Abstract: The general equation
governing the mass, spin and angular momentum of a Kerr-Newman black
hole applies equally well to a proton when the gravitational coupling
constant predicted by a discrete fractal paradigm is used in the equation,
along with the standard mass, spin and angular momentum of the proton.

Paper #11 of the “Selected Papers” section of this website demonstrated that subatomic particles such as protons and alpha particles can be successfully modeled in terms of Atomic Scale black holes if an appropriate gravitational “constant” is used in the calculations. We have also seen (e.g., Paper #12 of the “Selected Papers” section) that the discrete fractal scaling of the SSCP leads to the following expression for the coupling constants that characterize the gravitational interactions on different cosmological Scales:

G_{Ψ}
= [Λ^{1-D}]^{Ψ}
G_{0} , (1)

where G_{0} is the
conventional Newtonian gravitational coupling constant, which is appropriate
for Stellar Scale calculations and is equal to 6.67 x 10^{-8}
cm^{3}/g sec^{2}. The terms Λ (= 5.2 x 10^{17}) and D (=
3.174) are the dimensionless scaling constants that are fundamental
to the SSCP. The term Ψ (= …-2, -1, 0, 1, 2…) designates
specific cosmological Scales, and Ψ = 0 is assigned to the Stellar Scale.

Using Eq. (1) we can determine that the gravitational coupling constants on neighboring cosmological Scales are related by the general equation:

G_{Ψ}_{-1} = Λ^{2.174} G_{Ψ} . (2)

In the specific case of Atomic Scale (Ψ = -1) systems,

G_{-1 }
= Λ^{2.174 }
G_{0} = 2.18 x 10^{31} cm^{3}/g sec^{2}. (3)

Here we will show in a little
more detail than was given in Paper #11 that an empirical mass/spin/angular
momentum relationship determined for the proton is in good agreement
with the mass/spin/angular momentum relationship of a Kerr-Newman black
hole.

Solutions of the Einstein field equations of General Relativity for spinning and charged black holes were achieved by Kerr and Newman several decades ago. An important and well known relationship that applies to Kerr-Newman black holes is:

J = a_{*}[G_{Ψ} M^{2}/c] . (4)

The symbol J designates the
angular momentum of the object, a_{*} is referred to as the
dimensionless spin parameter, G_{Ψ} is the appropriate gravitational coupling
constant, M is the mass of the object, and c is the velocity of light.
Below we will show that this equation, which was derived primarily for
Stellar Scale black holes, also applies to the proton when the appropriate
Atomic Scale values for J, G_{Ψ} and M are inserted.

Conventional physics has determined
that the angular momentum of the proton (J_{p}) is:

J_{p}
= [j(j + 1)]^{1/2} ħ , (5)

where j is the proton’s dimensionless spin parameter, which equals ˝, and ħ is Planck’s constant divided by 2π. The SSCP asserts (New Developments 2007 – Fine Structure Constant) that

ħ = G_{-1}M^{2}/c
, (6)

where M is the revised Planck mass based on
G_{-1}, and is equal to 1.20 x 10^{-24} g.

Therefore,

J_{p}
= [1/2(1/2 + 1)]^{1/2 }[G_{-1}
M^{2}/c] = 0.866
[G_{-1} M^{2}/c] . (7)

If the proton is correctly modeled in terms of a Kerr-Newman black hole, then the following relationship should hold true in accordance with Eq. (4):

0.866 [G_{-1} M^{2}/c]
= a_{*}[G_{-1}m^{2}/c] , (8)

where m is the mass of the
proton. Eq. (8) can be simplified since the G_{-1} and
c terms cancel out. We can then insert values for M
(= 1.20 x 10^{-24} g) and m (= 1.67 x 10^{-24} g) into
the remaining equation and solve for a_{*}.

a_{*}
= 0.866 (M/m)^{2} = 0.866 (0.72)^{2}
= 0.45 . (9)

The fact that a_{*}
= 0.45 ≈
˝ is encouraging since this agrees with the proton’s empirically
and theoretically determined dimensionless spin parameter at the 90%
level. Small but currently unavoidable uncertainties involved
in determining the fundamental self-similar scaling constants Λ
and D of the SSCP preclude a more exact quantitative test at present.

The main implication of the above results is that the equation

J_{p}
= a_{*}[G_{Ψ}m^{2}/c] (10)

models the proton’s mass/spin/angular
momentum relationship correctly when the appropriate value of G_{Ψ}
is used in the calculations, in close analogy to Eq. (4). Within
the context of the SSCP, the proton appears to be an Atomic Scale Kerr-Newman
black hole.