From the previous post "\(\psi\) Gets Inverted And Fourier Xformed" dated 22 Jun 2016, it was suggested that the extend of \(\psi\) is such that from \(x=a_{\psi}\) to the center of the particle, the acceleration due to \(F_{\rho}\) achieve light speed at the center. That is to say,
\(\cfrac{1}{2}mc^2=\int_{0}^{a_{\psi}}{F_{\rho}}dx\)
\(F_{ \rho }=i\sqrt { 2{ mc^{ 2 } } } \, G.tanh\left( \cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } } (x-x_{ z }) \right)\)
with \(x_z=0\),
\(\cfrac{1}{2}mc^2=i2{ mc^{ 2 } }.ln(cosh(\cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } } (a_{ \psi }))\)
this implies,
\(ln(cosh(\cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } }a_{\psi}))=\cfrac{1}{4}\) --- (*)
If this is so, there is a new discrepancy in the calculations for \(\varepsilon_o\) and \(G\) that had previously assumed,
\(ln(cosh(\cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } }a_{\psi}))=ln(cosh(\pi))=2.450311\) --- (**)
Since, the function \(ln(cosh(x))\) is monotonously increasing, the assumed value of \(a_{\psi}\) given by (**) results in light speed before reaching the center under the action of \(F_{\rho}\). The particle is then hollow at the center.
If (*) is correct, then immediately after \(a_{\psi}\), the force in the field around the particle does not obey coulomb's inverse square law but increases until \(ln(cosh(\pi))\) or \(\cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } }a_{\psi}=\pi\), where (**) holds true. Furthermore, if (*) is true,
\(\varepsilon _{ o }=\cfrac { 2 }{ c^{ 2 }}\)
and
\(G_o=\cfrac { c^{ 2 } }{ 8\pi }.\cfrac{3}{4\pi(2c)^3}.\sqrt{3}=6.86e-12\)
A lower value for \(a_{\psi}\) as given by (*) allows for two particles to interact as waves without merging below the particle limit,
\(a_{\psi}=\pi\cfrac { \sqrt { 2{ mc^{ 2 } } } }{ G }\)
had (**) been assumed. But for the case of \(G_o\), the gravitational constant, (**) seems more appropriate.
Have a nice day.
Note: \(\cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } }a_{\psi} =acosh(e^{\cfrac{1}{4}})=0.736904590621\)
