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 \(\small{\sqrt{3}}\).
