265 lines
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265 lines
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Plaintext
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(word processor parameters LM=8, RM=75, TM=2, BM=2)
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Taken from KeelyNet BBS (214) 324-3501
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Sponsored by Vangard Sciences
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PO BOX 1031
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Mesquite, TX 75150
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There are ABSOLUTELY NO RESTRICTIONS
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on duplicating, publishing or distributing the
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files on KeelyNet except where noted!
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October 26, 1991
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CASGRAV1.ASC
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--------------------------------------------------------------------
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This EXCELLENT file shared with KeelyNet courtesy of
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Darrell Moffitt.
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--------------------------------------------------------------------
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A Derivation of Newton's Constant via Casimir Potentials
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and Quantum Fluctuation Effects in Vacuum
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Darrell Moffitt
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In recent times, numerous authors have explored the possibility that
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zero-point energy (z.p.e.), the observable consequence of quantum
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fluctuations in vacuum, may in some manner give rise to the
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phenomena called gravity. (1-4)
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Various arguments have been invoked, some suggesting that symmetry
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breaking effects similar to those observed in the Standard Model
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play a dominant role. (1-3)
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Other authors suggest that the feebleness of gravitational coupling
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reflects a natural cut-off in the frequency of electromagnetic waves
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composing the vacuum. (4) These arguments, while useful, fail to
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generate a straightforward derivation of Newton's constant. Be that
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as it may, there are mechanisms which produce close approximations.
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Two of these approximations derive from arguments based on Casimir
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potentials. (5) Both approximations make use of Casimir's polar-
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polar potential,
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((h/2<>c^5)(w^6/6)(P1*P2)(1/R)),
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describing the interaction between two polarizable systems.
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The frequency cut-off is determined naturally by the dimensions of
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the systems, w=(c/r); the volume polarizations (P1, P2) are
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determined similarly. The factor of (1/6) indicates an integral over
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w^5.
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One form of Newton's constant, related elsewhere (6), produces a
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value within one percent of experiment by relating the ground-state
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orbital frequency of hydrogen to a cubic electron density in the
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polar-polar potential, thus arriving at the expression,
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G = ((hc/<2F> m#^2) (<28>^3/4<>)^6),
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Page 1
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where "m#" is the electron mass; "h" is Planck's constant; "c" is
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the speed of light, and "<22>" is the electromagnetic coupling
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constant.
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A more accurate derivation will reveal the relation of zero-point
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processes to the appearance of a universal effect, while avoiding
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reference to a specific mass scale. This derivation makes use of two
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Casimir expressions, the polar-polar potential quoted before, and
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the wall-wall potential, (hc/2<> r^4), with an explicit form, to
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first order,
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F = (<28>hc/480),
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also known as the zero-point constant.
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A curious feature of this second derivation is dual frequency
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dependence, the terms of which originate in well measured attributes
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of the quantum vacuum.
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Known by a different interpretation as the vacuum conductivity, the
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first frequency,
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(<28>=2.65441873*10^-3/t),
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is the product of c and <20>, the vacuum dielectric constant. The
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second frequency term is the Lamb shift, w&, which measures the
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effect of z.p.e. on the orbit of an electron in the ground state of
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free hydrogen.
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Its numeric value, by latest measurement, is
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(2<>*1.0578458*10^9/t).
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One may better understand the role of these two frequency scales by
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conducting a dimensional analysis of Newton's constant, which can be
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interpreted as
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((d/t^2)(1/d1d2)),
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the ratio between a volume density oscillation, (d/t^2), and two
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interacting volume densities, (d1, d2). Thus, what is sought here is
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some form of vacuum source density oscillation and vacuum source
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density.
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When one considers the vacuum conductivity to be a plasma frequency,
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and divides the zero-point constant by the square of this frequency,
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a small but significant virtual density factor results,
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(F/(<28>^2*(cm)^6))= d0
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with a numeric value of
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(1.845214465*10^-13 (gm/cm^3)).
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Virtual density oscillations (v.d.o.s) in the quantum vacuum are a
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contentious issue, as no acoustic analog process has ever been
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directly observed.
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A simple way to conceive of such oscillations is to consider the
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Page 2
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vacuum polarization effects of sub-atomic particles. In this light,
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a v.d.o. represents an interaction of polarization and virtual
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particle currents averaged over a finite region of space.
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Consider, for example, a characteristic density oscillation defined
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by the core term of Casimir's polar-polar potential,
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((h/2<>c^5)(4<>w&/<2F>)^6),
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numerically equal to
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(2.41567929*10^-33 (gm/cm^3*t^2)).
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According to the dimensional analysis of Newton's constant performed
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earlier, an expression for gravitational coupling could be written
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as the ratio of this density oscillation and the virtual plasma
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density given above:
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((h/2<>c^5)(4<>w&/<2F>)^6/(d0)^2),
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yielding the quantity
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(7.09488846*10^-8 (cm^3/gm t^2)),
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slightly in excess of the measured value of Newtonian gravity. A
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correction factor, based in part on the zero-point constant
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derivation, produces the near approximation:
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G = ((h/2<>c^5)(4<>w&/<2F>)^6/(d0)^2)*(1+(<28>^2/240))^-1.5
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*(1+16<31>^2)^-1*(1+(<28>^2/2)),
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with a numeric value of (6.67319759*10^-8 (cm^3/gm t^2)).
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This may be favorably compared to the experimental value of Newton's
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constant,
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(6.6732....*10^-8 (cm^3/gm t^2)).
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Rigorous treatment of the derivation above requires a deep and
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prolonged evaluation by quantum electrodynamic techniques and their
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younger sibling, stochastic electrodynamics.
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Particular attention must be given to the general nature of v.d.o.s
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and their relation to pair-formation and vacuum polarization. One
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must also question the origin of the reduction factor in the Lamb
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shift, (4<>/<2F>), which might be construed as a secondary result of
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virtual pair orbits.
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The answer to these, and similar questions, is by no means clear.
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The larger question to answer, "why" there are physical constants,
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is likely to find its answer in the rich structure of the quantum
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vacuum itself, the "nothing" which more and more appears to be the
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source of everything we term "the universe".
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Page 3
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Appendix
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All quantities used in this paper are taken from p. 700, " Quantum
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Electrodynamics", the "Advanced Series on Directions in High Energy
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Physics", Vol. 7, 1987, edited by T. Kinoshita, World Scientific.
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A partial list of these quantities is quoted below.
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m# electron mass 9.1093897(54)*10^-31 kg
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e electron charge 1.60217733(49)*10^-19 C
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h/2<> Planck constant/2<> 1.05457266(63)*10^-34 Js
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<20>^-1 inverse fine structure 1.37059895(61)*10^2
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constant
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c speed of light 2.99792458*10^8 ms^-1
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Bibliography
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1. A. Zee, "Broken-Symmetric Theory of Gravity", Phys. Rev. Lett.,
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42, 7, 1979
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2. A. Zee, "Horizon Problem and the Broken-Symmetric Theory of
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Gravity", Phys. Rev. Lett., 44, 11, 1980
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3. A. Zee, "Spontaneously generated gravity", Phys. Rev. D, 23, 4,
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1981
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4. H.E. Puthoff, "Gravity as a zero-point fluctuation force", Phys.
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Rev. A, 39, 5, 1989
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5. Larry Spruch, "Retarded, or Casimir, long-range potentials",
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Physics Today, 11/86
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6. Darrell Moffitt, "CPEDOG", KeelyNet file, 9/91
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--------------------------------------------------------------------
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If you have comments or other information relating to such topics
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as this paper covers, please upload to KeelyNet or send to the
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Vangard Sciences address as listed on the first page.
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Thank you for your consideration, interest and support.
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Jerry W. Decker.........Ron Barker...........Chuck Henderson
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Vangard Sciences/KeelyNet
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--------------------------------------------------------------------
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If we can be of service, you may contact
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Jerry at (214) 324-8741 or Ron at (214) 242-9346
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--------------------------------------------------------------------
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Page 4
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