Technical Notes 2006

Discrete Self-Similar Scaling For Charge And Angular Momentum [October 2006]

I. Scaling For Electromagnetic Charge

Given that

e² = a ħ c,

where a is the fine structure constant (≈ 1/137.036), ħ is Planck’s constant divided by 2π (≈ 1.054 x 10^-27 erg sec) and c is the velocity of light (≈ 2.99 x 10¹⁰ cm/sec), the dimensionality of e² is

(ML²/T²)(T)(L/T),

or ML³/T².

According to the SSCP, e² would scale as Λ^DΛ³/Λ² ≈ Λ^D+1.

Therefore, e would scale as Λ^(D+1)/2, and since D ≈ 3.174,

e_Ψ ≈ Λ^2.087 e_Ψ-1    

   e_Ψ ≈ 9.40 x 10³⁶ e_Ψ-1

Note that whereas mass scales as Λ^D ≈ 1.70 x 10⁵⁶ between neighboring Scales, charge increases by the much smaller factor of Λ^(D+1)/2 ≈ 10³⁷ when we go “up” one cosmological Scale.  This implies that the ratio of unbalanced to balanced Ψ = -1 charges in a Ψ = 0 system is about 10³⁷/10⁵⁶ ≈ 5.4 x 10^-20.  So only a very small fraction of the Ψ = 0 mass has a net charge, but this appears to be enough to produce Ψ = 0 electromagnetic phenomena that is self-similar to Ψ = -1 electromagnetic phenomena.

II. Scaling For Angular Momentum

The dimensions of angular momentum (J) are: ML²/T.

According to the SSCP, J should scale as Λ^DΛ²/Λ ≈ Λ^D+1 ≈ Λ^4.174 ≈ 8.84 x 10⁷³.

Therefore:

J_Ψ ≈ 8.84 x 10⁷³ J_Ψ-1

for discrete self-similar analogues on neighboring Scales.

III. The Stellar Scale Planck's "Constant"

Angular momentum in the quantum realm is usually found in multiples of h or ħ, which have values of 6.626 x 10^-27 erg sec and 1.054 x 10^-27 erg sec, respectively.

The SSCP predicts that when we have enough high-quality observational data, the Stellar Scale will manifest a comparable discreteness in angular momentum phenomena with:

h_Ψ=0 ≈ Λ^D+1 h_Ψ=-1 ≈ 5.860 x 10⁴⁷ erg sec ,

and

ħ_Ψ=0 ≈ Λ^D+1 ħ_Ψ=-1 ≈ 9.325 x 10⁴⁶ erg sec.