Consider the wave equation,
\(\cfrac{\partial^2\psi}{\partial\,t^2}=c^2\cfrac{\partial^2\psi}{\partial\,x^2}=\cfrac{\partial\,x}{\partial\,t}\cfrac{\partial\,x}{\partial\,t}\cfrac{\partial^2\psi}{\partial\,x\partial\,x}\)
when one of the space dimension has been replaced by a time dimension. The particle exists along the charge time \(t_c\) axis or the gravitational time \(t_g\) axis, from which its velocity is based. And as a wave, travels down the other orthogonal time axis.
\(\cfrac{\partial^2\psi}{\partial\,t^2_c}=\cfrac{\partial\,x}{\partial\,t_c}\cfrac{\partial\,t_g}{\partial\,t_c}\cfrac{\partial^2\psi}{\partial\,t_g\partial\,x}\)
\(\cfrac{\partial^2\psi}{\partial\,t^2_g}=\cfrac{\partial\,x}{\partial\,t_g}\cfrac{\partial\,t_c}{\partial\,t_g}\cfrac{\partial^2\psi}{\partial\,t_c\partial\,x}\)
Since,
\(t_c=\cfrac{1}{\sqrt{2}}t.e^{-i\pi/4}\) and \(t_g=\cfrac{1}{\sqrt{2}}t.e^{+i\pi/4}\)
\(t_c=t_ge^{-i\pi/2}\) --- (a)
\(t_c=-it_g\) and \(\cfrac{\partial\,t_c}{\partial\,t_g}=-i\)
\(t_g=it_c\) and \(\cfrac{\partial\,t_g}{\partial\,t_c}=i\)
and,
\(\cfrac{\partial\,x}{\partial\,t_c}=\cfrac{\partial\,x}{\partial\,t_g}=ic\)
this definition of velocity has a explicit direction notation \(i\) to be consistent with the explicit orthogonality between the time axes in (a) above
The equations for 'time' waves are then given by,
\(\cfrac{\partial^2\psi}{\partial\,t^2_c}=ic.i\cfrac{\partial^2\psi}{\partial\,x\partial\,t_g}=ici\cfrac{\partial^2\psi}{\partial\,x\partial\,(it_c)}=ic\cfrac{\partial^2\psi}{\partial\,x\partial\,t_c}\)
\(\cfrac{\partial^2\psi}{\partial\,t^2_g}=ic.(-i)\cfrac{\partial^2\psi}{\partial\,x\partial\,t_c}=ic.(-i)\cfrac{\partial^2\psi}{\partial\,x\partial\,(-it_g)}=ic\cfrac{\partial^2\psi}{\partial\,x\partial\,t_g}\)
from the post "How Time Flies". \(t_c\) and \(t_g\) are equivalent. A mass particle is associated with \(t_g\) and an electron is associated with \(t_c\).
Consider the Lagrangian,
\(L=T-V\)
\(\cfrac{d\,}{d\,t}\left(\cfrac{\partial\, L}{\partial\,\dot{x}}\right)=\cfrac{\partial\, L}{\partial\,x}\)
If all dimension are equivalent, then the Lagrangian and Euler-Lagrange Equation is still applicable, when \(t\) is either \(t_c\) or \(t_g\).
\(\psi=T+V=\cfrac{1}{2}m_p\dot{x}^2+V\)
\(L=\cfrac{1}{2}m_p\dot{x}^2-(\psi-\cfrac{1}{2}m_p\dot{x}^2)\)
where \(\psi\) is the total energy between the two space dimensions and \(\dot{x}\) the velocity of the particle.
\(L=m_p\dot{x}^2-\psi\)
We consider existence along \(t_c\),
\(\cfrac{d\,}{d\,t_c}\left(\cfrac{\partial\, L}{\partial\,\dot{x}}\right)=\cfrac{d\,}{d\,t_c}\left(\cfrac{\partial}{\partial\,\dot{x}}\left\{m_p\dot{x}^2-\psi\right\}\right)\)
\(\cfrac { d\, }{ d\, t_c } \left( \cfrac { \partial \, L }{ \partial \, \dot { x } } \right) =\cfrac { d\, }{ d\, t_c } \left( 2m_{ p }\dot { x } -\cfrac { \partial \psi }{ \partial \, \dot { x } } \right) =2m_{ p }\ddot { x } -\cfrac { \partial }{ \partial \, \dot { x } } \left\{ \cfrac { d\, \psi }{ d\, t_c } \right\} \)
and,
\(\cfrac{\partial\, L}{\partial\,x}=\cfrac{\partial}{\partial\,x}\left\{m_p\dot{x}^2-\psi\right\}=-\cfrac{\partial\,\psi}{\partial\,x}\)
So,
\(\cfrac { \partial }{ \partial \, \dot { x } } \left\{ \cfrac { d\, \psi }{ d\, t_c } \right\}=2m_{ p }\ddot { x } +\cfrac{\partial\,\psi}{\partial\,x}\) --- (*)
Differentiate with respect to \(x\),
\(\cfrac { \partial}{ \partial \, x } \left\{ \cfrac { \partial }{ \partial \, \dot { x } } \left\{ \cfrac { d\, \psi }{ d\, t_c } \right\} \right\} =\cfrac { \partial ^{ 2 }\psi }{ \partial \, x\partial \, x } \)
\(\cfrac { \partial }{ \partial \, \dot { x } } \left\{ \cfrac { \partial ^{ 2 }\psi }{ \partial \, t_c\partial \, x } \right\} =\cfrac { \partial }{ \partial \, \dot { x } } \left\{ -\cfrac { i }{c } \cfrac { \partial ^{ 2 }\psi }{ \partial \, t^{ 2 }_c } \right\} =\cfrac { \partial ^{ 2 }\psi }{ \partial \, x^{ 2 } } \)
\(\cfrac { \partial }{ \partial \, \dot { x } } \left\{ \cfrac { \partial ^{ 2 }\psi }{ \partial \, t^{ 2 }_c } \right\} =ic\cfrac { \partial ^{ 2 }\psi }{ \partial \, x^{ 2 } } \) --- (b)
but,\(\require{cancel}\)
\(\cfrac { \partial \, \psi }{ \partial \, x } =\cfrac { \partial \, \psi }{ \partial \, \dot { x } }\cancelto{0}{ \cfrac { d\, \dot { x } }{ d\, x }} +\cfrac { \partial \, \psi }{ \partial \, t_c } \cfrac { d\, t_c }{ d\, x } \)
So (*) becomes,
\(\cfrac { \partial }{ \partial \, \dot { x } } \left\{ \cfrac { d\, \psi }{ d\, t_c } \right\} =2m_{ p }\ddot { x } +\cfrac { \partial \, \psi }{ \partial \, t_c } \cfrac { d\, t _c}{ d\, x } \)
Consider,
\(\cfrac { \partial }{ \partial \, t_c } \cfrac { \partial }{ \partial \, \dot { x } } \left\{ \cfrac { d\, \psi }{ d\, t_c } \right\} = \cfrac { \partial }{ \partial \, \dot { x } } \left\{\cfrac { \partial ^{ 2 }\psi }{ \partial \, t^{ 2 } _c}\right\}\)
\(= \cfrac { \partial }{ \partial \, t_c } \left\{ 2m_{ p }\ddot { x } +\cfrac { \partial \, \psi }{ \partial \, t _c} \cfrac { d\, t_c }{ d\, x } \right\} =2m_{ p }\dddot { x } +\cfrac { \partial ^{ 2 }\psi }{ \partial \, t^{ 2 }_c } \cfrac { d\, t_c }{ d\, x } +\cfrac { \partial \, \psi }{ \partial \, t_c } \cfrac { \partial }{ \partial \, t _c} \left\{ \cfrac { 1 }{ \cfrac { d\, x }{ d\, t _c} } \right\} \)
\(\cfrac { \partial }{ \partial \, \dot { x } } \left\{\cfrac { \partial ^{ 2 }\psi }{ \partial \, t^{ 2 }_c }\right\}= 2m_{ p }\dddot { x } +\cfrac { \partial ^{ 2 }\psi }{ \partial \, t^{ 2 }_c } \cfrac { d\, t_c }{ d\, x } -\cfrac { \partial \, \psi }{ \partial \, t_c } \left\{ \cfrac { 1 }{ \left( \cfrac { d\, x }{ d\, t _c} \right) ^{ 2 } } \cfrac { d^{ 2 }x }{ d\, t^{ 2 }_c } \right\} \)
So, since \(ic\) is a constant, substitute in (b)
\(ic\cfrac { \partial ^{ 2 }\psi }{ \partial \, x^{ 2 } } =2m_{ p }\dddot { x } +\cfrac { \partial ^{ 2 }\psi }{ \partial \, t^{ 2 }_c } \cfrac { d\, t _c}{ d\, x } -\cfrac { \partial \, \psi }{ \partial \, t_c } \left\{ \cfrac { 1 }{ \left( \cfrac { d\, x }{ d\, t _c} \right) ^{ 2 } } \cfrac { d^{ 2 }x }{ d\, t^{ 2 }_c } \right\}\)
\(ic\cfrac { \partial ^{ 2 }\psi }{ \partial \, x^{ 2 } } =2m_{ p }\dddot { x } +\cfrac { \partial ^{ 2 }\psi }{ \partial \, t^{ 2 }_c } \cfrac { 1 }{ \dot { x } } -\cfrac { \partial \, \psi }{ \partial \, t _c} \left\{ \cfrac { 1 }{ \dot { x } ^{ 2 } } \ddot { x } \right\} \)
Rearranging the terms,
\(\ddot { x } \cfrac { \partial \, \psi }{ \partial \, t_c } =-ic\dot { x } ^{ 2 }\cfrac { \partial ^{ 2 }\psi }{ \partial \, x^{ 2 } } +\dot { x } \cfrac { \partial ^{ 2 }\psi }{ \partial \, t^{ 2 }_c } +2m_{ p }\dot { x } ^{ 2 }\dddot { x } \)
If the field forces are constants in time given location \(x\), \(\dddot { x }=0\)
\(\ddot { x } \cfrac { \partial \, \psi }{ \partial \, t _c} =-ic\dot { x } ^{ 2 }\cfrac { \partial ^{ 2 }\psi }{ \partial \, x^{ 2 } } +\dot { x } \cfrac { \partial ^{ 2 }\psi }{ \partial \, t^{ 2 }_c } \) --- (1)
Consider,
\(\cfrac { \partial ^{ 2 }\psi }{ \partial \, t^{ 2 } _c} =\cfrac { \partial \, }{ \partial \, t_c} \left\{ \cfrac { \partial \, \psi }{ \partial \, t_c } \right\} =\cfrac { \partial \, }{ \partial \, x } \left\{ \cfrac { \partial \, \psi }{ \partial \, t_c } \right\} \cfrac { d\, x }{ d\, t_c } +\cfrac { \partial \, }{ \partial \, \dot { x } } \left\{ \cfrac { \partial \, \psi }{ \partial \, t_c } \right\} \cfrac { d\, \dot { x } }{ d\, t_c } \)
From previously,
\( \cfrac { \partial }{ \partial \, \dot { x } } \left\{ \cfrac { d\, \psi }{ d\, t_c } \right\} =2m_{ p }\ddot { x } +\cfrac { \partial \, \psi }{ \partial \, t_c } \cfrac { d\, t_c }{ d\, x } \)
Therefore,
\( \cfrac { \partial ^{ 2 }\psi }{ \partial \, t^{ 2 }_c } =\cfrac { \partial \, }{ \partial \, x } \left\{ \cfrac { \partial \, \psi }{ \partial \, t_c } \right\} \cfrac { d\, x }{ d\, t_c } +\left\{ 2m_{ p }\ddot { x } +\cfrac { \partial \, \psi }{ \partial \, t _c} \cfrac { d\, t _c}{ d\, x } \right\} \cfrac { d\, \dot { x } }{ d\, t _c} \)
\( \cfrac { \partial ^{ 2 }\psi }{ \partial \, t^{ 2 }_c } =\cfrac { \partial \, }{ \partial \, t_c } \left\{ \cfrac { \partial \, \psi }{ \partial \, x } \right\} \dot{x} +2m_{ p }\ddot { x } ^{ 2 }+\cfrac { \partial \, \psi }{ \partial \, t _c} \cfrac { \ddot { x } }{ \dot { x } } \)
\(\dot { x } \cfrac { \partial ^{ 2 }\psi }{ \partial \, t^{ 2 }_c } =\cfrac { \partial \, }{ \partial \, t_c } \left\{ \cfrac { \partial \, \psi }{ \partial \, x } \right\} \dot { x } ^{ 2 }+2m_{ p }\dot { x } \ddot { x } ^{ 2 }+\ddot { x } \cfrac { \partial \, \psi }{ \partial \, t _c} \)
\( \dot { x } \cfrac { \partial ^{ 2 }\psi }{ \partial \, t^{ 2 } _c} =\cfrac { \partial \, }{ \partial \, t _c} \left\{ \cfrac { \partial \, \psi }{ \partial \, x } \right\} \dot { x } ^{ 2 }+\ddot { x } \cfrac { \partial \, }{ \partial \, t _c} \left\{ m_{ p }\dot { x } ^{ 2 } \right\} +\ddot { x } \cfrac { \partial \, \psi }{ \partial \, t_c } \)
Since,
\( \psi =\cfrac { 1 }{ 2 } m_{ p }\dot { x } ^{ 2 }+V\)
\( m_{ p }\dot { x } ^{ 2 }=2(\psi -V)\)
We have,
\( \dot { x } \cfrac { \partial ^{ 2 }\psi }{ \partial \, t^{ 2 }_c } =\cfrac { \partial \, }{ \partial \, t_c } \left\{ \cfrac { \partial \, \psi }{ \partial \, x } \right\} \dot { x } ^{ 2 }+2\ddot { x } \cfrac { \partial \, }{ \partial \, t _c} \left\{ \psi -V \right\} +\ddot { x } \cfrac { \partial \, \psi }{ \partial \, t_c} \)
\(\dot { x } \cfrac { \partial ^{ 2 }\psi }{ \partial \, t^{ 2 }_{ c } } =-\cfrac {i }{ c } \cfrac { \partial ^{ 2 }\psi }{ \partial \, t^{ 2 }_{ c } } \dot { x } ^{ 2 }+2\ddot { x } \cfrac { \partial \, }{ \partial \, t_{ c } } \left\{ \cfrac { 3 }{ 2 } \psi -V \right\} \)
\(\dot { x } \cfrac { \partial ^{ 2 }\psi }{ \partial \, t^{ 2 }_{ c } } \left( 1+i\cfrac { \dot { x } }{ c } \right) =2\ddot { x } \cfrac { \partial \, }{ \partial \, t_{ c } } \left\{ \cfrac { 3 }{ 2 } \psi -V \right\} \)
\(\dot { x } \cfrac { \partial ^{ 2 }\psi }{ \partial \, t^{ 2 }_{ c } } =\cfrac { 2\ddot { x } }{ \left( 1+i\cfrac { \dot { x } }{ c } \right) } \cfrac { \partial \, }{ \partial \, t_{ c } } \left\{ \cfrac { 3 }{ 2 } \psi -V \right\} \)
And (1) becomes,
\( \ddot { x } \cfrac { \partial \, \psi }{ \partial \, t _c} =-ic\dot { x } ^{ 2 }\cfrac { \partial ^{ 2 }\psi }{ \partial \, x^{ 2 } } +\cfrac { 2\ddot { x } }{ \left( 1+i\cfrac { \dot { x } }{ c } \right) } \cfrac { \partial \, }{ \partial \, t _c} \left\{ \cfrac { 3 }{ 2 } \psi -V \right\}\)
\( \ddot { x } \cfrac { \partial \, \psi }{ \partial \, t_{ c } } \left\{ 1-\cfrac { 3 }{ \left( 1+i\cfrac { \dot { x } }{ c } \right) } \right\} =-ic\dot { x } ^{ 2 }\cfrac { \partial ^{ 2 }\psi }{ \partial \, x^{ 2 } } -\cfrac { 2\ddot { x } }{ \left( 1+i\cfrac { \dot { x } }{ c } \right) } \cfrac { \partial V\, }{ \partial \, t_{ c } } \)
\(\ddot { x } \cfrac { \partial \, \psi }{ \partial \, t_{ c } } \left\{ \cfrac { -2+i\cfrac { \dot { x } }{ c } }{ \left( 1+i\cfrac { \dot { x } }{ c } \right) } \right\} =-ic\dot { x } ^{ 2 }\cfrac { \partial ^{ 2 }\psi }{ \partial \, x^{ 2 } } -\cfrac { 2\ddot { x } }{ \left( 1+i\cfrac { \dot { x } }{ c } \right) } \cfrac { \partial V\, }{ \partial \, t_{ c } } \)
\(\ddot { x } \left( 2-i\cfrac { \dot { x } }{ c } \right) \cfrac { \partial \, \psi }{ \partial \, t_{ c } } =ic\left( 1+i\cfrac { \dot { x } }{ c } \right) \dot { x } ^{ 2 }\cfrac { \partial ^{ 2 }\psi }{ \partial \, x^{ 2 } } +2\ddot { x } \cfrac { \partial V\, }{ \partial \, t_{ c } } \) --- (**)
This is not a wave in space and time \(t\).
When \(\ddot{x}=0\),
\(ic\left( 1+i\cfrac { \dot { x } }{ c } \right) \dot { x } ^{ 2 }\cfrac { \partial ^{ 2 }\psi }{ \partial \, x^{ 2 } } =0\)
which implies,
\(\cfrac { \partial ^{ 2 }\psi }{ \partial \, x^{ 2 } } =0\)
which means,
\(\cfrac { \partial \psi }{ \partial \, x }=constant=A\) --- (2)
\(\psi(x)=Ax+C\)
\(\psi(x)\) is linear in space! If at \(x=0\), \(\psi(0)=C=0\)
\(\psi(x)=Ax \)
\(A\) is then a force and we have simply,
\(Energy=\psi(x)=A.x= Force . distance=Work\)
We started with a particle that might be responsible for gravity or electrostatic force and ended with a simple work done equation. This means the energy of this particle is just work done against a force.
Here the force is given the definition as rate of change of energy, \(\psi\) with distance \(x\), from (2).
When \(\dot{x}=c\), which is the case for charges, expression (**) becomes,
\(\ddot { x } \left( 2-i \right) \cfrac { \partial \, \psi }{ \partial \, t_{ c } } =ic^3\left( 1+i \right)\cfrac { \partial ^{ 2 }\psi }{ \partial \, x^{ 2 } } +2\ddot { x } \cfrac { \partial V\, }{ \partial \, t_{ c } } \)
\(\ddot { x } \left( 2-i \right) \cfrac { \partial \, \psi }{ \partial \, t_{ c } } =c^3(-1+i)\cfrac { \partial ^{ 2 }\psi }{ \partial \, x^{ 2 } } +2\ddot { x } \cfrac { \partial V\, }{ \partial \, t_{ c } } \)
Comparing real and imaginary parts,
\( 2\ddot { x } \cfrac { \partial \, \psi }{ \partial \, t_{ c } } =-c^3\cfrac { \partial ^{ 2 }\psi }{ \partial \, x^{ 2 } } +2\ddot { x } \cfrac { \partial V\, }{ \partial \, t_{ c } } \) --- (2)
\( -i\ddot { x } \cfrac { \partial \, \psi }{ \partial \, t_{ c } } =ic^3\cfrac { \partial ^{ 2 }\psi }{ \partial \, x^{ 2 } } \)
\( \ddot { x } \cfrac { \partial \, \psi }{ \partial \, t_{ c } } =-c^3\cfrac { \partial ^{ 2 }\psi }{ \partial \, x^{ 2 } } \) substitute into (2)
\( \cfrac { \partial \, \psi }{ \partial \, t_{ c } } =2\cfrac { \partial V\, }{ \partial \, t_{ c } } \) and \(\cfrac { \partial \, \psi }{ \partial \, t } =2\cfrac { \partial V\, }{ \partial \, t }\)
Since, \(\psi=T+V\)
\( \cfrac { \partial \, \psi }{ \partial \, t_{ c } } =2\cfrac { \partial T\, }{ \partial \, t_{ c } } \) and \(\cfrac { \partial \, \psi }{ \partial \, t } =2\cfrac { \partial V\, }{ \partial \, t }\)
\( \cfrac { \partial \, \psi }{ \partial \, t_{ c } } \) is divided evenly between \(\cfrac { \partial T\, }{ \partial \, t_{ c } } \) and \(\cfrac { \partial V\, }{ \partial \, t_{ c } } \)
From \(t_c=\cfrac{1}{\sqrt{2}}t.e^{-i\pi/4}\) and \( \ddot { x } \cfrac { \partial \, \psi }{ \partial \, t_{ c } }=-c^3\cfrac { \partial ^{ 2 }\psi }{ \partial \, x^{ 2 } } \)
\( \ddot { x } \cfrac { \partial \, \psi }{ \partial \, t } =-\cfrac{c^3}{\sqrt{2}}\cfrac { \partial ^{ 2 }\psi }{ \partial \, x^{ 2 } }e^{-i\pi/4} \)
This expression is unit dimension correct. However, in space and time \(t\), \(\psi\) is not a wave along \(x\) even when \(\dot{x}=c\).
