User:Thepigdog/General Relativity - 02 - Lorentz transforms

The assumptions are,
 * The two axis of the reference frames are parallel.
 * The transformation between frames depends only the velocity.
 * The transformation from frame to the primed frame uses a velcoity v.
 * The transformation from primed frame to the frame uses a velcoity -v.
 * The speed of light as measured in the two frames is constant.
 * The difference in position divided by the difference in time of a light particle recorded at two positions is always the constant c.
 * In the transformation, each component in the reference frame is a linear combination of the components in the primed frame.

The last condition is expressed as,
 * $$x'_\mu = \sum_\nu m_{\mu,\nu} x_\nu$$

where $$m_{\mu,\nu}$$ are the components of the transformation matrix.

One approach would be two put this transformation into the Minkowski metric,
 * $$\sum_\mu g_{\mu,\mu} x_\mu^2 = \sum_\mu g_{\mu,\mu} {x'}_\mu^2 $$
 * $$\sum_\mu g_{\mu,\mu} x_\mu^2 = \sum_\mu g_{\mu,\mu} (\sum_\nu m_{\mu,\nu} x_\nu)^2 $$
 * $$ = \sum_\mu g_{\mu,\mu} (\sum_\nu m_{\mu,\nu} x_\nu)(\sum_\rho m_{\mu,\rho} x_\rho) $$
 * $$ = \sum_\mu \sum_\nu \sum_\rho g_{\mu,\mu} m_{\mu,\nu} x_\nu m_{\mu,\rho} x_\rho $$
 * $$ = \sum_\nu \sum_\rho x_\nu x_\rho \sum_\mu g_{\mu,\mu} m_{\mu,\nu} m_{\mu,\rho} $$
 * $$ = \sum_{\nu,\rho} x_\nu x_\rho \sum_\mu g_{\mu,\mu} m_{\mu,\nu} m_{\mu,\rho} $$
 * $$ = (\sum_{\nu,\nu} x_\nu^2 \sum_\mu g_{\mu,\mu} m_{\mu,\nu}^2) + (\sum^{\nu \ne \rho}_{\nu,\rho} x_\nu x_\rho \sum_\mu g_{\mu,\mu} m_{\mu,\nu} m_{\mu,\rho})$$

Gives two equations,
 * $$\sum_\nu g_{\nu,\nu} x_\nu^2 = \sum_{\nu,\nu} x_\nu^2 \sum_\mu g_{\mu,\mu} m_{\mu,\nu}^2$$
 * $$ 0 = \sum^{\nu \ne \rho}_{\nu,\rho} x_\nu x_\rho \sum_\mu g_{\mu,\mu} m_{\mu,\nu} m_{\mu,\rho})$$

From the first equation,
 * $$g_{\nu,\nu} = \sum_\mu g_{\mu,\mu} m_{\mu,\nu}^2$$

From the second equation,
 * $$ 0 = \sum^{\nu > \rho}_{\nu,\rho} x_\nu x_\rho \sum_\mu g_{\mu,\mu} m_{\mu,\nu} m_{\mu,\rho}$$
 * $$ \nu > \rho \implies 0 = \sum_\mu g_{\mu,\mu} m_{\mu,\nu} m_{\mu,\rho}$$