Cosmetics Attracting Attentions

From the post ""the post "Not Exponential, But Hyperbolic And Positive Gravity!" dated 22 Nov 2014,

$F_{\rho}=e^{ i3\pi /4 }D\sqrt { 2{ mc^{ 2 } } } .tanh\left( \cfrac { { D } }{ \sqrt { 2{ mc^{ 2 } } }  }( x-x_o).e^{ i\pi /4 } \right)$

where we insist that,

$G=D.e^{ i\pi /4 }$

is real,

But given,

$tanh(x)=\cfrac{e^x-e^{-x}}{e^x+e^{-x}}=\cfrac{1-e^{-2x}}{1+e^{-2x}}$

D can be real.

Consider,

$e^{ i\pi /4 }=(e^{ i\pi /2 })^{1/2}=\sqrt{i}$

that follows from Euler's

$e^{i\pi}+1=0$

An so,

$tanh(h\sqrt{i})=\cfrac{1-e^{-2h\sqrt{i}}}{1+e^{-2h\sqrt{i}}}$

where

$h=\cfrac { { D } }{ \sqrt { 2{ mc^{ 2 } } }  }( x-x_o)$

and

$e^{ i3\pi /4 }=i^{3/2}=i\sqrt{i}$

$F_{\rho}=i\sqrt{i}.D\sqrt { 2{ mc^{ 2 } } } .\cfrac{1-e^{-2h\sqrt{i}}}{1+e^{-2h\sqrt{i}}}$

and if we define,

$\varphi=\sqrt{i}$

$F_{\rho}=\varphi^3.D\sqrt { 2{ mc^{ 2 } } } .\cfrac{1-e^{-2\varphi h}}{1+e^{-2\varphi h}}$

where

$\varphi$ rotates $45^o$ and $\varphi^3$ rotates $135^o$ from the direction of $x$.

Cosmetics to bring attention to the these two rotations; they have significance, but what?

Furthermore,

 $\sqrt{2mc^2}=\sqrt{m(\sqrt{2}c)^2}$

also has significance; it implies that across two orthogonal dimensions, in place of $c$ is $\sqrt{2}c$.  This suggests that if entanglement is the reason for light speed limit, then entanglement is dimension specific, a different type of entanglement occurs along each orthogonal dimension; a different entanglement of the specific energy defining that dimension.  Particles can be limited separately along $t_c$ where in entanglement with other particles share electrostatic energy, and at the same time limited along $t_g$ where they share gravitational energy; ditto for $t_T$.

Since we normally dealt with energy of a particular sort, $\sqrt{2}c$ has never arise until we cross between orthogonal dimensions.

In space,

$c^2+c^2=2c^2$

$2c^2+c^2=3c^2$

And the speed limit we encountered first was,

$c=\sqrt{3}k$

where $k$ is a constant, $c$ light speed.  Three groups of particles entangled separately across three space dimensions with the one particle under observation.

Note: $\small{\sqrt{3}=1.7320508075688773...}$  is not rational.  Looking for more significant numbers to $c$ may just be chasing after the tail of $\small{\sqrt{3}}$.