hc
Ε = --
λ
F λ X
Ε = -----
2π
2π hc
F = ------
X λ**2
This force is applied over a surface in space, proportional to the wavelength squared times some constant. This is normalized across a region of space quantified as λ**2, and we compute a resistance factor, which is independent of the wavelength.
2π hc
R = ------
X
where X is an as of yet unknown constant. Next, we compute the attractive force factor between the 'plates' of the equivalent photon capacitor by considering it a parallel plate capacitor, computing the work required to increase the spacing by a small amount and then normalize this to the region of space it's acting on.
q**2
Rcr = ----
2 ε0
Setting them equal, we can then solve for X.
4π ε0 c h
X = ----------
q**2
2π
X = ---
α
Once more, the fine structure constant appears as another independent constant. Substituting the computed value of X into the expression for R, we arrive at the resistance of the Universe to the curvature, as seen by a photon is then given by,
R = α h c = 1.4495E-27 nt m**2
This constant isn't really completely independent since we can show that it's a consequence of the fine structure constant relationships indicated by the geometric photon model and the energy photon model. The real purpose of calculating this is to quantify the intrinsic resistance of the Universe to curvature. Additionally, we can derive the force equation for charge from these relationships, thus providing a linkage between curvature and charge. The occurrence of the fine structure constant in the expression for the Universes intrinsic opposition to curvature is at the root of why it keeps showing up in other places.
From Part 3, we derived the force of the gravitational attraction between two equal mass particles as,
K R m**2
F(r) = --- * ----
2 r**2
where KR/2 is equal to the gravitational constant, G. The R we computed earlier is the same R that appears in this equation. When we plug in the computed value of R and the Gravitational constant, G, we can solve for K.
R = α h c
2 G
K = ----- = 5.78445E17 1/kg**2
α h c
Substituting for α, we get,
c μ0 q**2
α = ---------
2 h
c**2 μ0 q**2
R = ------------
2
2 G
K = ------------
c**2 μ0 q**2
There may be a way, to derive K independently which would make the Gravitational constant, G, a computed constant. There also appears to be a relationship between the expression of K and Einstein's Theory of General Relativity. Considering just the real part for distances >> r0, we can restate G(r) from Part 3 and contrast it with Einsteins equation. Note that K' is the mass times the constant K.
/ r**2 - r0**2 \
G(r) = K' sqrt( ----------------- )
\(r**2 + r0**2)**2/
8 G m sqrt(r**2 - r0**2)
G(r) = ---- ------- ------------------
c**2 μ0 q**2 (r**2 + r0**2)
We can compare this to Einsteins equation of GTR. Note that in this form, the G on the left is a tensor representation of curvature and the G on the right is the Gravitational constant G. The Gravitational constant is often represented as k in order to avoid this confusion.
8 π G
G = ------ T
c**2
For this form of Einsteins equation, the units of T (the stress energy tensor) are mass density (kg/m**3) which makes the units of G (the curvature tensor) the reciprocal of an area (1/m**2). For the case of CTR curvature, it's units are 1/kg*m. The difference is that GTR curvature represents the sum of curvature components contributed from multiple particles, while CTE curvature is the contribution of only a single particle. Combining curvature from multiple particles is not a simple summation since the curvature function is expressed relative to the reference frame of the particle and the reference frame of the resulting sum is the center of mass of the collection of particles.
To convert CTE curvature to GTR curvature, we start by considering a uniformly packed sphere of N particles whose radius is r'. Based on spherical symmetry arguments, we can consider that the effects of the particles are uniform at distances more than a couple of sphere diameters away. We can compute an adjustment factor to convert the effect of a single particle into the effect of the collection by scaling it based on a relative mass density ratio for the collection of particles.
Md = (N π m)/r'
where N is the number of particles of mass m and r' is the radius of the sphere containing the particles. For r >> r' and r' >> r0 and N sufficiently large, we can restate the CTE curvature as,
8 π G N m**2 1
G(r) = ------ - ------- -
c**2 r' μ0 q**2 r
This seems to provide a way to describe the stress energy tensor for for a collection of particles as a function of the effect of individual particles.
Another interesting form arises when we substitute μ0 with 1/ε0c**2.
N m**2 1
G(r) = 8 π G ε0 - ---- -
r' q**2 r
If we consider m to be the average mass of protons and neutrons, CTE curvature seems related to the relative force of charge times the relative force of gravity for individual particles.
From the CTE expression of curvature, we can derive an expression for the curvature associated with a photon by substituting the fine structure constant α, the energies associated with mass and equivalent photons, considering r0 to be the wavelength and translating the coordinate system from a polar system to a Cartesian system where x > 0 is where the photon is going and x < 0 is where the photons been.
8 π G c m sqrt(r**2 - r0**2)
G(r) = ------ ------- ------------------
c**2 2 h α (r**2 + r0**2)
E = m c**2
E = hc/λ
8 π G E
G(x) = ------ ------- f(x)
c**2 2 E α λ
8 π G 1
G(r) = ------ ----- f(x)
c**2 2 α λ
Part 5 summarizes what we have determined and describes how the physical model of the photon can only be the result of equal and opposite amounts of curvature and anticurvature.
(C) 1997-2004 George White, All Rights ReservedCTE Home, Doc map