Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: Rossetta Stones of Physics ( Black Holes)

I'll start the ball rolling by observing that your equation for the escape
velocity is correct, but your words leading up to it are not. You have set
the total energy equal to zero, not the Lagrangian (recall that the
potential energy here is negative). Also two language/grammar nitpicks :
Several times you refer to the "faith" of a star - perhaps you meat
"fate"? - You also say that the stiffness of a body "effects" the speed of
sound - that should be "affects"?


Bob Sciamanda (W3NLV)
Physics, Edinboro Univ of PA (em)
----- Original Message -----
From: "Robert B Zannelli" <Spinoza321@AOL.COM>
To: <>
Sent: Sunday, February 18, 2001 2:13 PM
Subject: Rossetta Stones of Physics ( Black Holes)

It is certainly not my intention to lecture to the many eminent Physicist
this list especially at such a basic level. However for those non
or non professional physicist who are members of this list I thought this
posting, which I posted to another Physics list earlier, might be
interesting. The subject matter of this posting is hardly new as these
were developed circa 1970s. Nevertheless I welcome comments or criticism
anyone. If list members feel this type of post is inappropriate for this
I request they inform me of that by private email or list posting.

Dear List Members:
Perhaps nothing in science has captured the popular imagination
than what John Wheeler has named Black Holes. These are stars which due to
their great mass have shrunk to what are called singularities once they
exhausted their fusion fuel. While the life cycle of stars is a
story, in itself, I will only touch upon that story briefly and limit this
post to talk about the amazing properties of black holes.
Stars that have reached the end of their fusion energy production
lives have one of three possible faiths. Stars which have a residual mass
less than about 1.3 solar masses become what are called white dwarfs which
are immensely dense remnants of the original star. For this class of dead
star further collapse is prevented by what is called the electron
the internal pressure due the Pauli exclusion principal preventing
from all being in the same Quantum State. For those stars that retain
than 1.3 solar masses after burnout electron degeneracy is insufficient to
generate the pressure to prevent further collapse and these stars become
neutron stars. Here the internal balancing pressure is generated by
degeneracy and the star is called a neutron star.
However for the those stars which retain greater than about 2 solar
masses there is no force in the universe that can stop their collapse.
is due to the strictures of special relativity. There are two equivalent
to explain this. First in special relativity we must remember that mass
energy are the same thing. This means that as the stars internal pressure
increases to combat the gravitational force which is collapsing the star
very same pressure which is a form of energy begins itself to increase the
gravitational force. At a sufficiently high pressure the gravitational
effects of this pressure become greater than the pressure effects
and the faith of the star is sealed. It must collapse to a singularity.
The second way to look at this is that the stiffness of any body
the speed of sound through that body. At about 2 solar masses the
factor would have to have to generate sound velocities greater than the
of light which is prohibited by SR.
What makes a black hole such a unique entity is the fact that when a
black hole forms it produces an event horizon which causes an unbeatable
information barrier which actually cuts off the interior of the black hole
from our universe. Let us see why. For any gravitational body the escape
velocity is (The velocity necessary for any thing within the confines of
gravitational force of the body to escape to infinity) is given by setting
the lagrangian (The difference of kinetic and potential energy) to zero
solving for the velocity in the equation. Therefore we get:

m*(v^2)/2=GM*m/r which gives us v=(2*G*M/r)^.5

From this we can see that as r gets smaller the escape velocity
increases. In
fact at some point this velocity reaches the speed of light. The surface
of this star that corresponds to the radius where the escape velocity is
speed of light is called the event horizon. This is where the barrier is
divides the unobservable interior of the black hole from our Universe.
This is given by the equation:


Which is derived by some simple algebra from the above
(This is for a non spinning zero charged black hole only. While a zero
charged black hole is likely a nonspinning one is not. However this will
serve for the purposes of this post)

So it would appear that we could say that a black hole could be described
a cold absolute zero temperature entity. ( Since it emits zero energy into
the Universe) However this is not really true. This is where the story of
black holes gets even more interesting. As anyone who knows anything at
about physics knows that general relativity and Quantum mechanics are like
oil and water. However thanks to the work of Steven Hawking and others we
have the first small convergence of these two theories. Where these two
theories may converge if ever so slightly is at the surface of the event
horizons of black holes.
To understand why this is so we must look a little closer at the nature
black holes. First it is believed that black holes retain only three
properties after their collapse. These are mass, electrical charge and
angular momentum. This is known in Cosmology as the "black holes have no
theorem." This means that during a black hole collapse there is a great
of information which in the language of physics means there is a great
entropy increase. ( Entropy is the measure of disorder in any system) This
interesting fact caused physicist to look at black holes from a
The first hints came in 1970 from with the realization that surface area
a black hole always increases when additional matter falls into it. Since
this represents a further loss of information it means a further increase
entropy. This led to Jacob D. Berkenstein an Israeli Physicist to propose
that the area of the event horizon of a black hole was proportional to
entropy. S=K*A Where K is a constant of proportion, S is Entropy and A is
the area of the event horizon. However if this true then it follows that
black holes cannot really be so black. Let us see why. ( The following is
to the work of Stephan Hawking.)
In thermodynamics the following relationship holds dS=dE/T where S is
entropy and E is the energy of the system and T is the temperature of the
system. ( This relationship is due to the work of Rudolf Clausius
Now we already know that E and M are proportional due to special
and given that according to Berkenstein S and A are also proportional we
derive the following relationship.

F=GM/(r^2) and A=4*pi(r^2) we have F=4*pi*G*M/F where F is a
of the gravitational field,A is the area of the event horizon, and M is
mass of the black hole. Given this relationship is easy to see that
Where K is a constant of proportionality. This of course means that F is
proportional to the temperature of the black hole at the surface of the
horizon. This is a certainly not zero. Given that we have determined that
temperature of the black hole is not zero we are left with the conclusion
that black hole must radiate energy.
Of course given the limitations of classical physics there is no way
that any thing can breach the event horizon light speed barrier. But the
black hole must emit energy since it's temperature is not zero. However
thanks to the work of Stephan Hawking we have a way out of this quandary.
Quantum Mechanics we have what are known as virtual processes. This is due
the uncertainty principal which is on the order of:


Where delta E is the uncertainty of energy for any system and deltat is
time interval under consideration and hbar is Planck's constant divided by
2*pi. What this means is that the conservation of mass energy may be
but only for a time deltat=(hbar/2)/deltaE where deltaE the energy
uncertainty in question. The larger the mass energy violation the shorter
time it last for. Now let us look at the boundary of the event horizon.
mass energy violations produce particle pairs a particle and it's anti
particle since all other conservation laws must hold. This means that a
possible virtual particle pair may be an electron and an anti electron.
would mean that the time interval where this could occur would be on the
order of:


Where M is the mass of the electron. Therefore this process
could occur on the order of x=t*C where C is the speed of light and x is
maximum separation of the particle and anti particle. This processes are
occurring continuously everywhere in the Universe. In fact this process
has a
measurable effect known as the Casimir effect named after it's discoverer.
Normally these virtual particle pairs recombine within the time
defined by the uncertainty principal. However when this process happens
the surface of an event horizon there is a finite chance that the particle
pair will end up on opposite sides of he event horizon. Since no particle
exceed the speed of light this means that this particle pair can never
recombine. The particle outside the event horizon escapes acquiring the
energy of the particle pair while the particle that has penetrated the
horizon carries an equal negative energy to insure that mass energy is
conserved. Effectively the black hole has radiated energy.
The probability for this process to occur is proportional to the
curvature of space time in the vicinity of the event horizon which is
indirectly proportional to the area of the event horizon therefore
proportional to the mass of the black hole. Hawking has shown that we may
calculate the temperature of a black hole with the following equation;


Where K is Boltzmann's constant which may be simplified to:

T(Degress Kelvin)=6E-8/M(solar masses)

From this it is obvious that as black hole radiant energy their
increases until as M approaches zero they explode with incredible energy.
This is a truly remarkable result.
However given the likely masses of black holes this process takes a long
time. Hawking has shown that:

t=(approx) 1E66 *(M^3) with M in solar masses and t in

No black hole in our Universe assuming it had a mass of at least
solar masses has enough time to have exploded. However it is possible that
were processes which could have produced black holes of very small mass
values during the early history of our Universe and these black holes if
exist could be exploding even now.

Bob Zannelli