\(G=\cfrac { \sqrt { 2c } }{ a_{ \psi } } \theta _{ \psi }\)
and
\( G=\cfrac { 1 }{ \sqrt { 2 } } \left( \cfrac { 1 }{ c } \right) ^{ 3/2 }.\cfrac { 1 }{ tanh(\theta _{ \psi }) } \)
An illustrative plot of 1/tanh(x) and 2x is shown below,
where both positive and negative \(G\) are admissible because of the \(\sqrt{2}\) factor in both equations. This provides the option to make \(F_{\rho}\) positive or negative and both.
And we have another particle zoo.
What if \(E\overset{Fourier}{\longrightarrow }B\)? That the \(B\) field is the Fourier transform of an \(E\) field and \(E\) field is the Inverse Fourier Transform of \(B\). Or that they are of one entity that is oscillating around light speed. Above light speed it present itself as a \(B\) field in space, below light speed it is an \(E\) field in space. An entity we shall call \(EB\). At the threshold of \(v=c\), we apply the Fourier Transform when we move above light speed \(F_f[E]=B\) and the Inverse Fourier Transform \(F_f^{-1}[B]=E\)when we move below light speed.
When \(E=cos(2\pi f_o x)\),
\(F_f[E]=B=\cfrac{1}{2}\{\delta(f+f_o)+\delta(f-f_o)\}\) --- (*)
where \(f\) is defined as part of \(e^{-i2\pi f x}\), a complex sinusoidal in the product of space and time, spacetime.
This suspicion arises from the postulate that in the time domain relevant parameters are periodic and is defined over a per unit time interval (per second, \(Hz\)).
Expression (*) introduces negative frequency \(-f_o\) and complex wave part \(isin(2\pi fx)\). And a factor of one half that divides equally the magnitude (coefficient) of \(E\) between the positive and negative frequency, \(f_o\) and \(-f_o\).
The average power of the wave, over one period is with a factor \(P_f=\cfrac{1}{2}.1+\cfrac{1}{2}.\left(\cfrac{1}{4}+\cfrac{1}{4}\right)=\cfrac{3}{4}\)
The average value of \(E\) or \(B\) is with a factor \(D_{f}=\sqrt{\cfrac{3}{4}}=0.8660\). And \(\cfrac{1}{D_f}=1.15470\)
\(D_f\) or \(\cfrac{1}{D_f}\) might amplify the discrepancies in \(\varepsilon_o\) and \(\mu_o\); calculated vs experimental values.
If we cannot measure negative frequency \(f_o\) and its power then,
The average power of the wave is with a factor \(P_f=\cfrac{1}{2}.1+\cfrac{1}{2}.\cfrac{1}{4}=\cfrac{5}{8}\)
Which gives \(D_{f}=\sqrt{P_f}=\sqrt{\cfrac{5}{8}}=0.79057\) and \(\cfrac{1}{D_f}=1.26491\).
However if instead we allow,
\(E=cos(2\pi f_o x)+isin(2\pi f_o x)\) --- (**)
where
\(F_f[isin(2\pi f_o x)]=i.\cfrac{1}{2}i\{\delta(f+f_o)-\delta(f-f_o)\}=\cfrac{1}{2}\{\delta(f-f_o)-\delta(f+f_o)\}\)
then,
\(F_f[E]=B=\delta(f-f_o)\) --- (*)
\(B\) field is at \(f_o\) with no complex part, and there is no extra factor due to the transform, \(P_f=D_f=1\).
Does this mean a complex wave part must exist, either in space or in time for conservation of energy between space and time, through the transform? No.
Expression (**) implies that \(E\) is also circular.
This means extra energy is required to set \(B\) into circular motion. When \(E\) is already in circular motion, no extra energy is needed for \(B\) to coil around the space dimension. Circular motion itself stores potential energy, \(P_{cir}\). This extra energy required results in a discrepancy factor \(P_f\ne1\). When energy is required,
\(P_{cir}=1-P_f=\cfrac{1}{4}\)
one fourth of the total energy is required. \(P_{cir}\) could be the reason why Fourier Transform was not admitted. Without accounting for \(P_{cir}\), Fourier Transform will result in energy discrepancy.
Negative frequency, \(-f_o\) would suggest that \(B\) is travelling back in time. How did that happen?
Note: Fourier transform is defined as,
\(F_{ f }[f(x)]=F(f)=\int_{-\infty}^{\infty} { f(x).e^{ -i2\pi fx } } dx\)
and the Inverse Fourier transform is,
\(F_{ f}^{-1}[F(f)]=f(x)=\int_{-\infty}^{\infty} { F(f).e^{ i2\pi fx } } df\)
where all parameters are explicit.
\(P_f=\cfrac{1}{2}\).(power in \(E\))\(+\cfrac{1}{2}\).(power in \(B\))
With reference to power in \(E\) part of the wave (ie. taken to be \(1\)), power in \(B\) is reduced by a factor of \(\left(\cfrac{1}{2}\right)^{2}=\cfrac{1}{4}\) because its magnitude was reduced by \(\cfrac{1}{2}\) when it splits into positive and negative frequency. Both positive and negative frequency contributes equally to the power of the wave, \(\cfrac{1}{4}+\cfrac{1}{4}\)
