User:Thepigdog/Play2

Consider light travelling from the origin, along any any axis $\mu$. This light will have traveled a distance,

$x_{\mu }=ct$

$x'_{\mu }=ct'$

Substituting in (Equation 4a and 4b),

$x_{\mu }=e_{\mu }\alpha (-vt'+\sum _{\nu }n_{\mu ,\nu }x'_{\nu })$ (equation 5a)

$x'_{\mu }=e_{\mu }\alpha (vt+\sum _{\nu }n_{\mu ,\nu }x_{\nu })$ (equation 5a)

gives

$ct=e_{\mu }\alpha t'(n_{\mu ,\mu }c-v)$

$ct'=e_{\mu }\alpha t(n_{\mu ,\mu }c+v)$

Multiplying each sides of these equations gives,

$c^{2}tt'=e_{\mu }^{2}\alpha ^{2}t't(n_{\mu ,\mu }^{2}c^{2}-v^{2})$

$1=e_{\mu }^{2}\alpha ^{2}(n_{\mu ,\mu }^{2}-{\frac {v^{2}}{c^{2}}})$

$e_{\mu }^{2}\alpha ^{2}n_{\mu ,\mu }^{2}=1+e_{\mu }^{2}\alpha ^{2}{\frac {v^{2}}{c^{2}}}$

$e_{\mu }^{2}\alpha ^{2}n_{\mu ,\mu }^{2}=1-e_{\mu }^{2}+e_{\mu }^{2}({\frac {1-{\frac {v^{2}}{c^{2}}}+{\frac {v^{2}}{c^{2}}}}{1-{\frac {v^{2}}{c^{2}}}}})$

$e_{\mu }^{2}\alpha ^{2}n_{\mu ,\mu }^{2}=1-e_{\mu }^{2}+e_{\mu }^{2}({\frac {1}{1-{\frac {v^{2}}{c^{2}}}}})$

$e_{\mu }^{2}\alpha ^{2}n_{\mu ,\mu }^{2}=1-e_{\mu }^{2}+e_{\mu }^{2}\alpha ^{2}$

$e_{\mu }\alpha n_{\mu ,\mu }={\sqrt {1-e_{\mu }^{2}+e_{\mu }^{2}\alpha ^{2}}}$

Light traveling out in any directions

Consider light travelling from the origin, in any direction f. This light will have traveled a distance,

$x_{\nu }=ctf_{\nu }$

$x'_{\nu }=ct'f_{\nu }+vt'f_{\nu }(1-f.e)$????

Substituting in (Equation 4a and 4b),

$ctf_{\mu }=e_{\mu }\alpha (-vt'+\sum _{\nu }n_{\mu ,\nu }ct'f_{\nu })$ (equation 5a)

$ct'f_{\mu }=e_{\mu }\alpha (vt+\sum _{\nu }n_{\mu ,\nu }ctf_{\nu })$ (equation 5a)

gives

$ctf_{\mu }=e_{\mu }\alpha t'(kc-v)$

$ct'f_{\mu }=e_{\mu }\alpha t(kc+v)$

where,

$k=\sum _{\nu }n_{\mu ,\nu }f_{\nu }$

Multiplying each sides of these equations gives,

$c^{2}tt'f_{\mu }^{2}=e_{\mu }^{2}\alpha ^{2}t't(k^{2}c^{2}-v^{2})$

$f_{\mu }^{2}=e_{\mu }^{2}\alpha ^{2}(k^{2}-{\frac {v^{2}}{c^{2}}})$

$e_{\mu }^{2}\alpha ^{2}k^{2}=f_{\mu }^{2}+e_{\mu }^{2}\alpha ^{2}{\frac {v^{2}}{c^{2}}}$

$e_{\mu }^{2}\alpha ^{2}k^{2}=f_{\mu }^{2}-e_{\mu }^{2}+e_{\mu }^{2}({\frac {1-{\frac {v^{2}}{c^{2}}}+{\frac {v^{2}}{c^{2}}}}{1-{\frac {v^{2}}{c^{2}}}}})$

$e_{\mu }^{2}\alpha ^{2}k^{2}=f_{m}u^{2}-e_{\mu }^{2}+e_{\mu }^{2}({\frac {1}{1-{\frac {v^{2}}{c^{2}}}}})$

$e_{\mu }^{2}\alpha ^{2}k^{2}=f_{\mu }^{2}-e_{\mu }^{2}+e_{\mu }^{2}\alpha ^{2}$

$e_{\mu }\alpha \sum _{\nu }n_{\mu ,\nu }f_{\nu }={\sqrt {f_{\mu }^{2}-e_{\mu }^{2}+e_{\mu }^{2}\alpha ^{2}}}$

Setting $f_{\mu }=1$ gives,

$e_{\mu }\alpha n_{\mu ,\mu }={\sqrt {1-e_{\mu }^{2}+e_{\mu }^{2}\alpha ^{2}}}$

Setting $\rho \neq \mu \implies f_{\rho }=1$ gives,

$e_{\mu }\alpha n_{\mu ,\rho }=e_{\mu }{\sqrt {\alpha ^{2}-1}}=e_{\mu }\alpha {\frac {v}{c}}$

$x_{\mu }=e_{\mu }\alpha (-vt'+{\frac {v}{c}}\sum _{\nu \neq \mu }x'_{\nu })+x'_{\mu }{\sqrt {1-e_{\mu }^{2}+e_{\mu }^{2}\alpha ^{2}}}$ ???? not correct ????