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 The True Most Fundamental Gravitational Constant
  Posted: May 16 2014, 03:13 PM

Casual Member

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Joined: 7-February 14

Newton's law of gravitation is:

$$F = G m_1 m_2 \frac{1}{r^2}$$

It looks simple and natural.

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

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

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

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

$$F = G^* m_1 m_2 \frac{1}{4 \pi r^2}$$

Immediately we recognize that $$4 \pi r^2$$ 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:

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

$$2 \pi r$$ is the surface area of a 2 dimensional sphere of radius r.

$$2 \pi^2 r^3$$ is the surface area of a 4 dimensional sphere of radius r.

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

$$\pi^3 r^5$$ is the surface area of a 6 dimensional sphere of radius r.

Newton's law of gravitation in n dimensions is:

$$F = G^* m_1 m_2 \frac{1}{S_n}$$

Where $$S_n$$ is simply the surface area of a n dimensional sphere of radius r.

This proves that $$G^* = 4 \pi G$$ 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.

$$c = \hbar = 4 \pi G = \epsilon_0 = k_B = 1$$
Posted: May 17 2014, 07:03 AM

Dozens Demigod

Group: Members
Posts: 2,432
Member No.: 655
Joined: 11-July 12

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=þQ, g=E/þ etc. Maxwell's equations apply, but B/þ 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, \( k_G = c/\gamma þ^2 \) and \( \epsilon_g = þ^2/c \).

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

Posted: Jul 17 2014, 10:29 AM


Gravitational constant?!?!?!
Posted: Jul 17 2014, 11:21 PM

Dozens Demigod

Group: Members
Posts: 2,432
Member No.: 655
Joined: 11-July 12

It's the G in Newton's equation f = G Mm/r². Its value is 1.023e-9 ft³/s² lb or 6.672e-10 dm³ / ds² kg.

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