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                                 October 26, 1991

                                   CASGRAV1.ASC
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               This EXCELLENT file shared with KeelyNet courtesy of
                                 Darrell Moffitt.
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             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

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