\({v}^{2}_{t}+{v}^{2}_{s} = A-B{d}_{s}(x)\)
\({v}^{2}_{t}+{v}^{2}_{s} = c^2-(c^2-\cfrac{{c}^{2}}{{n}^{2}}){d}_{s}(x)\)
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.
\({V}_{c}= \cfrac{c}{n}(sin{\theta}+i.cos{\theta})\), 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.
