons to
a neutron with the emission of a neutrino.
This process of electron capture was observed
and reported in 1937
by Luiz Alvarez.
The implication was that in a collapsing star
all electrons and protons might combine in
this manner leaving an object composed of
nothing but neutrons.
What we now call a neutron star.
This led to a consideration of the stability
of neutron stars.
In 1939 Oppenheimer and Volkoff analyzed the
problem and showed that there is an upper
limit to the mass of a neutron star, beyond
which it will undergo gravitational collapse.
They concluded, actual stellar matter after
the exhaustion of thermonuclear sources of
energy will, if massive enough, contract indefinitely.
This limit is not as well defined as the Chandrasekhar
limit, because the physics is more extreme
and not as well understood.
It is thought to be less than about three
solar masses.
Currently, the largest known neutron star
has an estimated mass about twice that of
the sun.
A neutron star over this limit might collapse
to a black hole.
Another, hypothetical, possibility is the
transformation of neutrons into a collection
of free quarks, forming a quark star.
Currently there is no substantial evidence
of quark stars.
Even if such things did exist they would have
their own upper mass limit.
Since quarks are elementary particles there
should be no further transformations possible
and we would finally have a collapse into
a black hole.
So, it seems that there is no theoretical
process that can stop a large enough star
from ultimately forming a black hole through
gravitational collapse.
Someone who pushed back against the idea of
black holes was Albert Einstein.
Somewhat ironically since it was his theory
that predicted them.
In 1939 he published a paper claiming
The Schwarzschild singularity does not appear
for the reason that matter cannot be concentrated
arbitrarily.
And this is due to the fact that otherwise
the constituting particles would reach the
velocity of light.
His argument was based on assuming the particles
of a collapsing star would follow essentially
circular orbits.
He remarked, Although the theory given here
treats only clusters whose particles move
along circular paths it does not seem to be
subject to reasonable doubt that more general
cases will have analogous results.
But, Einstein was wrong, and we�ve already
seen why in our previous discussion of orbits
around a black hole.
It�s true that there is a radius outside
the event horizon where the velocity of a
circular orbit is the speed of light.
And, it�s true that no circular orbits smaller
than this are possible.
However, it does not follow that an in-falling
particle passing within this radius would
have to travel faster than light.
In fact, we�ve already calculated such orbits.
They are not circular.
They�re spirals.
And, as we�ve shown, a particle can fall
in radially.
Both of these processes are well described
by relativity theory.
Regardless of how bizarre they may have seemed
to Einstein and others, no known physical
process has been found that could keep a black
hole from forming.
Given that they are theoretically possible.
The obvious question is, can we detect black
holes?
By definition we can not see a black hole.
Classically they do not emit any radiation
for us to detect.
So, any evidence for their existence will
have to be circumstantial.
What they do produce is a very strong gravitational
field.
If a visible object moves through that field
it will exhibit an acceleration that indicates
both the mass and distance of the black hole.
Due to breakthroughs in observational technology,
astronomers are now able to view and tracks
stars near the center of our Milky Way galaxy.
What they�ve seen are several stars orbiting
an invisible central object.
From these orbits the object�s mass has
been determined to be some four million solar
masses.
Moreover, the orbits place an upper limit
on its size � roughly the size of our solar
system.
A super-massive black hole is the only known
object consistent with these observations.
Other evidence for black holes can come from
an accretion disk.
If material falls toward a black hole it can
accelerate to near the speed of light.
These high speeds and violent collisions produce
tremendous heat which in turn generates radiation.
Temperatures of millions of degrees are possible
which can produce intense x-ray emissions.
One source of an accretion disk can be a companion
star in mutual orbit with the black hole.
The radiation burst associated with material
falling into an accretion disk is called an
x-ray nova.
However, this by itself is not evidence of
a black hole.
The intense gravity of a neutron star can
also produce an accretion disk and x-ray nova.
But there is a key difference, illustrated
in this figure.
As material spirals in through the accretion
disk, the radiation it produces is ever-more
red-shifted.
If the central object is a black hole this
red-shifting continues without end and the
material disappears into the event horizon.
However, a neutron star has a solid surface,
and the in-spiraling material ultimately strikes
this producing an additional characteristic
x-ray burst.
It�s the lack of this final burst that provides
evidence that the central object has no solid
surface.
The only massive object without a surface
is a black hole.
Finally, gravitational waves were recently
observed by the laser interferometer gravitational-wave
observatory.
These bear the characteristic signature of
merging black holes, and cannot be explained
by any other known process.
