Photons Really Matter

From the post "Photons Matter"

${v}^{2}_{t}+{v}^{2}_{s} = A-B{d}_{s}(x)$

${v}^{2}_{t}+{v}^{2}_{s} = c^2-(c^2-\cfrac{{c}^{2}}{{n}^{2}}){d}_{s}(x)$

And that in an uniform photon-speed-slowing medium,

${v}_{s} =\cfrac{c}{n}.sin{\theta}$

where $\theta = sin^{-1}{ \sqrt{1-\cfrac{{n}_{o}}{c^2}}}$

From the energy conservation equation,

${v}^{2}_{t}+{v}^{2}_{s}=\cfrac{{c}^{2}}{n^2}$, if  ${v}_{s} =\cfrac{c}{n}.sin{\theta}$ then,

${v}^{2}_{t} = \cfrac{{c}^{2}}{n^2}(1-sin^{2}{\theta}) = (\cfrac{c}{n}.cos{\theta})^2$

${v}_{t} = \cfrac{c}{n}.cos{\theta}$

where $cos{\theta}$ is given by $\cfrac{\sqrt{{n}_{o}}}{c}$ and ${n}_{o}$ is related to the refractive index of the medium by

${n}_{o}=c^2(1-\cfrac{1}{n^2})$

Obviously,

$\cfrac{{v}_{t}}{c} = \gamma =  \cfrac{1}{n}cos{\theta}=  \cfrac{1}{n}\sqrt{1-\cfrac{1}{n^2}}$

where $n$ is the refractive index of the medium slowing photons down.  A photon that passing through a medium that slow it down in space also experiences time dilation.  Photons begin to experience time in the denser medium.  Denser space is an optically denser medium as far as a photon is concern.

We can also formulate a complex speed expression,

${V}_{c}= \cfrac{c}{n}(sin{\theta}+i.cos{\theta})$,    that

${v}_{s} =Re[\cfrac{c}{n}(sin{\theta}+i.cos{\theta})]$    and

${v}_{t} =Im[\cfrac{c}{n}(sin{\theta}+i.cos{\theta})]$

In other words,

${V}_{c}= {v}_{s}+i{v}_{t}$

where the real part is space and the imaginary part is time.  And that space and time is orthogonal.