PX155 - H1 - lorentz transformation

from RP1 - linear transformations from $S$ to $S^{'}$ :
$x^{'} = γ (x - u t)$
$y^{'} = y$
$z^{'} = z$
$t^{'} = γ [t + \frac{1 - γ^{2}}{γ^{2}} \frac{x}{u}]$
- some galilean transformations if $γ = 1$
instead use RP2 - pulse of light fired from origin at $t = t^{'} = 0$ along x direction:
- in $S$ : $x = c t$
- in $S^{'}$ : $x^{'} = c t^{'}$
  $γ (x - u t) = c γ [t + \frac{1 - γ^{2}}{γ^{2}} \frac{x}{u}]$
  $γ (x - u t) = c γ [t + \frac{1 - γ^{2}}{γ^{2}} \frac{c t}{u}]$
  $- u^{2} t = c^{2} \frac{1 - γ^{2}}{γ^{2}} t$
  $\frac{1 - γ^{2}}{γ^{2}} = - \frac{u^{2}}{c^{2}}$
  $\frac{1}{γ^{2}} = 1 - \frac{u^{2}}{c^{2}}$

∴ γ = \frac{1}{\sqrt{1 - \frac{u^{2}}{c^{2}}}}

the lorentz transformations are:
$x^{'} = γ (x - u t)$ and $x = γ (x^{'} + u t^{'})$
$y^{'} = y$ and $y = y^{'}$
$z^{'} = z$ and $z = z^{'}$
$t^{'} = γ (t - \frac{u x}{c^{2}})$ and $t = γ (t^{'} + \frac{u x^{'}}{c^{2}})$
values of $γ$ :

	u	$γ$
speed of sound	$340 m s^{- 1}$	$1 + 6.4 \times 10^{- 13}$
low earth orbit	7.8 $k m s^{- 1}$	$1 + 3.4 \times 10^{- 10}$
$e^{-}$ in old TV	$1.2 \times 10^{8} m s^{- 1}$	$1.078$
LHC $7 T e V$	$0.999999 c$	$7500$
UHECR protons	$\leq c$	$5 \times 10^{10}$

UHECR - ultra high energy cosmic ray
$γ$ is very close to $1$ for every day speeds, so, galilean transforms are good enough