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Pinbacker 
Posted: May 16 2014, 03:13 PM

Casual Member Group: Members Posts: 117 Member No.: 795 Joined: 7February 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. $$c = \hbar = 4 \pi G = \epsilon_0 = k_B = 1$$ 
wendy.krieger 
Posted: May 17 2014, 07:03 AM

Dozens Demigod Group: Members Posts: 2,432 Member No.: 655 Joined: 11July 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 'overdefine' 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. http://z13.invisionfree.com/DozensOnline/i...view=getnewpost 
PiotrGrochowski 
Posted: Jul 17 2014, 10:29 AM

Unregistered 
Gravitational constant?!?!?!


wendy.krieger 
Posted: Jul 17 2014, 11:21 PM

Dozens Demigod Group: Members Posts: 2,432 Member No.: 655 Joined: 11July 12 
It's the G in Newton's equation f = G Mm/r². Its value is 1.023e9 ft³/s² lb or 6.672e10 dm³ / ds² kg.
