Chapter 10. Gravitational Constant, Revisited

In this chapter, we create Newton's gravitational-force equation by using SG units of measure. Again, the historical SI version is:

F_gm = G m₁ m₂ r^-2 (84

where: G = 6.7 × 10^-11 m³·kg^-1·s^-2.

A. Gravitational Constant in SG Units

We have seen that, in SE units (see Equation 71), the comparable value for G is:

G = 2 (β θ² 4 π)^-1 λ_e³·m_e^-1·t_e^-2 (85

Moving the factor θ^-2 into the SE units of measure converts them into SG units, as follows:

G = 2 (β 4 π)^-1 (θ^-2 λ_e³·m_e^-1·t_e^-2) (86

G = 2 (β 4 π)^-1 (θ^-1 λ_e)³ (θ m_e)^-1 (θ^-1 t_e)^-2 (87

G = 2 (β 4 π)^-1 λ_g³·m_g^-1·t_g^-2 (88

Rearranging the factors in preparation for removing them from the gravitational constant, G, (making it disappear by converting its value to dimensionless unity):

G = 2 (β^-1 m_g·λ_g·t_g^-2)(m_g^-2·λ_g²)(4 π)^-1 (89

B. Newton's Force Equation in SG Units

To see these numbers in context, let us put them in place of the historical value of the gravitational constant, G, in the SG version of Newton's force equation as follows:

F_gg = [2 (β^-1 m_g·λ_g·t_g^-2)(m_g^-2·λ_g²)(4 π)^-1] ×

[m₁ m₂ (r²)^-1] (90

where the first set of square brackets encloses the SG value of the gravitational constant, G, and the second set, the rest of the right side of the equation. Of course, the variables, m₁, m₂, and r, use SG units of measure. Separating these variables' SG values from their SG units of measure gives:

F_gg = [2 (β^-1 m_g·λ_g·t_g^-2)(m_g^-2·λ_g²)(4 π)^-1] ×

[|m₁ m₂|_g (|r|_g²)^-1 (m_g²·λ_g^-2)] (91

C. Eliminating the Gravitational Constant

In Equation 91, the SG-based gravitational constant, G, consists of four groups of factors: (2), (β^-1 m_g·λ_g·t_g^-2), (m_g^-2·λ_g²), and (4 π). We eliminate each group in turn.

The Factor (4 π)

The factor, (4 π), belongs with the r² factor for the area of the surface of the sphere of influence of the gravitational force. Placing it with r² gives:

F_gg = [2 (β^-1 m_g·λ_g·t_g^-2) (m_g^-2·λ_g²)] ×

[|m₁ m₂|_g (4 π |r|_g²)^-1 (m_g²·λ_g^-2)] (92

The Number Two

The number, 2, belongs with the m₁ and m₂ factors. In essence, the m₁ mass contributes (m₁ m₂) to the force, and the m₂ mass does the same for a total of (2 m₁ m₂). Placing the number, 2, with the two masses gives:

F_gg = [(β^-1 m_g·λ_g·t_g^-2)(m_g^-2·λ_g²)] ×

[2 |m₁ m₂|_g (4 π |r|_g²)^-1 (m_g²·λ_g^-2)] (93

Factors of Units of Force

Each of the two remaining groups of factors in the gravitational constant, G, represents the SG unit of gravitational force, but is expressed in terms of the action that created the force reaction and not the unit of force, itself, as it should be. The SG unit of inertial force, f_ig, replaces (m_g·λ_g·t_g^-2), and the SG unit of gravitational force, f_gg, replaces (m_g²·λ_g^-2) to give:

F_gg = [(β^-1 f_ig)(f_gg^-1)] ×

[2 |m₁ m₂|_g (4 π |r|_g²)^-1 (f_gg)] (94

which shows more-easily that [(β^-1 f_ig)(f_gg^-1)] exists to convert the unit of force that is used in the equation from gravitational to inertial; therefore:

(β^-1 f_ig) = (f_gg) (95

and

[(β^-1 f_ig)(f_gg^-1)] = 1 (96

and the gravitational constant, G, no longer exists as such.

SG Gravitational-Force Equation

Newton's SG gravitational-force equation is now:

F_gg = 2 |m₁ m₂|_g (4 π |r|_g²)^-1 f_gg (97

or, by converting the reaction, f_gg, into the action that caused it, m_g²·λ_g^-2, and reinserting these units back into |m₁ m₂|_g and |r|_g², the final form becomes

F_gg = 2 m₁ m₂ (4 π r²)^-1 (98

Gravitational-Force Comparison

We compare the SI- and SG-based gravitational-force equations using the SI newton, N, as the comparison unit of force, where m₁ and m₂ each equals m_g, and r equals λ_g. The four steps of this procedure are as follows:

1) Determine the force F_gm (in newtons), using the historic form of Newton's force equation:

F_gm = G m_g² (λ_g)^-2 = (6.7 × 10^-11 kg ^-1·m³·s^-2) ×
(1.9 × 10^-9 kg)² (1.2 × 10^-23 m)^-2 = 1.6 × 10³⁸ N (99

2) Determine the force F_gg (in f_gg) using the new SG-based form of Newton's force equation (without a constant of proportionality):

F_gg = 2 m_g² (4 π λ_g²)^-1 = 2(1 m_g²) [4 π(1 λ_g²)]^-1 =

0.16 m_g²·λ_g^-2 (100

or, converting from the action to reaction form,

F_gg = 0.16 f_gg (101

3) Convert the SG value of f_gg to the SI value:

f_gg = (4 π) 2^-1 F_gm = [2 π (1.6 × 10³⁸ N)] =

1.0 × 10³⁹ N (102

4) Convert the SG value of F_gg to the SI value:

F_gm = f_gg F_gg = [(1.0 × 10³⁹ N) (·f_gg^-1)] (0.16 f_gg) =

1.6 × 10³⁸ N (103

and Equations 99 and 103 equal each other.

Conclusion

Similar to the conclusion about Coulomb's electromagnetic equation in Chapter 5, the presentation in this chapter proves mathematically that the two quantum actions, (m_g·λ_g·t_g^-2) and (m_g²λ_g^-2), which are based upon the quantum attributes of the masson, create, respectively, the gravitational quantum units of inertial and gravitational force. The most salient discovery is that the ratio between the magnitudes of these two units of force is 137.036..., which is the fine-structure constant, β.

Notice that the fine-structure constant enters into Newton's gravitational equation in the same exact manner as it does with Coulomb's electromagnetic equation. This opens a whole new area of speculation as to its significance, which I pursue in the subjective and metaphysical Part Two.