(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