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</o:shapelayout></xml><![endif]--></head><body lang=EN-US link="#0563C1" vlink="#954F72"><div class=WordSection1><p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif;color:#444444'>Hello Everyone,<o:p></o:p></span></p><p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif;color:#444444'><o:p> </o:p></span></p><p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif;color:#444444'>Since the discussion is on de Broglie waves, I wanted to point out that it is possible to characterize the amplitude and frequency of the underlying standing waves that produce the interference effect that we know as de Broglie waves in a moving frame of reference. As John Williamson has pointed out, waves at the particle’s Compton frequency propagating both away from a particle and towards a particle produce “standing waves” in the surrounding volume. These standing waves interfere with each other when viewed in a moving frame of reference producing de Broglie waves. <o:p></o:p></span></p><p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif;color:#444444'><o:p> </o:p></span></p><p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif;color:#444444'>I am here to say that it is possible to go much further in this analysis and ultimately derive a particle’s curvature of spacetime, the gravitational force between particles and the electrostatic force between particles with the help of a manual insertion of the fine structure constant. All we have to do is assume that the Compton frequency waves responsible for de Broglie are related to gravitational waves and look to gravitational wave equations for guidance. The recent discovery of gravitational waves supports this approach because now there is no doubt that gravitational waves exist and the gravitational wave equations are correct. Key to adopting the gravitational wave equations is the adoption of the impedance of spacetime <i>Z</i><sub>s</sub> = <i>c</i><sup>3</sup>/<i>G </i>as a fundamental characteristic of waves propagating in the spacetime field. In other words, the particle’s Compton frequency waves that are responsible for de Broglie waves also are assumed to be quantifiable waves in spacetime. <o:p></o:p></span></p><p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif;color:#444444'><o:p> </o:p></span></p><p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif;color:#444444'>We know that gravitational waves distort spacetime and encounter the tremendously large impedance of spacetime <i>Z</i><sub>s</sub> = <i>c</i><sup>3</sup>/<i>G</i>. The amplitude of gravitational waves is often specified in dimensionless strain amplitude such as <i>A</i> = </span><i><span style='font-size:12.0pt;line-height:106%;font-family:"Cambria Math",serif;color:#444444'>Δ</span></i><i><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif;color:#444444'>L</span></i><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif;color:#444444'>/<i>L</i> where </span><i><span style='font-size:12.0pt;line-height:106%;font-family:"Cambria Math",serif;color:#444444'>Δ</span></i><i><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif;color:#444444'>L</span></i><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif;color:#444444'> is the distortion of spacetime over distance <i>L</i>. The use of </span><i><span style='font-size:12.0pt;line-height:106%;font-family:"Cambria Math",serif;color:#444444'>Δ</span></i><i><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif;color:#444444'>L</span></i><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif;color:#444444'>/<i>L</i> is an approximation that assumes that the distance<i> L</i> is much shorter than 1 wavelength. Strain amplitude is actually the maximum slope of a wave and the more accurate way of expressing this maximum slope is </span><i><span style='font-size:12.0pt;line-height:106%;font-family:"Cambria Math",serif;color:#444444'>Δ</span></i><i><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif;color:#444444'>L</span></i><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif;color:#444444'>/<i><s>λ</s></i> where lambda bar is <i><s>λ</s></i> = <i>c</i>/</span><span style='font-size:12.0pt;line-height:106%;font-family:"Cambria Math",serif;color:#444444'>ω</span><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif;color:#444444'> = λ/2π. <o:p></o:p></span></p><p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif;color:#444444'><o:p> </o:p></span></p><p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif;color:#444444'>Now the next step is interesting. If the medium of spacetime has finite characteristics such as a finite impedance, then there should be boundary conditions which in turn implies that the medium must be nonlinear to accommodate these boundary conditions. Waves propagating in a nonlinear medium result in a linear term and nonlinear terms. The first and largest of the nonlinear terms scales with wave amplitude squared. For example, light passing through any transparent material (including glass and air) exhibits the optical Kerr effect which has the index of refraction increase with a constant times the electric field squared (plus higher order terms which can usually be ignored). The increase in index of refraction is the generation of a non-oscillating term caused by an oscillating term. This is well known and experimentally proven in laser physics.<o:p></o:p></span></p><p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif;color:#444444'><o:p> </o:p></span></p><p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif;color:#444444'>Is it possible that something like that happens in nature to generate gravity? Gravity bends light something like an increase in the index of refraction. There are also differences between an increase in the index of refraction and gravity, but suppose that we check out this lead. We will also be attempting to tie gravity to the underlying cause of de Broglie waves. We know <i><s>λ</s></i><sub>c</sub> so the missing term in all of this is the </span><i><span style='font-size:12.0pt;line-height:106%;font-family:"Cambria Math",serif;color:#444444'>Δ</span></i><i><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif;color:#444444'>L</span></i><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif;color:#444444'>. How much does nature allow spacetime to be distorted by particles? I am going to skip to the answer which is that the displacement allowed in the spatial domain is equal to Planck length (<i>L</i><sub>p</sub>) and in the temporal domain the displacement if Planck time (<i>T</i><sub>p</sub>). These are both very small quantities, but they are going to be put into equations with the tremendously large impedance of spacetime. <o:p></o:p></span></p><p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif;color:#444444'><o:p> </o:p></span></p><p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif;color:#444444'>For review, De Broglie waves are low frequency, long wavelength beats that occur in a moving frame of reference. The underlying source of these beats are standing waves in the surrounding spacetime field at the particle’s Compton frequency (for an electron </span><span style='font-size:12.0pt;line-height:106%;font-family:"Cambria Math",serif;color:#444444'>ω</span><sub><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif;color:#444444'>c</span></sub><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif;color:#444444'> = 7.7 x 10<sup>20</sup> s<sup>-1</sup>) and Compton wavelength. (for an electron <s>λ</s><sub>c</sub> = 3.86 x 10<sup>-13</sup> m). This high frequency and small amplitude is undetectable as a wave but the effects of these waves are detectable. I can specify the strain amplitude (L<sub>p</sub>/<s>λ</s><sub>c</sub>) of these waves and the impedance of spacetime is known. It is also possible to quantify the amplitude of these waves at any distance. The amplitude should decrease with distance as 1/<i>r</i>. However, rather than specifying the scale in meters, the particle’s natural scale is its reduced Compton wavelength (<i><s>λ</s></i><sub>c</sub> = </span><span style='font-size:12.0pt;line-height:106%;font-family:"Cambria Math",serif;color:#444444'>ħ</span><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif;color:#444444'>/mc). Therefore, I specify the distance from a particle measured in the number of reduced Compton wavelengths as: <i>N</i> = r/<i><s>λ</s></i><sub>c</sub>. </span><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif'><o:p></o:p></span></p><p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif;color:#444444'><o:p> </o:p></span></p><p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif;color:#444444'>When I generate force wave equations and insert the nonlinear wave amplitude, <i>Z<sub>s</sub></i> and <i><s>λ</s></i><sub>c</sub><sup>2</sup> as an area term I get the Newtonian gravitational equation. When I examine the effect on the surrounding spacetime field I get the weak gravity curvature of spacetime produced by a particle (<i>Gm</i>/<i>c<sup>2</sup>r</i>). <o:p></o:p></span></p><p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif;color:#444444'><o:p> </o:p></span></p><p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif;color:#444444'>When I examine the first order effect of these waves, I get a much larger effect that corresponds to the electric field and electrostatic force that would be produced by Planck charge <i>q</i><sub>p</sub> = <i>e </i>/(α)<sup>1/2</sup> ≈ 11.7 <i>e</i>). Even this is reasonable because Planck charge is based on the permeability of space ε<sub>o</sub> and Planck charge has a coupling constant to photons equal to 1. Charge e is can be thought of as a degraded form of Planck charge which compensates for various outside influences including vacuum polarization shielding. <o:p></o:p></span></p><p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif;color:#444444'><o:p> </o:p></span></p><p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt'><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif;color:#444444'>I have explained some of this in previous emails. This next part was actually a repeat of a previous email. I have talked about being able to specify amplitudes to achieve gravitational and electrostatic forces. There is only one wave, but it is possible to break this wave into four different amplitude terms (<i>A</i><sub>e</sub>, <i>A<sub>E</sub></i>, <i>A<sub>g</sub></i>, and <i>A<sub>G</sub></i>). You will not be able to understand all the implications of these amplitudes without reading other things that I have written, but you can get an overall impression. The implication of this is that both electric fields and gravitational fields are distortions of the spacetime field which have both a non-oscillating term and an oscillating term. The oscillating term (<i>A</i><sub>e</sub>) associated with an electric field gives the correct energy density in an electric field. The oscillating term associated with a gravitational field (<i>A</i><sub>g</sub>) implies that a gravitational field also has energy density just like an electric field. This is supported because it is possible to generate the curvature of spacetime using this oscillating energy density. It is also possible to show how this distributed energy density in a gravitational field results in the generation of gravitational waves when massive bodies are accelerated. Here are the 4 wave amplitudes.<o:p></o:p></span></p><p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt;line-height:normal'><span style='font-size:12.0pt;font-family:"Times New Roman",serif;color:#20188C'> <o:p></o:p></span></p><p class=MsoNormal style='text-align:justify;line-height:115%'><i><span style='font-size:11.5pt;line-height:115%;font-family:"Cambria Math",serif;color:#CC0000'>A<sub>e</sub> </span></i><span style='font-size:11.5pt;line-height:115%;font-family:"Cambria Math",serif;color:#CC0000'>= </span><span style='font-size:11.5pt;line-height:115%;font-family:"Cambria Math",serif;color:#C00000'>α<sup>1/2</sup><i>A<sub>β</sub></i>/<i>N<sup>2</sup></i> </span><span style='font-size:11.5pt;line-height:115%;font-family:"Cambria Math",serif'>electromagnetic standing wave amplitude oscillating at <i>ω<sub>c</sub></i> (charge <i>e</i>). This amplitude at the particle’s Compton angular frequency gives the energy density for the electric field surrounding the particle with charge e <o:p></o:p></span></p><p class=MsoNormal style='text-align:justify;line-height:115%'><i><span style='font-size:11.5pt;line-height:115%;font-family:"Cambria Math",serif;color:#CC0000'>A</span></i><sub><span style='font-size:11.5pt;line-height:115%;font-family:"Cambria Math",serif;color:#CC0000'>𝔼</span></sub><!--[if gte msEquation 12]><m:oMath><i><span style='font-size:14.0pt;line-height:115%;font-family:"Cambria Math",serif;color:#CC0000'><m:r> </m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:106%;font-family:"Calibri",sans-serif;position:relative;top:4.5pt;mso-text-raise:-4.5pt;mso-fareast-language:EN-US'><img width=5 height=29 id="_x0000_i1025" src="cid:image003.png@01D167F6.B7E72690"></span><![endif]><span style='font-size:11.5pt;line-height:115%;font-family:"Cambria Math",serif;color:#CC0000'>= </span><span style='font-size:11.5pt;line-height:115%;font-family:"Cambria Math",serif;color:#C00000'>α<sup>1/2</sup><i>A<sub>β</sub></i>/<i>N</i> </span><span style='font-size:11.5pt;line-height:115%;font-family:"Cambria Math",serif'>electromagnetic non-oscillating strain amplitude (charge <i>e</i>) This gives the voltage in dimensionless Planck units and can be converted to electric field strength.<o:p></o:p></span></p><p class=MsoNormal style='text-align:justify;line-height:115%'><i><span style='font-size:11.5pt;line-height:115%;font-family:"Cambria Math",serif;color:#CC0000'>A<sub>g</sub> </span></i><span style='font-size:11.5pt;line-height:115%;font-family:"Cambria Math",serif;color:#C00000'>= <i>A<sub>β</sub><sup>2</sup></i>/<i>N<sup>2</sup> </i> </span><span style='font-size:11.5pt;line-height:115%;font-family:"Cambria Math",serif'>gravitational standing wave amplitude oscillating at <i>2ω<sub>c</sub></i>. This implies that gravitational fields have observable energy density and creates curved spacetime. <o:p></o:p></span></p><p class=MsoNormal style='text-align:justify;line-height:115%'><i><span style='font-size:11.5pt;line-height:115%;font-family:"Cambria Math",serif;color:#CC0000'>A<sub>G</sub> </span></i><span style='font-size:11.5pt;line-height:115%;font-family:"Cambria Math",serif;color:#C00000'>=<i>A<sub>β</sub><sup>2</sup></i>/<i>N</i> </span><span style='font-size:11.5pt;line-height:115%;font-family:"Cambria Math",serif'>gravitational non-oscillating strain amplitude (spacetime curvature). This is equal to Gm/c<sup>2</sup>r which is the curvature of spacetime produced by a fundamental particle.<o:p></o:p></span></p><p class=MsoNormal style='text-align:justify;line-height:115%'><span style='font-size:11.5pt;line-height:115%;font-family:"Cambria Math",serif'> <o:p></o:p></span></p><p class=MsoNormal><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif'>John M.<o:p></o:p></span></p></div></body></html>