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"Black hole" is the name given to a body that has gravity excessive enough to prevent light from escaping. It does this by pulling in the nether at a velocity equal to or more than "c", the velocity of light.
The Schwarzschild radius (some spell the name Schwartzschild) shows the necessary specifications for a black hole to exist. The Schwarzschild radius need not be the radius of the body that is producing the gravity. It is the distance from the center of the body at which the velocity of the incoming nether equals "c". When the velocity of the incoming nether equals "c", light cannot escape. It is like a river with a current moving at a velocity equal to what a motorboat can go. The boat stays in one place.
When the velocity of the current is greater than the boat's velocity, the boat is pulled backward. When the incoming nether velocity exceeds "c", light cannot even get started.
The Schwarzschild radius can be derived from the equation for escape velocity which is the same as nether velocity downward for any given location within a gravity funnel.
R = radius
G = the gravitational constant
M = mass of the body producing the gravity funnel
c = the velocity of light
m = mass of a smaller body in the gravity funnel
g = gravity at a point within the gravity funnel
F = force
F = GMm/R2 is the equation using the gravitational constant to show the force between two masses.
F = mg is the equation for force downward for body within a gravity funnel.
From the above, F = mg = GMm/R2. So:
gm = GMm/R2
g = GM/R2
v = (2Rg)1/2 is the equation for escape velocity or nether velocity downward at any point in a gravity funnel.
Let v be equal to c. Then
c = (2Rg)1/2
c = [2R(GM/R2)]1/2 by substitution
c = (2GM/R)1/2
c2 = 2GM/R
R = 2GM/c2 which is the Schwarzschild radius.
It is interesting to go to a website that shows how Karl Schwarzschild originally derived this radius. He used Einstein's theory of gravity as a beginning. Without knowing the true nature of gravity, he was able to use a lot of complicated theory and equations to arrive at the correct solution which the physicists called his "metric" or the "Schwarzschild solution" so that they could introduce another barrier between themselves and the lay person.
Considering how little Schwarzschild knew of the true nature of gravity when he derived his theory in 1916, he did a great job. But notice the difference when one understands what gravity actually is. Once one knows the true nature of gravity, the simple equation for escape velocity along with the equation for the gravitational constant are enough to complete the derivation in the form shown above.
The Schwarzschild radius was placed here to show how black hole radii can be found within certain limits (these radii will always be equal to or less than their corresponding Schwarzschild radii) - and for some other calculations to be shown regarding the electron. I did not bother with the above derivation before now because the equation for escape velocity is so adequate, that a separate derivation seemed too simple and too trivial to be shown.
Now I realize that part of nether theory's problem in being accepted has to do with it (1) not being complicated enough to impress and confuse the intelligent lay person, and (2) not showing the precise derivations for things already accepted. Then there were comments on why I did not choose to use calculus. Obviously, the simple solution above is more elegant than the original two plus pages of calculus that Schwarzschild used in his derivation.
There is a theory that after enough energy and matter (which is energy also - but in a different form) enter a black hole, they begin to "evaporate" (Stephen Hawking). This makes sense if all the facts are not known. However, in my opinion, the observations of Halton Arp (Seyfert galaxies) indicate that something else happens. Of course, Halton Arp has not been allowed by the majority of astronomers and astrophysicists to properly air his findings because they upset the theories of some of the controlling prestigious members of their club.
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