User:Thepigdog/Play

Distance and the moving frames V2
The two reference systems are moving with a relative constant velocity v in a direction e (given by a unit vector). So at, $$\mathbf{x} = (0, 0, 0)$$, $$\mathbf{x}' = v t' \mathbf{e}$$. This equation may be written,
 * $$x'_\nu = v t' e_\nu$$

Note: There is an assumption here that $$\mathbf{e}$$ and $$\mathbf{e}'$$ are the same. This assumes that the co-ordinate axis are parallel.


 * $$x_\mu = e_\mu (p_{\mu,0} t' + \sum_\nu p_{\mu,\nu} x'_\nu) + (1 - e_\mu) x'_\mu$$ (equation 1)

(Equation 1) which applies for any $$x'_\mu$$. The general formula may be applied to describe the origin of the co-ordinate systems. Substituting,
 * $$x_\nu$$ for $$0$$
 * $$x'_\nu$$ for $$v t' e_\nu$$

in (equation 1) gives,
 * $$0 = e_\mu (p_{\mu,0} t' + \sum_\nu p_{\mu,\nu} v t' e_\nu) + (1 - e_\mu) v t' e_\mu$$

which simplifies to,
 * $$0 = p_{\mu,0} + v ((1 - e_\mu) + \sum_\nu p_{\mu,\nu} e_\nu) $$

and then,
 * $$-\frac{p_{\mu,0}}{v} = (1 - e_\mu) - \sum_\nu m_{\mu,\nu} e_\nu $$

This suggests a more natural constant for $$(\mu,0)$$. Define, $$n_{\mu,0}$$ as,
 * $$ n_{\mu,0} = -\frac{p_{\mu,0}}{v}$$ (definition 1)

then,
 * $$ n_{\mu,0} = (1 - e_\mu) + \sum_\nu p_{\mu,\nu} e_\nu $$

Then define $$n_{\mu,\nu}$$ as,
 * $$p_{\mu,\nu} = n_{\mu,0} n_{\mu,\nu} $$ (definition 2)

then,
 * $$ n_{\mu,0} = (1 - e_\mu) + \sum_\nu n_{\mu,0} n_{\mu,\nu} e_\nu $$


 * $$1 - \frac{1 - e_\mu}{n_{\mu,0}} = \sum_\nu n_{\mu,\nu} e_\nu$$ (equation 3)

Substituting the (definition 1) and (definition2) into (equation 1) gives,
 * $$x_\mu = e_\mu n_{\mu,0} (- v t' + \sum_\nu n_{\mu,\nu} x'_\nu) + (1 - e_\mu) x'_\mu$$

Assuming that displacements in one direction in one co-ordinate system, result only in displacements in the same direction in the primed system,
 * $$\mu \ne \nu \implies n_{\mu,\nu} = 0$$

Then equation 3 gives,
 * $$1 - \frac{1 - e_\mu}{n_{\mu,0}} = n_{\mu,\mu} e_\mu$$

Solve for $$n_{\mu,\mu}$$ gives,
 * $$n_{\mu,\mu} = \frac{1 - \frac{1 - e_\mu}{n_{\mu,0}}}{e_\mu} = 1 + (1 - \frac{1}{n_{\mu,0}})(\frac{1}{e_\mu} - 1) $$

gives,
 * $$x_\mu = e_\mu n_{\mu,0} (- v t' + \frac{1 - \frac{1 - e_\mu}{n_{\mu,0}}}{e_\mu} x'_\mu) + (1 - e_\mu) x'_\mu$$


 * $$x_\mu = e_\mu n_{\mu,0} (- v t' + (1 + (1 - \frac{1}{n_{\mu,0}})(\frac{1}{e_\mu} - 1)) x'_\mu) + (1 - e_\mu) x'_\mu$$
 * $$x_\mu = e_\mu n_{\mu,0} (- v t' + x'_\mu + (1 - \frac{1}{n_{\mu,0}})(\frac{1}{e_\mu} - 1) x'_\mu) + (1 - e_\mu) x'_\mu$$
 * $$x_\mu = e_\mu n_{\mu,0} (- v t' + x'_\mu) + e_\mu n_{\mu,0} (1 - \frac{1}{n_{\mu,0}})(\frac{1}{e_\mu} - 1) x'_\mu + (1 - e_\mu) x'_\mu$$
 * $$x_\mu = e_\mu n_{\mu,0} (- v t' + x'_\mu) + (n_{\mu,0} - 1)(1 - e_\mu ) x'_\mu + (1 - e_\mu) x'_\mu$$
 * $$x_\mu = e_\mu n_{\mu,0} (- v t' + x'_\mu) + (n_{\mu,0} - 1 + 1)(1 - e_\mu ) x'_\mu $$
 * $$x_\mu = e_\mu n_{\mu,0} (- v t' + x'_\mu) + n_{\mu,0}(1 - e_\mu ) x'_\mu $$
 * $$x_\mu = e_\mu n_{\mu,0} (- v t') + n_{\mu,0} x'_\mu $$
 * $$x_\mu = n_{\mu,0} (x'_\mu - v t' e_\mu ) $$

???? same result - seems broken ????