User:Thepigdog/Play2

Consider light travelling from the origin, along any any axis $$\mu$$. This light will have traveled a distance,
 * $$x_\mu = c t$$
 * $$x'_\mu = c t'$$

Substituting in (Equation 4a and 4b),
 * $$x_\mu = e_\mu \alpha (-v t' + \sum_\nu n_{\mu,\nu} x'_\nu)$$ (equation 5a)
 * $$x'_\mu = e_\mu \alpha (v t + \sum_\nu n_{\mu,\nu} x_\nu)$$ (equation 5a)

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

Multiplying each sides of these equations gives,
 * $$c^2 t t' = e_\mu^2 \alpha^2 t' t(n_{\mu,\mu}^2 c^2 - v^2)$$
 * $$1 = e_\mu^2 \alpha^2 (n_{\mu,\mu}^2 - \frac{v^2}{c^2})$$
 * $$e_\mu^2 \alpha^2 n_{\mu,\mu}^2 = 1 + e_\mu^2 \alpha^2 \frac{v^2}{c^2}$$
 * $$e_\mu^2 \alpha^2 n_{\mu,\mu}^2 = 1 - e_\mu^2 + e_\mu^2(\frac{1 - \frac{v^2}{c^2} + \frac{v^2}{c^2}}{1 - \frac{v^2}{c^2}})$$
 * $$e_\mu^2 \alpha^2 n_{\mu,\mu}^2 = 1 - e_\mu^2 + e_\mu^2(\frac{1}{1 - \frac{v^2}{c^2}})$$
 * $$e_\mu^2 \alpha^2 n_{\mu,\mu}^2 = 1 - e_\mu^2 + e_\mu^2 \alpha^2$$
 * $$e_\mu \alpha n_{\mu,\mu} = \sqrt{1 - e_\mu^2 + e_\mu^2 \alpha^2}$$

Light traveling out in any directions
Consider light travelling from the origin, in any direction f. This light will have traveled a distance,
 * $$x_\nu = c t f_\nu$$
 * $$x'_\nu = c t' f_\nu + v t' f_\nu (1 - f.e) $$????

Substituting in (Equation 4a and 4b),
 * $$c t f_\mu = e_\mu \alpha (-v t' + \sum_\nu n_{\mu,\nu} c t' f_\nu)$$ (equation 5a)
 * $$c t' f_\mu = e_\mu \alpha (v t + \sum_\nu n_{\mu,\nu} c t f_\nu)$$ (equation 5a)

gives
 * $$c t f_\mu = e_\mu \alpha t' (k c - v)$$
 * $$c t' f_\mu = e_\mu \alpha t (k c + v)$$

where,
 * $$k = \sum_\nu n_{\mu,\nu} f_\nu$$

Multiplying each sides of these equations gives,
 * $$c^2 t t' f_\mu^2 = e_\mu^2 \alpha^2 t' t(k^2 c^2 - v^2)$$
 * $$f_\mu^2 = e_\mu^2 \alpha^2 (k^2 - \frac{v^2}{c^2})$$
 * $$e_\mu^2 \alpha^2 k^2 = f_\mu^2 + e_\mu^2 \alpha^2 \frac{v^2}{c^2}$$
 * $$e_\mu^2 \alpha^2 k^2 = f_\mu^2 - e_\mu^2 + e_\mu^2(\frac{1 - \frac{v^2}{c^2} + \frac{v^2}{c^2}}{1 - \frac{v^2}{c^2}})$$
 * $$e_\mu^2 \alpha^2 k^2 = f_mu^2 - e_\mu^2 + e_\mu^2(\frac{1}{1 - \frac{v^2}{c^2}})$$
 * $$e_\mu^2 \alpha^2 k^2 = f_\mu^2 - e_\mu^2 + e_\mu^2 \alpha^2$$
 * $$e_\mu \alpha \sum_\nu n_{\mu,\nu} f_\nu = \sqrt{f_\mu^2 - e_\mu^2 + e_\mu^2 \alpha^2}$$

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

Setting $$\rho \ne \mu \implies f_\rho = 1$$ gives,
 * $$e_\mu \alpha n_{\mu,\rho} = e_\mu\sqrt{ \alpha^2 - 1} = e_\mu \alpha \frac{v}{c}$$


 * $$x_\mu = e_\mu \alpha (-v t' + \frac{v}{c} \sum_{\nu \ne \mu} x'_\nu) + x'_\mu \sqrt{1 - e_\mu^2 + e_\mu^2 \alpha^2}$$ ???? not correct ????