How Time Flies

From the definition of \(\psi\) in the post "Maybe Not",

\(\psi=\cfrac{1}{2\mu_o}B^2=-\cfrac{1}{2}\varepsilon_oE^2\)

\(\cfrac{\partial\,\psi}{\partial\,t}=\cfrac { 1 }{ 2\mu _{ o } } \cfrac { \partial B^{ 2 } }{ \partial \, t }=-\cfrac { \varepsilon_o }{ 2 } \cfrac { \partial E^{ 2 } }{ \partial \, t }\)

If \(\psi=e^{iwt}\), just the time component to represent time itself.

\(\cfrac{\partial\,\psi}{\partial\,t}=i\omega e^{i\omega t}=\cfrac { 1 }{ 2\mu _{ o } } \cfrac { \partial B^{ 2 } }{ \partial \, t }=-\cfrac { \varepsilon_o }{ 2 } \cfrac { \partial E^{ 2 } }{ \partial \, t }\)

\(\omega e^{i\omega t}=-i\cfrac { 1 }{ 2\mu _{ o } } \cfrac { \partial B^{ 2 } }{ \partial \, t }=i\cfrac { \varepsilon_o }{ 2 } \cfrac { \partial E^{ 2 } }{ \partial \, t }\)

\(\omega e^{i\omega t}=-i\cfrac{i}{i}\cfrac { 1 }{ 2\mu _{ o } } \cfrac { \partial B^{ 2 } }{ \partial \, t }=i\cfrac{i}{i}\cfrac { \varepsilon_o }{ 2 } \cfrac { \partial E^{ 2 } }{ \partial \, t }\)

\(\omega e^{i\omega t}=\cfrac { 1 }{ 2\mu _{ o } } \cfrac { \partial B^{ 2 } }{ \partial \, it }=-\cfrac{ \varepsilon_o }{ 2 } \cfrac { \partial E^{ 2 } }{ \partial \, it }\)

where \(it\) is the orthogonal charge-time component postulated in the post "Temperature, Space Density And Gravity".

\(\because\,\omega =2\pi f=2\pi \cfrac { c }{ \lambda_t  } \)

\( e^{ i\omega t }=\cfrac { \lambda_t  }{ 4\pi \mu _{ o }c } \cfrac { \partial B^{ 2 } }{ \partial \, it } =-\cfrac { \lambda_t \varepsilon _{ o } }{ 4\pi c } \cfrac { \partial E^{ 2 } }{ \partial \, it } \)

\( c=\cfrac { 1 }{ \sqrt { \mu _{ o }\varepsilon _{ o } }  } \quad Z_{ o }=\sqrt { \cfrac { \mu _{ o } }{ \varepsilon _{ o } }  } \)

\( e^{ i\omega t }=\cfrac { \lambda_t  }{ 4\pi \mu _{ o }c } \cfrac { \partial B^{ 2 } }{ \partial \, it } =-\cfrac { \lambda_t \varepsilon _{ o } }{ 4\pi c } \cfrac { \partial E^{ 2 } }{ \partial \, it } \)

\( e^{ i\omega t }=\cfrac { \lambda_t  }{ 4\pi  } \cfrac { 1 }{ Z_{ o } } \cfrac { \partial B^{ 2 } }{ \partial \, it } =-\cfrac { \lambda_t  }{ 4\pi  } Z_{ o }\cfrac { \partial E^{ 2 } }{ \partial \, it } \)

Let  \(it=t_c\),

\( e^{ i\omega t }=\cfrac { \lambda_t  }{ 4\pi  } \cfrac { 1 }{ Z_{ o } } \cfrac { \partial B^{ 2 } }{ \partial \, t_c } =-\cfrac { \lambda_t  }{ 4\pi  } Z_{ o }\cfrac { \partial E^{ 2 } }{ \partial \, t_c } \)

or,

\(e^{ i\omega t }=\cfrac { \lambda_t }{ 2\pi c } \cfrac { \partial \psi  }{ \partial \, t_{ c } }  \)

which requires,

\(1=\cfrac { \lambda_t }{ 2\pi c } . \omega \)

that is always true.

We see that time as we experience it, \(e^{iwt}\), is the rate of change of \(\psi\) with charge time.  If Pythagoras Theorem holds true for the time domain, and that time as we experience it is the resultant of two orthogonal time components, charge time, \(t_c\) and gravitational time, \(t_g\) then,

\(\overrightarrow {t}^2=\overrightarrow {t}^2_c+\overrightarrow {t}^2_g\)

This implies,

\(e^{i\omega_ct_c}=\cfrac{1}{\sqrt{2}}e^{i\omega t}.e^{-i\pi/4}=\cfrac{1}{\sqrt{2}}e^{i(\omega t-\pi/4)}\)

\(e^{i\omega_gt_g}=\cfrac{1}{\sqrt{2}}e^{i\omega t}.e^{i\pi/4}=\cfrac{1}{\sqrt{2}}e^{i(\omega t+\pi/4)}\)

and of course,

\(e^{iw_gt_g}=ie^{iw_ct_c}\)

There is no reason not be believe that,

\(\omega_g=\omega_c=\omega\)  by symmetry.

And so,

\(\cfrac{\partial}{\partial\,\omega}\left\{e^{i\omega_ct_c}\right\}=\cfrac{\partial}{\partial\,\omega}\left\{e^{i\omega t_c}\right\}=\cfrac{\partial}{\partial\,\omega}\left\{\cfrac{1}{\sqrt{2}}e^{i\omega t}.e^{-i\pi/4}\right\}\)

which implies,

\(t_c=\cfrac{1}{\sqrt{2}}t.e^{-i\pi/4}\)

Generalizing, we have

\(t_g=\cfrac{1}{\sqrt{2}}t.e^{+i\pi/4}\)

\(\cfrac{1}{\sqrt{2}}|t|=|t_c|=|t_g|\)

Which makes us behind \(t_c\) and \(t_g\) in the absolute sense.  \(t_c\) lag \(t\) by \(\cfrac{\pi}{4}\) and \(t_g\) leads \(t\) by \(\cfrac{\pi}{4}\) in phase.  Time as a wave is a natural vector.