Deriving G : Universal Gravitational Constant

[muon200]

Derived Value
G = 6.48 x 10^-11 m^3/(s^2 x kg)

Experimental Value
G = 6.67 x 10^-11 m^3/(s^2 x kg)

Formula Derived
G = D x r^2 x (dv/dr)/M

D is the diffusivity of Herenowium ( here and now ee um)
Herenowium is the 4 dimensional continuum with new features
D = (1.4 x 10^-4)/(m x sec x sec)

r is radius of Earth
r = 6.3 x 10^6 meters

v is volume of space in a frustum of a cone above Earth's core

dv/dr is the gradient of continuum near Earth
dv/dr = 7 x 10^4 m^2

M is mass of Earth
M = 6 x 10^24 kg

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by Alan C. Folmsbee
December 18, 2014
Wailuku, Hawaii

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acknowledgements
Thanks go to Dr. Joy Christian for correcting my mistakes about time. Time is growing.

[muon200]

This effort to derive G is based on the idea that gravity is caused by the masses of hadrons. Also, hadrons are blamed for shrinking space and growing time in a continuum that is conserved. The continuum is also endowed with capabilities to accelerate hadrons while they remain at rest relative to a local velocity ether called Herenowium.

Past essays on this theory of gravity have succeeded in calculating the size of the universe by asserting that each hadron shrinks space by its own volume every 5.5 nanoseconds. That result shows that time grew by 14 billion years and space shrunk by 14 billion light years. I was so happy with that result that I decided to attempt to derive G using the balance of continuum as:

5.5 nanoseconds balanced 4 x 10^-45 cubic meters.

Those "balance numbers" conserve the space-time continuum.

There are several ways to attempt this, and this treatment has the risk of descending into a tautology where circular reasoning can occur. But since I have no physics reputation to protect, I will risk being wrong and proceed with courage.

Diffusion is explored as the mechanism of gravity. As space shrinks into Earth's hadrons, diffusion of external continuum moves to fill the space void, carrying with it any masses, or just making a force if motion is prevented by a mechanism. Diffusion of growing time moves outward as space moves inward. G is a constant used in equations of force and acceleration and so those old equations are available for this treatment of G and gravity.

a = GM/(r^2)
a is acceleration
M is Earth's mass
r is Earth's radius

The Earth is used as a laboratory in which its measured gravity effects can be employed with confidence.

A cone from the center of the Earth is cut into frustums so that a finite element analysis is begun. Each conic section (frustum) has a different density of Herenowium so that a diffusion, like in Fick's Law, can be calculated. Two adjacent frusta are evaluated numerically to derive the diffusion magnitudes due to the gradient of space density and a gradient of time density flowing from hadrons.

The derivative dv/dr is calculated and a diffusivity is calculated for the continuum. Both time and space share the same derivative, so that acceleration is proposed to occur, instead of a flux as in Fick's Law. Volume is v and r is radius. The derivative is calculated from the two volumes of adjacent frusta. The diffusivity D is calculated to fit the balance numbers. This constitutes a derivation from basic principles. It is asserted that G is being derived from this theory in which no more fundamental work is needed, only the "balance numbers" are needed to derive G.

The balance numbers are employed now, to anchor the new theory into the derivation of G, the Universal Gravitational Constant.

In one second, the hadrons in the cone shrink space by their own volume every 5.5 nanoseconds. That is 182 megahertz, inverted.

Radius of Earth 6.371 x 10^6 meters
Volume of the Earth 1.5 x 10^19 cubic meters
Earth’s number of protons and neutrons is 3.5 x 10^51 (Aluminum)
V0 = Sum of hadron volumes for Earth 1.4 x 10^7 cubic meters

A cone is defined from the center of Earth to a mass held 300km above the surface. The mass is dropped for one second and it falls 4.8 meters. The cone radius is 2.4 meters at the mass so that a frustum is defined to be almost a square cylinder.

V1 Volume of the cone up to Earth's radius

Ratio of cone volume V1 to Earth volume V2 is X12
V1/V2 = X12 =
Number of hadrons in V1 is
.
.
.

to be completed soon, unless someone else derives G first...

[muon200]

There is one curious coincidence which I can promote with humble trepidation. Remember my "balance numbers" for conservation of continuum...

5.5 nanoseconds balances 4 x 10^-45 cubic meters
5.5 ns balances proton volume

The Balance of Continuum can be manipulated to make G's magnitude:

Let 5.5 ns be called Tau
Let 4x 10^-45 cubic meters be Sigma (proton volume space)

(Magnitude of G) = pi x cube root(Sigma) / sqrt(Tau)

cube root of Sigma = 1.60 x 10^-15 meters
sqrt(Tau) = 7.42 x 10^-5 sqrt(seconds)

So (Magnitude of G) = 6.79 x 10^-11 meters per sqrt(seconds)

The units of G are = 6.67 x 10^-11 meters^2/(kg x seconds^2)

The ratio of this (Magnitude of G)/G is kg x sec^1.5/meter


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Coincidence? Again? Yes. Just a coincidence. But hold that thought for future reference...

[muon200]

In an effort to replace the empirical Universal Gravitational Constant G with a more meaningful constant H, the following equations are evaluated :

The Force of Gravity is f, according to Newton...
f = (Mm/r^2) x G

The new equation is more meaningful, according to Folmsbee...
f = (Mm/r^2) x H x (cube root(sigma)) / sqrt(tau)

Where
H is the Universal Herenowium Constant
H = pi meters^2/sqrt(seconds)
sigma is the space volume of a proton or neutron
tau is the time growth for a sigma to be shrunk from space
tau and sigma combine to conserve continuum

sigma is 4 x 10^-45 meters^3
tau is 5.5 nanoseconds
H is pi (square meters)/sqrt(seconds)

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Verify that the units are correct...

The units of G must equal units of
H x (cube root(sigma)) / sqrt(tau)

units of G
meter^3/(kg second^2)

units of H
pi meter^2/(kg second^1.5)

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confirm force units in the Folmsbee Gravitational Equation

f = (Mm/r^2) x H x (cube root(sigma)) / sqrt(tau)
f = (kg^2/meter^2) x (meter^2/(kg sec^1.5) ) x meter / sqrt(second)

cancel factors to see force units
f = kg^(2-1) x meter^(-2+2+1) x second^(-1.5 - .5)
f = kg meter/second^2

It is confirmed.

[muon200]

Let me spell it out for you...
The Universal Gravitational Constant G is today, for the first time in history, defined by fundamental facts. It is no longer just a weird lumped fudge factor that Newton failed to explain. It combines the volume of a proton with 5.5 nanoseconds...

G = (pi (proton space)^(1/3) / (5.5 ns)^(1/2)) square meters per (kg seconds^1.5))

Newton never understood what causes gravity. He lumped a G factor to fit experiments.
Einstein never understood what causes gravity. He quoted Newton.
Folmsbee understands gravity. Atoms shrink space and grow time. This is gravity.

The 5.5ns and proton volume have already been used to calculate the size of the universe and how far a rock falls in Hawaii in one second. The Folmsbee-Christian Gravitational Theory is on the table. Talk to me... assail me...humble me or pay me. I want a chair. There is more to do... I am ready go.

Alan Charles Folmsbee
Wailuku, Hawaii