International Journal of Astronomy and Astrophysics, 2012, 2, 125-128
http://dx.doi.org/10.4236/ijaa.2012.23017 Published Online September 2012 (http://www.SciRP.org/journal/ijaa)
Convergent Calculations That Dark Solutions Are
Reflective of Mass-Energy yet to Occur
Michael A. Persinger
Laurentian University, Sudbury, Canada
Received February 4, 2012; revised March 15, 2012; accepted March 22, 2012
ABSTRACT
The discrepancy between the observed and expected estimates from universal constants of mass-energy within the uni-
verse is in the order of a factor of ~10. Discrepancies between numerical solutions between the models of Dirac, Szy-
dowski-Godlowski, and Friedman could be accommodated by the gnomonic solution of 0.44 for a square that displays
both linearity and curvature (flare). This value is also reflected in dimensionless parameter A, the term for 4D-G (gravi-
tational constant) transformation, and the optimal k in Friedman’s universe. One interpretation from G∂ (density), as-
suming an effective average mass of 1 proton/m3 as a universal, one-particle force, is that dark solutions reflect the
matter yet to occur in the open cold matter model of ~90 billion years.
Keywords: Dark Matter; Gravity; Physical Cosmology; Square Gnomon; Age of Universe
1. Introduction
The accommodation of dark matter and energy within
contemporary values for the constraints from constants
and known space-time boundaries has been subjected to
multiple formulations [1,2]. Two central contradictions
are the discrepancy by a factor of ~10 between age of the
universe and its estimated mass and density and the si-
multaneous requirement for both curvature and non-
curvature for its shape. In the present paper a potential
explanation for the “origin” of dark matter and how it
might relate to a mathematical form that intrinsically
contains properties of both curvature and linearity is
explored.
2. The Dimensionless Constant in Dynamic
Pressure to G Conversion
Paul Dirac [3] suggested that within a zero-curvature
universe the mass would be finite and could be described
as a large dimensionless number of 1078. Assuming a unit
proton per m3, which is consistent with a pressure of an
effective one-particle force [1], Persinger [4] estimated
the intrinsic universal pressure (∂c2), where ∂ is density
and c = velocity of light) of 1.5 × 10−10 kg/m·s2. The
transform required for an equivalence to G, the gravi-
tational constant (m3 kg·s), was m4/kg2. Assuming the
width of the universe to be ~1026 m (1, the cos-
mological constant), the value for m4 is 10104 (an extra
dimension to 1078) and hence the mass would be 1052 kg.
If Dirac’s number represented three-dimensional space
and there was an average of 1 proton/m3 the mass would
be about 10% (1051 kg) of that estimate.
However to obtain the equivalence of coefficients be-
tween the estimated intrinsic “dynamic pressure” and G,
the transformation term was 0.44 m4/kg2 which is re-
markably similar to the dimensionless parameter A [4]
which has been calculated to average 0.46 with a lower
range that would include 0.44. The parameter has been
employed to constrain models for dark energy and re-
flects a peak observed for baryonic acoustic oscillations
[2]. One the consequences of this term when included in
various models of the density parameter Ω, the ratio of
the observed density (∂) to the critical density ∂c of a
Friedmann-type universe, is that the universe is almost
flat. The typical equation for the density parameter is:


2
8π3HG  (1)
where H is Hubble’s parameter.
In numbers the quantity 8 × 3.14 × 6.67 × 10−11 m3/kg
s2 × 0.2 × 1.67 × 10−27 kg/m3 divided by 3 × 5.89 × 10−36
s−2 (assuming 75 km/s/MParsec), i.e. h = 2.4 × 10−18 s−1,
is 0.032. This is a factor of 10 smaller than the precise
model of Syzdlowski and Godlowski [5] whose Ωm,0 so-
lution was a concentric centroid of 0.30. Due to pri-
mordial nucleosynthesis they assumed that for h = 1 (H0 =
100 km/s/MParsec) baryonic mater would constitute a
term of 0.05 indicating that the greatest proportion of
their Ωm,0 solution is from non-baryonic dark matter. This
approximately 10 fold difference is consistent with the
discrepancy between the mass-energy equivalent pre-
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M. A. PERSINGER 126
dicted by Dirac’s number and the value derived from
density matched to G.
3. Friedman’s Expansion and ~0.44
Friedman’s conceptualization of the expansion of the
universe, Hubble’s constant, was


2
H8π3kaG 


2


(2)
(the latter term being the product of the inverse of Lamb-
da-the cosmological constant) and k, intrinsic curva-
ture). The square root of H is 9 × 107 m/s. However
Hubble’s constant is per MegaParsec and there are 3.1 ×
1022 m/MPsec. From an estimated universal width of 1026
m, the universe is ~3000 MPsec. H (9 × 107 m/s) di-
vided by 3 × 103 MPsec, is 3 × 104 m/s or 30 km/s. The
current estimated mean value for Hubble’s constant,
empirically, is 75 km/s. The curvature value is 30 km/s
divided by 75 or 0.40. This value is within measurement
variability of 0.44. If 0.44 were employed as the re-
ference the velocity in this model would be ~68 km/s/
MParsec.
4. The Congruent Gnomon Solution
There have been many thoughtful explorations of com-
plex geometries to accommodate the universe’s shape
[e.g., 6-11]. A less mathematically complex potential
solution is the gnomon, which is a form that when added
to some form results in a new form similar but not
identical to the original [12]. The flare (or curvature of
spiral) solution to a square winkle’s gnomon is λ =
2/π·ln2 = 0.44, a logarithmic spiral that circumscribes the
self-similarity of an infinite constructive process. Al-
though infinite, the process is bounded with finite peri-
meters. This value is also within the range of the dimen-
sionless constant.
The consideration of a “flat” square and its intrinsic
curvature might also be a candidate to explain the pro-
perties of basic forces. The essence of squares is their
symmetrical right angles which dominate the concep-
tualization of both Euclidean and non-Euclidean space.
Superimposed upon this primary would be the non-linear
component of an expanding curvature perceived as a
spiral whose radius doubles every 90˚ that would be re-
quired for the exactness of any calculations. Depending
upon the assumptions of the organization of structure
between Planck’s length and that occupied by matter
(~10−15 m) the integrated finite perimeter would be be-
tween about a factor of 16 and 8π longer with 6 addi-
tional levels but wrapped within four-dimensional space.
5. Dark Matter as Potential Matter
Solutions that assume limits or boundaries of a system
can be more amenable to its parsimonious representation
because the dynamics become static. Across levels of
scientific discourse [13] there is a clear relationship
between the absolute values of the ∆s (the increment of
space) being measured to discern a phenomenon and the
optimal ∆t (increment of time) required for it to be
observed (measured) as an integral unit. As defined by
the Nyquist limit the threshold for discerning a process
must be >2 ∆t. At the maximum boundary, the universe,
where both ∆s and ∆t = 1, this would not occur. In other
words there would be no process operations that usually
complicate the geometry and the temporal properties or
relationships between units of matter. The universe
would be static (fixed) because there would be no time.
The most parsimonious relationship that defines when
this maximum time might occur is ∂G. Assuming a
density of one proton per unit m3 the quantity is (1.61 ×
10−27 kg/m3) × 6.67 × 10−11 m3/kg·s2 or 10.74 × 10−38 Hz2
or the equivalent of 3 × 1018 s (90 billion years). This
value is similar to estimates by Hoffman et al. [6] who
calculated that in the open cold matter (OCDM) model
the boundary for the fate of the universe (the final epoch)
would be 89.2 billon years. Assuming the contemporary
outer range of the age of the universe to be 13 billions of
years or 4.1 × 1017 s, this would indicate that the current
formation is ~14% of the ultimate boundary condition.
Ordinary baryonic matter accounts for 10% to 20% [14]
of the masses of major galactic clusters which have been
attributed to “dark matter”.
One interpretation of this congruence is that the
energy-mass equivalence attributed to smaller increments
of matter might be applied to the entire space occupied
by all mass. If there is potential energy then the presence
of potential mass, not yet manifested, is one implication.
From this perspective the influences attributed to “dark
matter” reflect the matter yet to be formed within an
expanding universe. The involvement of the total time of
the universe within which there is no process would also
be consistent with the suggestion by Balakin et al. [1]
that tachyon matter is a candidate for dark matter and
energy.
6. The Casimir Contribution
One of the assumptions of the Casimir effect [15] is
matter is formed from virtual particles within the vacuum
potential if the boundary of an electromagnetic field is
expanding. The challenge to an expanding universal
boundary and the matter within it is often facilitated by
demonstrating a coupling between gravitational and elec-
trodynamic processes. Assuming the intrinsic pressure
within the universe to be 15 × 10−11 Pa [4] and to equate
∂c2 with G and the existence of a concentric second
boundary around the universal boundary that acts against
passive expansion, the separation between the two neu-

M. A. PERSINGER 127
tral boundaries would be:


14
2
aπcS 240F



 (3)
where ћ is the modified Planck’s constant, F is the force
derived from ∂c2 and S is the surface area of the universe
assuming a circumference is between 1026 to 1027 m. The
resulting separation between these two concentric sur-
faces of the universal boundary would be 54 µm [16].
Assuming this thin shell is a black body the equivalent
temperature from Wein’s law is 53˚K and according to
Stephan’s law, the power density would be the product of
T4 and the constant 5.67 × 10−8 W/m2, or 0.45 W/m2.
If Varshni’s [17] assessment of the distribution of red-
and blue-shift data are applicable, that is our solar system
is near the center of distributions of galaxies and by
inference the universe, then one could assume that the
energy generated by and throughout the black body shell
produced by the Casimir effect would decrease as a
function of r−1 rather than r−2 in all directions. Hence, the
power density reaching radio telescopes would be (0.45
W/m2)/~1026 m or in the order of hundreds of mJy (1
Janksy = 10−26 W/m2 Hz). If the Casimir pressure (1.5 ×
10−10 Pa or kg/m·s2) is multiplied by the estimated vo-
lume of the universe (~1078 m3) the energy is 1.5 × 1068 J
or about 10% of the total energy-mass equivalence based
upon 1052 kg or 8πG/c4·Tuv.
The implications for these solutions is that with the
presence of 0.45 W/m2 and an estimated universal surface
area of 4.5 × 1053 m2 there would ~2 × 1053 J/s available
with a mass equivalence of 2 × 1036 kg or about 1 million
solar masses per sec. At this rate the current mass would
be matched within 1016 s or within a factor of 0.1 of the
current age of the universe. This rate of acceleration is
well within the range expected within the Szydlowski-
Godlowski model [5]. The transformation of virtual par-
ticles to “real” particles by the Casimir process not only
reiterates the intricate connection between it and G [18]
but suggests that dark matter and energy would be our
present inferential measurement of the virtual condition.
7. Conclusion
The accommodation of the approximately 10% discre-
pancy between expected and observed mass-energy equi-
valents has been considered the basis for the presumption
of dark matter and energy which has been interconnected
with spatial curvature and one particle forces. The simu-
ltaneous accommodation of linear and non-linear geo-
metry might be accomplished by the systematic appli-
cation of the gnomonic solution for the square of 0.44
which is a value that solves for several proportions
including the Friedmann curvature k and dimensionless
parameter A. One possible interpretation of the results
developed in this paper is that dark matter and energy are
manifestations from Casimir virtual particles of what is
yet to occur within a system determined by G within a
OCDM model of the universe whose fate is ~90 billion
years.
8. Acknowledgements
Thanks to Blake T. Dotta, Lucas Tessaro and Ghislaine F.
Lafreniere for technical comments concerning the manu-
script.
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