(word processor parameters LM=8, RM=75, TM=2, BM=2)
Taken from KeelyNet BBS (214) 324-3501
Sponsored by Vangard Sciences
PO BOX 1031
Mesquite, TX 75150
There are ABSOLUTELY NO RESTRICTIONS
on duplicating, publishing or distributing the
files on KeelyNet except where noted!
October 26, 1991
CASGRAV1.ASC
--------------------------------------------------------------------
This EXCELLENT file shared with KeelyNet courtesy of
Darrell Moffitt.
--------------------------------------------------------------------
A Derivation of Newton's Constant via Casimir Potentials
and Quantum Fluctuation Effects in Vacuum
Darrell Moffitt
In recent times, numerous authors have explored the possibility that
zero-point energy (z.p.e.), the observable consequence of quantum
fluctuations in vacuum, may in some manner give rise to the
phenomena called gravity. (1-4)
Various arguments have been invoked, some suggesting that symmetry
breaking effects similar to those observed in the Standard Model
play a dominant role. (1-3)
Other authors suggest that the feebleness of gravitational coupling
reflects a natural cut-off in the frequency of electromagnetic waves
composing the vacuum. (4) These arguments, while useful, fail to
generate a straightforward derivation of Newton's constant. Be that
as it may, there are mechanisms which produce close approximations.
Two of these approximations derive from arguments based on Casimir
potentials. (5) Both approximations make use of Casimir's polar-
polar potential,
((h/2ãc^5)(w^6/6)(P1*P2)(1/R)),
describing the interaction between two polarizable systems.
The frequency cut-off is determined naturally by the dimensions of
the systems, w=(c/r); the volume polarizations (P1, P2) are
determined similarly. The factor of (1/6) indicates an integral over
w^5.
One form of Newton's constant, related elsewhere (6), produces a
value within one percent of experiment by relating the ground-state
orbital frequency of hydrogen to a cubic electron density in the
polar-polar potential, thus arriving at the expression,
G = ((hc/ã m#^2) (à^3/4ã)^6),
Page 1
where "m#" is the electron mass; "h" is Planck's constant; "c" is
the speed of light, and "à" is the electromagnetic coupling
constant.
A more accurate derivation will reveal the relation of zero-point
processes to the appearance of a universal effect, while avoiding
reference to a specific mass scale. This derivation makes use of two
Casimir expressions, the polar-polar potential quoted before, and
the wall-wall potential, (hc/2ã r^4), with an explicit form, to
first order,
F = (ãhc/480),
also known as the zero-point constant.
A curious feature of this second derivation is dual frequency
dependence, the terms of which originate in well measured attributes
of the quantum vacuum.
Known by a different interpretation as the vacuum conductivity, the
first frequency,
(å=2.65441873*10^-3/t),
is the product of c and î, the vacuum dielectric constant. The
second frequency term is the Lamb shift, w&, which measures the
effect of z.p.e. on the orbit of an electron in the ground state of
free hydrogen.
Its numeric value, by latest measurement, is
(2ã*1.0578458*10^9/t).
One may better understand the role of these two frequency scales by
conducting a dimensional analysis of Newton's constant, which can be
interpreted as
((d/t^2)(1/d1d2)),
the ratio between a volume density oscillation, (d/t^2), and two
interacting volume densities, (d1, d2). Thus, what is sought here is
some form of vacuum source density oscillation and vacuum source
density.
When one considers the vacuum conductivity to be a plasma frequency,
and divides the zero-point constant by the square of this frequency,
a small but significant virtual density factor results,
(F/(å^2*(cm)^6))= d0
with a numeric value of
(1.845214465*10^-13 (gm/cm^3)).
Virtual density oscillations (v.d.o.s) in the quantum vacuum are a
contentious issue, as no acoustic analog process has ever been
directly observed.
A simple way to conceive of such oscillations is to consider the
Page 2
vacuum polarization effects of sub-atomic particles. In this light,
a v.d.o. represents an interaction of polarization and virtual
particle currents averaged over a finite region of space.
Consider, for example, a characteristic density oscillation defined
by the core term of Casimir's polar-polar potential,
((h/2ãc^5)(4àw&/ã)^6),
numerically equal to
(2.41567929*10^-33 (gm/cm^3*t^2)).
According to the dimensional analysis of Newton's constant performed
earlier, an expression for gravitational coupling could be written
as the ratio of this density oscillation and the virtual plasma
density given above:
((h/2ãc^5)(4àw&/ã)^6/(d0)^2),
yielding the quantity
(7.09488846*10^-8 (cm^3/gm t^2)),
slightly in excess of the measured value of Newtonian gravity. A
correction factor, based in part on the zero-point constant
derivation, produces the near approximation:
G = ((h/2ãc^5)(4àw&/ã)^6/(d0)^2)*(1+(ã^2/240))^-1.5
*(1+16à^2)^-1*(1+(à^2/2)),
with a numeric value of (6.67319759*10^-8 (cm^3/gm t^2)).
This may be favorably compared to the experimental value of Newton's
constant,
(6.6732....*10^-8 (cm^3/gm t^2)).
Rigorous treatment of the derivation above requires a deep and
prolonged evaluation by quantum electrodynamic techniques and their
younger sibling, stochastic electrodynamics.
Particular attention must be given to the general nature of v.d.o.s
and their relation to pair-formation and vacuum polarization. One
must also question the origin of the reduction factor in the Lamb
shift, (4à/ã), which might be construed as a secondary result of
virtual pair orbits.
The answer to these, and similar questions, is by no means clear.
The larger question to answer, "why" there are physical constants,
is likely to find its answer in the rich structure of the quantum
vacuum itself, the "nothing" which more and more appears to be the
source of everything we term "the universe".
Page 3
Appendix
All quantities used in this paper are taken from p. 700, " Quantum
Electrodynamics", the "Advanced Series on Directions in High Energy
Physics", Vol. 7, 1987, edited by T. Kinoshita, World Scientific.
A partial list of these quantities is quoted below.
m# electron mass 9.1093897(54)*10^-31 kg
e electron charge 1.60217733(49)*10^-19 C
h/2ã Planck constant/2ã 1.05457266(63)*10^-34 Js
à^-1 inverse fine structure 1.37059895(61)*10^2
constant
c speed of light 2.99792458*10^8 ms^-1
Bibliography
1. A. Zee, "Broken-Symmetric Theory of Gravity", Phys. Rev. Lett.,
42, 7, 1979
2. A. Zee, "Horizon Problem and the Broken-Symmetric Theory of
Gravity", Phys. Rev. Lett., 44, 11, 1980
3. A. Zee, "Spontaneously generated gravity", Phys. Rev. D, 23, 4,
1981
4. H.E. Puthoff, "Gravity as a zero-point fluctuation force", Phys.
Rev. A, 39, 5, 1989
5. Larry Spruch, "Retarded, or Casimir, long-range potentials",
Physics Today, 11/86
6. Darrell Moffitt, "CPEDOG", KeelyNet file, 9/91
--------------------------------------------------------------------
If you have comments or other information relating to such topics
as this paper covers, please upload to KeelyNet or send to the
Vangard Sciences address as listed on the first page.
Thank you for your consideration, interest and support.
Jerry W. Decker.........Ron Barker...........Chuck Henderson
Vangard Sciences/KeelyNet
--------------------------------------------------------------------
If we can be of service, you may contact
Jerry at (214) 324-8741 or Ron at (214) 242-9346
--------------------------------------------------------------------
Page 4