# The True Most Fundamental Gravitational Constant

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Casual Member
Pinbacker
Casual Member
Joined: Feb 7 2014, 03:32 PM
Newton's law of gravitation is:

&#036;&#036;F = G m_1 m_2 \frac{1}{r^2}&#036;&#036;

It looks simple and natural.

But that's only in 3 dimensions. Let's look what happens in n dimensions:

&#036;&#036;n=2 : F = 2 G m_1 m_2 \frac{1}{r}&#036;&#036;
&#036;&#036;n=4 : F = \frac{2}{\pi} G m_1 m_2 \frac{1}{r^3}&#036;&#036;
&#036;&#036;n=5 : F = \frac{3}{2 \pi^2} G m_1 m_2 \frac{1}{r^4}&#036;&#036;
&#036;&#036;n=6 : F = \frac{4}{\pi^2} G m_1 m_2 \frac{1}{r^5}&#036;&#036;

Oh no! Newton's law of gravitation becomes ugly, weird and incomprehensible.

But by posing &#036;&#036;G^* = 4 \pi G&#036;&#036; Newton's law of gravitation can be reformulated as such:

&#036;&#036;F = G^* m_1 m_2 \frac{1}{4 \pi r^2}&#036;&#036;

Immediately we recognize that &#036;&#036;4 \pi r^2&#036;&#036; is simply the surface area of a sphere of radius r.

But that's only in 3 dimensions. Let's look what happens in n dimensions:

&#036;&#036;n=2 : F = G^* m_1 m_2 \frac{1}{2 \pi r}&#036;&#036;
&#036;&#036;n=4 : F = G^* m_1 m_2 \frac{1}{2 \pi^2 r^3}&#036;&#036;
&#036;&#036;n=5 : F = G^* m_1 m_2 \frac{1}{\frac{8}{3} \pi^2 r^4}&#036;&#036;
&#036;&#036;n=6 : F = G^* m_1 m_2 \frac{1}{\pi^3 r^5}&#036;&#036;

&#036;&#036;2 \pi r&#036;&#036; is the surface area of a 2 dimensional sphere of radius r.

&#036;&#036;2 \pi^2 r^3&#036;&#036; is the surface area of a 4 dimensional sphere of radius r.

&#036;&#036;\frac{8}{3} \pi^2 r^4&#036;&#036; is the surface area of a 5 dimensional sphere of radius r.

&#036;&#036;\pi^3 r^5&#036;&#036; is the surface area of a 6 dimensional sphere of radius r.

Newton's law of gravitation in n dimensions is:

&#036;&#036;F = G^* m_1 m_2 \frac{1}{S_n}&#036;&#036;

Where &#036;&#036;S_n&#036;&#036; is simply the surface area of a n dimensional sphere of radius r.

This proves that &#036;&#036;G^* = 4 \pi G&#036;&#036; is the true most fundamental gravitational constant, not G.

Am I a genius or what?

Now I will begin to create my new system of measurement, based on dozenal and natural units.

&#036;&#036;c = \hbar = 4 \pi G = \epsilon_0 = k_B = 1&#036;&#036;

wendy.krieger
wendy.krieger
The nature of rationalisation is best handled by a new radiation constant $$\gamma$$, and Gauss's flux constant $$\beta$$ This exists in gravity, electricity, and light. Without a subscript, it refers to solid space, but the dimension can be specified, eg $$\gamma_2 = 2\pi \beta$$, and $$\gamma = \gamma_3 = 4\pi\beta$$. Here $$\beta$$ is the constant in Gauss's law, eg $$\Phi = Q / \beta$$.

We assume Heavisides's equations of gravity to electricity, viz Q~M, with an equality that M=Ã¾Q, g=E/Ã¾ etc. Maxwell's equations apply, but B/Ã¾ is quite faint, and it will be still unresolved whether Heaviside or Einstein is correct here. It's a factor of 2.

We also note that cosmologists use c.g.s. gaussian equations, for both gravity and electricity, where we use the fpsc system, where eg $$\epsilon = \mu = 1/c$$.

G renders as two different constants in a rational system, because it lives in a set of equations that suppose $$\gamma=1$$, where in SI, $$\gamma=4\pi$$. So we write $$k_G$$ for coulomb's constant, and $$\epsilon_g$$ for the permittivity of gravity (ie your $$4\pi G$$).

In any case, $$k_G = c/\gamma Ã¾^2$$ and $$\epsilon_g = Ã¾^2/c$$.

Ã¾ is quite large, it's like 270,000,000 pounds/verber. This is nearly the value of c = 983574900 ft/s. Then, c/Ã¾ is nearly unity.

On Dirac's constant $$\hbar$$. As with c.g.s. units, it is based on the radian, the correct unit is ft.pdl.s/rad. In a rationalised system, we use h, measured in ft.pdl.s/cycle, the notional value is 1/2.

When the base between units is freely settable, we can 'over-define' the system, to produce a pure numeric. The base units of the GKO, which sets c=1, and c/Ã¾ near 1, are to suppose $$Q = e/\alpha^{20}, \ M = m_e/\alpha^{30}, \ \alpha^{-4}c, \ k_B=\alpha^{28} \ 2h=\alpha^{39}, \ \epsilon c = 1$$. This sets $$1/\alpha = 137.036$$.

Units are in powers of $$\alpha$$, but nearly everything in the CODATA tables is exact.

The value of c/Ã¾ here is 1.1441442, and G is variously 1/1.30 or 1/5.20 $$\pi$$.

The COF Booklet thread points to a PDF i am doing in Latex, the chapter on electricity, the bohr atom, and some notes on gravity, are largely there.

http://z13.invisionfree.com/DozensOnlin ... getnewpost

PiotrGrochowski
PiotrGrochowski
Gravitational constant?!?!?!

wendy.krieger
wendy.krieger
It's the G in Newton's equation f = G Mm/rÂ². Its value is 1.023e-9 ftÂ³/sÂ² lb or 6.672e-10 dmÂ³ / dsÂ² kg.