Passing Opposite In Love

Consider two similar particles one along the positive time line, $t_c$ and the other on the negative time line, $-t_c$,

$\cfrac { \partial \, \psi _{ 1 } }{ \partial \, t_c }=\sqrt{2}\cfrac { \partial \, \psi _{ 1 } }{ \partial \, t }e^{ i\pi /4 }$

$\cfrac { \partial \, \psi _{ 2 } }{ \partial \, (-t_c) }=-\sqrt{2}\cfrac { \partial \, \psi _{ 2 } }{ \partial \, t }e^{ i\pi /4 }$

given, $t_c=\cfrac{1}{\sqrt{2}}t.e^{-i\pi/4}$

When they are exchanging energy along the two orthogonal space dimension,

$\cfrac { \partial \, \psi _{ 1 } }{ \partial \, t _c} =\cfrac { \partial \, \psi _{ 2 } }{ \partial \left( -t _c\right)  }$

$\cfrac { \partial \, \psi _{ 1} }{ \partial \, t } =-\cfrac { \partial \, \psi _{ 2 } }{ \partial \, t }$

Here $\psi$ is a wave, oscillating between two space dimensions and traveling down a third dimension that is the time dimension, $t_g$.  The wave exist in a second time dimension, $t_c$ by which velocities and other rate of change in time are measured.

With these relationships we formulate,

$\ddot { x } \cfrac { \partial \, \psi _{ 1 } }{ \partial \, t } =\ddot { x } .\left(-\cfrac { \partial \, \psi _{ 2 } }{ \partial \, t }\right) =-\ddot { x } \cfrac { \partial \, \psi _{ 2 } }{ \partial \, t }$

Since,

$\ddot { x } \cfrac { \partial \, \psi _{ 1 } }{ \partial \, t } =-\cfrac { c^{ 3 } }{ \sqrt { 2 }  } \cfrac { \partial ^{ 2 }\psi _{ 1 } }{ \partial \, x^{ 2 } } e^{ -i\pi /4 }$

and

$-\ddot { x } .\cfrac { \partial \, \psi _{ 2 } }{ \partial \, t } =\cfrac { c^{ 3 } }{ \sqrt { 2 }  } \cfrac { \partial ^{ 2 }\psi _{ 2 } }{ \partial \, x^{ 2 } } e^{ -i\pi /4 }$

We have,

$-\cfrac { c^{ 3 } }{ \sqrt { 2 }  } \cfrac { \partial ^{ 2 }\psi _{ 1 } }{ \partial \, x^{ 2 } } e^{ -i\pi /4 }=\cfrac { c^{ 3 } }{ \sqrt { 2 }  } \cfrac { \partial ^{ 2 }\psi _{ 2 } }{ \partial \, x^{ 2 } } e^{ -i\pi /4 }$

and so,

$-\cfrac { \partial ^{ 2 }\psi _{ 1 } }{ \partial \, x^{ 2 } } =\cfrac { \partial ^{ 2 }\psi _{ 2 } }{ \partial \, x^{ 2 } }$

The gradient of the forces which are indicative of their directions,

$\cfrac { \partial \, F_{ 1 } }{ \partial \, x } =\cfrac { \partial ^{ 2 }\psi _{ 1 } }{ \partial \, x^{ 2 } } =-\cfrac { \sqrt { 2 }  }{ c^{ 3 } } \ddot { x } \cfrac { \partial \, \psi _{ 1 } }{ \partial \, t } e^{ i\pi /4 }$

$\cfrac { \partial \, F_{ 2 } }{ \partial \, x } =\cfrac { \partial ^{ 2 }\psi _{ 2 } }{ \partial \, x^{ 2 } } =\cfrac { \sqrt { 2 }  }{ c^{ 3 } } \ddot { x } \cfrac { \partial \, \psi _{ 1 } }{ \partial \, t } e^{ i\pi /4 }$

These forces satisfy the change in energy required of both particles as $\psi$ oscillates in $t_c$ time.  $\psi$ however is not oscillating in $t$.  In the diagram above, we consider a point on $\psi_2$ along the line joining the two centers of the particles.  The resultant force $F$ is towards $\psi_1$ drawing the particles closer and we see that the forces are attractive; that particles in opposing directions on the time axis $t_c$ attract each other.  It is assumed that the force decreases monotonously with $x$.

In a similar way,

when both particle are traveling in the same direction,

$\cfrac { \partial \, F_{ 1 } }{ \partial \, x } =\cfrac { \partial ^{ 2 }\psi _{ 1 } }{ \partial \, x^{ 2 } } =-\cfrac { \sqrt { 2 }  }{ c^{ 3 } } \ddot { x } \cfrac { \partial \, \psi _{ 1 } }{ \partial \, t } e^{ i\pi /4 }$

$\cfrac { \partial \, F_{ 2 } }{ \partial \, x } =\cfrac { \partial ^{ 2 }\psi _{ 2 } }{ \partial \, x^{ 2 } } =-\cfrac { \sqrt { 2 }  }{ c^{ 3 } } \ddot { x } \cfrac { \partial \, \psi _{ 1 } }{ \partial \, t } e^{ i\pi /4 }$

The resultant force $F$ is away from $\psi_1$ pushing the particles closer and we see that the forces are repuslive; that particles in the same directions on the time axis $t_c$ repel each other.  It is assumed that the force decreases monotonously with $x$; that $\cfrac { \partial \, \psi  }{ \partial \, t }$ does not flip sign.

It is better to solve for the force,

$F_{\psi}=-\cfrac { \partial\,\psi }{ \partial \, x}$

directly.  Just from the above analysis, attractive force is greater than the repulsive force.