22  The Schwarzschild Radius                                    Table of Contents     Previous       Next



Karl Schwarzschild (October 9, 1873 – May 11, 1916)

Click on picture of Schwarzschild for more information.
(quote below taken from website).

“One of his most famous and well known contribution to science was something called the Gravitational Radius, or the Schwarzschild radius. This is a characteristic radius associated with every mass. This “rule” states that if any spherical mass were to change and to acquire that radius, no known force could prevent it from collapsing into gravitational singularity. This gravitational singularity is also known as a black hole. Schwarzschild obtained this solution just a few months after Einstein discovered the theory of general relativity. The equation for this radius is rs =2Gm/c2. rs would be the Schwarzschild radius, G would be the gravitational constant, m is the mass of the gravitating object, and c is the speed of light. Although this equation proved the existence of black holes, their existence was still debated for decades. The surface at the Schwarzschild radius acts as an event horizon. He stated his discoveries in a paper written in Russia. He was the first to give exact solutions to Einstein’s equations of relativity. Einstein later stated: I had not expected that one could formulate the exact solution of the problem in such a simple way. Karl Schwarzschild was a very important contributor to the advancement of astronomy, and proved the existence of black holes.” 


This topic of the Schwarzschild Radius, and black holes was developed from Einstein’s gravitational equations. Here I will take a look at these concepts via Planck Units. Refer to section 15 “Planck Numerology” .

The Derivation of a Schwarzschild Wavelength:

1. E = Nhf       This is the Planck Einstein relation for photons. Energy comes in quanta. Where N is
                       the number of photons, h is the reduced Planck Constant and f is the frequency of the
                       photons.

2. E = 2Nhf     This is the Planck Einstein relation modified for gravitons. 2N is used since each
                       graviton hops back and forth to complete a cycle. See
Section 29 visualizing spin.

3. Total Energy of the mass = mc2      Where m is the mass and c is the max speed of space-time.

3.
If all the mass energy is converted into gravitational energy we have: 2Nhf = mc2

4.
Since f = c/λ , we have Nhc/λ = mc2, and λ = (2Nh) / (mc).

5. From Section 17 we have that N = m2/Pm2 and λ = (2hm2/Pm2) / (mc).

6. The Planck mass squared (Pm2) is equal to (hc)/G and solving for λ we get λ = 2mG/c2
       
(the Schwarzschild Radius expressed as a wavelength).

7. The Schwarzschild Radius is now a wavelength λ, very interesting!

Is a Schwarzschild Wavelength a Valid Concept?

When we think of a radius we have a context of a single mass that is very dense.  The Schwarzschild radius is held to be a small localized object.  When we change our perspective to that of wavelength we get a broader scope. Now the Schwarzschild wavelength (distance) does not necessarily represent a localized object, it can be distributed over many masses and be extended in space and time.  The mass in λ = 2mG/c2 does not need to be a single lump, the black hole can be an extended object.  We could even consider the universe as a whole and determine how close to a Schwarzschild wavelength it is. This is a reasonable speculation that may give an explanation of the question: Does a laser beam aimed out to the galaxies ever leave the universe? The answer is no if the light is trapped by a Schwarzschild wavelength.


Is the Radius of the Universe a Schwarzschild Wavelength?

   
Schwarzschild Radius = 2Gm/c2

   
The mass of the Universe mu is 1x1053 kg  (see Ref)
     
The radius of the Universe is 4.4x1026 m
     
The gravitational constant G is 6.673x10-11

    Let’s say that the estimated radius of the universe 4.4x1026 m is a Schwarzschild radius as   
    indicated above, then the mass needed to have a black hole would be 1.48 x10
53 kg. 

    It is very possible that the universe is a black hole, and all the light that originates in the universe
    stays in the universe.


A Visualization of the Schwarzschild Radius Using Planck Units:


1. The number of Planck masses in any given mass m is N = m/P
M. PM is the Planck mass.
2. The result of compressing a mass is to reduce the spacing of Planck masses.
3. The maximum compression results when the spacing between the Planck masses gets to two
    Planck lengths. At this compression the mass gets to its smallest possible radius, the
    Schwarzschild radius.
4. Can we get radiuses smaller than the Schwarzschild radius? Conventional science says yes. I
    believe this is true, but things will get tricky because the localized masses will be below the
    quantum mechanics threshold of the Planck mass and the entire mass may leave the realm of
    ordinary mass and be more like a quantum mechanical object.


We can solve for RS by making

the following ratio:

m/RS  =  PM /2PL
      

Where:

PM = (hc/G)1/2 , PL = (hG/c3)1/2


Solving for RS we get: RS = 2Gm/c2

the Schwarzschild radius (and yes it
would be more elegant with the Planck

mass having a radius of a single Planck

length).

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