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</o:shapelayout></xml><![endif]--></head><body lang=EN-US link="#0563C1" vlink="#954F72"><div class=WordSection1><h1 style='margin-top:0in;line-height:18.0pt;background:#F7F7F4'><span style='font-size:12.0pt;font-family:"Times New Roman",serif;color:windowtext'>Hello All,<o:p></o:p></span></h1><h1 style='margin-top:0in;line-height:18.0pt;background:#F7F7F4'><span style='font-size:12.0pt;font-family:"Times New Roman",serif;color:windowtext'><o:p> </o:p></span></h1><h1 style='margin-top:0in;line-height:18.0pt;background:#F7F7F4'><span style='font-size:12.0pt;font-family:"Times New Roman",serif;color:windowtext'>This second post shows that there is a previously unrecognized relationship between a particle’s reduced Compton wavelength <i><s>λ</s></i><sub>c</sub> and the particle’s Schwarzschild radius. This post will show the reason why this relationship exists. In my “Foundation” paper and “Aether” papers (attached to my previous post) I state that spacetime has a single field rather than the 17 overlapping fields of the standard model. The “spacetime field” has multiple resonances which correspond to the Compton frequencies of the various fundamental particles of the standard model. The spacetime field is a sea of dipole waves in spacetime with dimensionless strain amplitude: <o:p></o:p></span></h1><h1 style='margin-top:0in;line-height:18.0pt;background:#F7F7F4'><span style='font-size:12.0pt;font-family:"Times New Roman",serif;color:windowtext'><o:p> </o:p></span></h1><h1 style='margin-top:0in;line-height:18.0pt;background:#F7F7F4'><span style='font-size:12.0pt;font-family:"Times New Roman",serif;color:windowtext'> <i>A</i><sub>s</sub> = <i>L</i><sub>p</sub>/<i><s>λ</s></i> = <i>T<sub>p</sub>ω</i> <o:p></o:p></span></h1><h1 style='margin-top:0in;line-height:18.0pt;background:#F7F7F4'><span style='font-size:12.0pt;font-family:"Times New Roman",serif;color:windowtext'><o:p> </o:p></span></h1><h1 style='margin-top:0in;line-height:18.0pt;background:#F7F7F4'><span style='font-size:12.0pt;font-family:"Times New Roman",serif;color:windowtext'>These waves are all frequencies up to Planck angular frequency ω<sub>p</sub> but there are also various resonant frequencies which we know as virtual particles. In the Foundation paper I explain that I view gravity as a nonlinear effect which scales with wave amplitude squared (to a first approximation). Equations 15 to 21 of the Foundation paper support this prediction. For example, the only difference between the gravitational force Eq. 15 and the electrostatic force Eq. 16 is that the strain amplitude in Eq. 15 is <i>A<sub>s</sub></i><sup>2</sup> and in Eq. 16 this is <i>A<sub>s</sub></i><sub> </sub>(not squared). Other equations also show that gravity is a nonlinear effect which scales with amplitude squared <i>A<sub>s</sub></i><sup>2</sup>.<o:p></o:p></span></h1><h1 style='margin-top:0in;line-height:18.0pt;background:#F7F7F4'><span style='font-size:12.0pt;font-family:"Times New Roman",serif;color:windowtext'><o:p> </o:p></span></h1><h1 style='margin-top:0in;line-height:18.0pt;background:#F7F7F4'><span style='font-size:12.0pt;font-family:"Times New Roman",serif;color:windowtext'>Now I am ready to give my new insight. I have discovered that there is a fundamental relationship between a particle’s Compton radius <i><s>λ</s></i><sub>c</sub> and its Schwarzschild radius <i>R</i><sub>s</sub>. Most important, I have discovered the </span><b><span style='font-size:14.0pt;font-family:"Times New Roman",serif;color:windowtext'>reason</span></b><span style='font-size:12.0pt;font-family:"Times New Roman",serif;color:windowtext'> for this relationship and it perfectly supports my model of the universe. In my model of a fundamental particle (my rotar model) is a dipole wave in spacetime rotation at the speed of light at its Compton angular frequency </span><span style='font-size:12.0pt;font-family:"Cambria Math",serif;color:windowtext'>ω</span><sub><span style='font-size:12.0pt;font-family:"Times New Roman",serif;color:windowtext'>c</span></sub><span style='font-size:12.0pt;font-family:"Times New Roman",serif;color:windowtext'>. It is one Compton wavelength in radius which means that its mathematical radius is its reduced Compton wavelength <s>λ</s><sub>c</sub> = ħ/mc = c/ω<sub>c</sub> (single loop). This is the natural unit of length for a fundamental particle and is the reason that force equations become very simple when distance is expressed as the number (<i>N</i>) of reduced Compton wavelengths rather than meters (<i>N</i> = <i>r</i>/<i><s>λ</s></i><sub>c</sub>). In order to mathematically analyze the particle characteristics, it is necessary to also specify the wave amplitude and the impedance of spacetime. The particle’s strain amplitude is <i>A</i><sub>s</sub> = <i>L</i><sub>p</sub>/<i><s>λ</s></i><sub>c</sub> =<i> T<sub>p</sub>ω<sub>c</sub></i> and the impedance of spacetime is: Z<sub>s</sub> = c<sup>3</sup>/G ≈ 4x10<sup>35</sup> kg/s. This combination of component parts has been analyzed in the Foundation paper and the result is that the particle model has internal energy <i>E</i><sub>i</sub> = </span><span style='font-size:12.0pt;font-family:"Cambria Math",serif;color:windowtext'>ħω</span><sub><span style='font-size:12.0pt;font-family:"Times New Roman",serif;color:windowtext'>c</span></sub><span style='font-size:12.0pt;font-family:"Times New Roman",serif;color:windowtext'> which is correct. <o:p></o:p></span></h1><h1 style='margin-top:0in;line-height:18.0pt;background:#F7F7F4'><span style='font-size:12.0pt;font-family:"Times New Roman",serif;color:windowtext'><o:p> </o:p></span></h1><h1 style='margin-top:0in;line-height:18.0pt;background:#F7F7F4'><span style='font-size:12.0pt;font-family:"Times New Roman",serif;color:windowtext'>The Schwarzschild radius of this rotating wave particle model is: <i>R</i><sub>s</sub> = <i>Gm</i>/<i>c<sup>2</sup></i> rather than <i>r</i><sub>s</sub> = 2<i>Gm</i>/<i>c<sup>2</sup></i> . The reason for this difference is because the single fundamental particle is maximally rotating at the speed of light. The difference is a factor of 2 compared to the Schwarzschild radius of a non-rotating body made of multiple particles. I make a distinction by using the symbol <i>R</i><sub>s</sub> rather than <i>r</i><sub>s</sub>. <o:p></o:p></span></h1><h1 style='margin-top:0in;line-height:18.0pt;background:#F7F7F4'><span style='font-size:12.0pt;font-family:"Times New Roman",serif;color:windowtext'><o:p> </o:p></span></h1><h1 style='margin-top:0in;line-height:18.0pt;background:#F7F7F4'><span style='font-size:12.0pt;font-family:"Times New Roman",serif;color:windowtext'>The Foundation paper shows that there is a square relationship between the electrostatic force and the gravitational force when forces between single particles are expressed in their fundamental units. That paper also shows that there is a symmetrical relationship between the gravitational force, the electrostatic force and Planck force. Now I have determined that there is also a fundamental relationship between a fundamental particle’s radius <i><s>λ</s></i><sub>c</sub> = </span><span style='font-size:12.0pt;font-family:"Cambria Math",serif;color:windowtext'>ħ</span><span style='font-size:12.0pt;font-family:"Times New Roman",serif;color:windowtext'>/<i>mc</i> and its Schwarzschild radius <i>R<sub>s</sub></i> = <i>Gm</i>/<i>c<sup>2</sup></i>. Again, this relationship requires that these radii be expressed in the natural units which are dimensionless Planck units (designated with an underline) <o:p></o:p></span></h1><h1 style='margin-top:0in;line-height:18.0pt;background:#F7F7F4'><span style='font-size:12.0pt;font-family:"Times New Roman",serif;color:windowtext'> <o:p></o:p></span></h1><p class=MsoNormal><s><u><span style='font-size:14.0pt;line-height:106%;font-family:"Times New Roman",serif;color:#C00000'>λ</span></u></s><u><sub><span style='font-size:14.0pt;line-height:106%;font-family:"Times New Roman",serif;color:#C00000'>c</span></sub></u><u><span style='font-size:14.0pt;line-height:106%;font-family:"Times New Roman",serif;color:#C00000'>R<sub>s</sub> </span></u><span style='font-size:14.0pt;line-height:106%;font-family:"Times New Roman",serif;color:#C00000'>= 1</span><span style='font-family:"Times New Roman",serif'> <o:p></o:p></span></p><p class=MsoNormal><span style='font-family:"Times New Roman",serif'>where: <i><s><u>λ</u></s><u><sub>s</sub></u></i> = <i><s>λ</s><sub>s</sub>/L<sub>p</sub></i> and <i><u>R<sub>s</sub></u></i> = <i>R<sub>s</sub></i></span><span style='font-family:"Cambria Math",serif'>/</span><i><span style='font-family:"Times New Roman",serif'>L<sub>p </sub></span></i><span style='font-family:"Times New Roman",serif'> Planck length = <i>L<sub>p</sub> </i>= (</span><span style='font-family:"Cambria Math",serif'>ħ</span><i><span style='font-family:"Times New Roman",serif'>G</span></i><span style='font-family:"Times New Roman",serif'>/c<sup>3</sup>)<sup>1/2</sup><o:p></o:p></span></p><p class=MsoNormal><span style='font-family:"Times New Roman",serif'>In words, a fundamental particle’s Compton radius times its Schwarzschild radius equals 1 when both radii are expressed in dimensionless Planck units. Another way of expressing this is:<o:p></o:p></span></p><p class=MsoNormal><span style='font-family:"Times New Roman",serif;color:red'> </span><i><s><span style='font-size:14.0pt;line-height:106%;font-family:"Times New Roman",serif;color:#C00000'>λ</span></s></i><sub><span style='font-size:14.0pt;line-height:106%;font-family:"Times New Roman",serif;color:#C00000'>c</span></sub><span style='font-size:14.0pt;line-height:106%;font-family:"Times New Roman",serif;color:#C00000'>/<i>L</i><sub>p</sub> = <i>L</i><sub>p</sub>/<i>R</i><sub>s</sub></span><span style='font-family:"Times New Roman",serif;color:#C00000'><o:p></o:p></span></p><p class=MsoNormal><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif'>There is a type of symmetry between a particle’s Compton radius</span><span style='font-family:"Times New Roman",serif'> </span><i><s><span style='font-size:14.0pt;line-height:106%;font-family:"Times New Roman",serif'>λ</span></s></i><sub><span style='font-size:14.0pt;line-height:106%;font-family:"Times New Roman",serif'>c, </span></sub><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif'>a particle’s<sub> </sub></span><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif'>Schwarzschild radius and Planck length. For example, imagine a log scale of length. On this scale we designate a particle’s Compton radius <i><s>λ</s></i><sub>c</sub> and its Schwarzschild radius R<sub>s</sub>. Planck length is exactly half way between these two designation points on this log scale. There is another important connection between the radii and the forces. Suppose that we have two of the same particles, both with elementary charge <i>e</i>. If they are separated by an arbitrary distance, they will experience an electrostatic force (<i>F</i><sub>e</sub>) and a gravitational force (<i>F</i><sub>g</sub>). The ratio of these two forces is constant (independent of separation distance). This ratio exactly equals the ratio of radii <i>R</i><sub>s</sub>/</span><i><s><span style='font-size:14.0pt;line-height:106%;font-family:"Times New Roman",serif'>λ</span></s></i><sub><span style='font-size:14.0pt;line-height:106%;font-family:"Times New Roman",serif'>c</span></sub><span style='font-size:14.0pt;line-height:106%;font-family:"Times New Roman",serif'> </span><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif'>if we compensate for the electrostatic coupling constant by incorporating the inverse of the fine structure constant α<sup>-1</sup> = 137.036</span><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif'> </span><span style='line-height:106%;font-family:"Times New Roman",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'> </span><!--[if gte msEquation 12]><m:oMath><m:f><m:fPr><span style='font-size:14.0pt;font-family:"Cambria Math",serif;color:#C00000;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><m:sSub><m:sSubPr><span style='font-size:14.0pt;font-family:"Cambria Math",serif;color:#C00000;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span style='font-size:14.0pt;line-height:106%;font-family:"Cambria Math",serif;color:#C00000'><m:r>F</m:r></span></i></m:e><m:sub><i><span style='font-size:14.0pt;line-height:106%;font-family:"Cambria Math",serif;color:#C00000'><m:r>g</m:r></span></i></m:sub></m:sSub></m:num><m:den><m:sSub><m:sSubPr><span style='font-size:14.0pt;font-family:"Cambria Math",serif;color:#C00000;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span style='font-size:14.0pt;line-height:106%;font-family:"Cambria Math",serif;color:#C00000'><m:r>F</m:r></span></i></m:e><m:sub><i><span style='font-size:14.0pt;line-height:106%;font-family:"Cambria Math",serif;color:#C00000'><m:r>e</m:r></span></i></m:sub></m:sSub><m:sSup><m:sSupPr><span style='font-size:14.0pt;font-family:"Cambria Math",serif;color:#C00000;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i><span style='font-size:14.0pt;line-height:106%;font-family:"Cambria Math",serif;color:#C00000'><m:r>α</m:r></span></i></m:e><m:sup><i><span style='font-size:14.0pt;line-height:106%;font-family:"Cambria Math",serif;color:#C00000'><m:r>-</m:r><m:r>1</m:r></span></i></m:sup></m:sSup></m:den></m:f><i><span style='font-size:14.0pt;line-height:106%;font-family:"Cambria Math",serif;color:#C00000'><m:r>=</m:r></span></i><m:f><m:fPr><span style='font-size:14.0pt;font-family:"Cambria Math",serif;color:#C00000;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><m:sSub><m:sSubPr><span style='font-size:14.0pt;font-family:"Cambria Math",serif;color:#C00000;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><span style='font-size:14.0pt;line-height:106%;font-family:"Cambria Math",serif;color:#C00000'><m:r>R</m:r></span></i></m:e><m:sub><i><span style='font-size:14.0pt;line-height:106%;font-family:"Cambria Math",serif;color:#C00000'><m:r>s</m:r></span></i></m:sub></m:sSub></m:num><m:den><m:sSub><m:sSubPr><span style='font-size:14.0pt;font-family:"Cambria Math",serif;color:#C00000;font-style:italic'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i><s><span style='font-size:14.0pt;line-height:106%;font-family:"Cambria Math",serif;color:#C00000'><m:r>λ</m:r></span></s></i></m:e><m:sub><i><span style='font-size:14.0pt;line-height:106%;font-family:"Cambria Math",serif;color:#C00000'><m:r>c</m:r></span></i></m:sub></m:sSub></m:den></m:f></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:106%;font-family:"Calibri",sans-serif;position:relative;top:11.0pt;mso-text-raise:-11.0pt;mso-fareast-language:EN-US'><img width=95 height=46 id="_x0000_i1025" src="cid:image001.png@01D129E0.4CB57AB0"></span><![endif]><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",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'>For example, for an electron, both ratios equal 1.75 x 10<sup>-45</sup>. <o:p></o:p></span></p><p class=MsoNormal><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",serif'>However, I was looking for the reason that there should be a relationship between <i><s>λ</s></i><sub>c</sub> and <i>R</i><sub>s</sub>. These are both radii where <i>R</i><sub>s</sub> is the gravitational radius and <i><s>λ</s></i><sub>c</sub> is a particle’s radius (single loop). In my model, a black hole happens when the strain of spacetime reaches 100% for a particular wavelength or radius. Another way of saying this is a black hole happens when <i>A<sub>s</sub></i> = 1 for a particular wavelength. To explain the rest of the concept I am going to use an electron as an example. An electron has strain amplitude <i>A</i><sub>s</sub> = <i>ħ</i>/<i>mc</i> = <i>m</i>/<i>m</i><sub>p</sub> = ω<sub>c</sub>/ω<sub>p </sub>= <i>L</i><sub>p</sub>/<i><s>λ</s></i><sub>c</sub> ≈ 4.18 x 10<sup>-23</sup> (dimensionless). The standard model also recognizes the importance of this number because the standard model contains many dimensionless numbers and this is the number associated with electrons. In my model, the spatial distortion produced by the rotating dipole wave is <i>A</i><sub>s</sub> = <i>L</i><sub>p</sub>/<i><s>λ</s></i><sub>c</sub>. Gravitational effects scale with <i>A</i><sub>s</sub><sup>2</sup> because the spacetime field is finite and has boundary conditions which means that it is a nonlinear medium for wave propagation. To a first approximation the nonlinear effect scales with amplitude squared. There are higher order terms which are being ignored. Therefore, the oscillating gravitational amplitude is <i>A</i><sub>s</sub><sup>2</sup> ≈ 1.75x10<sup>-45</sup> for an electron. In order to reach the condition which would produce an gravitational amplitude of <i>A<sub>s</sub></i><sup>2</sup> = 1 we would have to reduce the rotar radius of the electron by a factor of <i>A<sub>s</sub></i><sup>2</sup>. A mechanical reduction of the radius would both increase the amplitude As and increase the Compton frequency <i>ω<sub>c</sub></i>. In the mathematical calculation of the Schwarzschild radius, there is no mechanical reduction of radius. Therefore there would be no increase in <i>ω<sub>c</sub></i>. Therefore it is necessary for the calculation to increase the amplitude by <i>A<sub>s</sub></i><sup>2</sup> in order to calculate the Schwarzschild radius with no change in frequency. This leads to the new insight which perfectly supports this model. Here is the equation: <o:p></o:p></span></p><h1 style='margin-top:0in;line-height:18.0pt;background:#F7F7F4'><i><span style='font-size:14.0pt;font-family:"Times New Roman",serif;color:#C00000'> R</span></i><sub><span style='font-size:14.0pt;font-family:"Times New Roman",serif;color:#C00000'>s</span></sub><span style='font-size:14.0pt;font-family:"Times New Roman",serif;color:#C00000'> = <i><s>λ</s></i><sub>c</sub><i>A<sub>s</sub></i><sup>2</sup></span><span style='font-size:14.0pt;font-family:"Times New Roman",serif;color:windowtext'> <o:p></o:p></span></h1><h1 style='margin-top:0in;line-height:18.0pt;background:#F7F7F4'><span style='font-size:12.0pt;font-family:"Times New Roman",serif;color:windowtext'> For an electron: <i>R<sub>s</sub></i> = 6.76x10<sup>-58</sup> m, <i><s>λ</s><sub>c</sub></i> = 3.86x10<sup>-13 </sup>m and <i>A<sub>s</sub><sup>2</sup></i> = 1.75x10<sup>-45</sup> <o:p></o:p></span></h1><h1 style='margin-top:0in;line-height:18.0pt;background:#F7F7F4'><span style='font-size:12.0pt;font-family:"Times New Roman",serif;color:windowtext'> <o:p></o:p></span></h1><p class=MsoNormal><span style='font-family:"Times New Roman",serif'>I have generated a specific quantifiable models of an electron, a photon, the energy density of spacetime, the curvature of spacetime, the gravitational force, the electrostatic force between particles, an electric field and a charged particle. I have found it particularly difficult to get anyone in this group to discuss any of these points. If you disagree with anything in my model, I would like you to point out the logical flaw. I would be happy to discuss and if necessary to debate any point in my model. It would actually be very helpful to me if you can point out a logical flaw. <o:p></o:p></span></p><p class=MsoNormal><span style='font-family:"Times New Roman",serif'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-family:"Times New Roman",serif'>John M. <o:p></o:p></span></p><p class=MsoNormal><span style='font-size:12.0pt;line-height:106%;font-family:"Times New Roman",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'><o:p> </o:p></span></p></div></body></html>