Energy Accounting With Fourier

When,

$G=\cfrac { \sqrt { 2c }  }{ a_{ \psi  } } \theta _{ \psi  }$

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, $\cfrac{1}{4}+\cfrac{1}{4}$