Adjusting For Unit Consistency, Lorentz's

With reference to the previous post, "Thought Experiment Gone Wrong" dated 20 Nov 2017, when we replace "units" with meters and uses light speed, $c$ as standard,

Instead of one unit,

$x^{'}cos(\theta)=x^{'}.\cfrac{\sqrt{c^2-v^2}}{c}=x^{'}.\sqrt{1-\cfrac{v^2}{c^2}}=x$

So,

$x^{'}=\cfrac{x}{\sqrt{1-\cfrac{v^2}{c^2}}}$ --- (*)

as the origin is still displaced by $vt$,

$x^{'}=x-vt$

but the dilation needed is now given by (*), so,

$x^{'}=\cfrac{x-vt}{\sqrt{1-\cfrac{v^2}{c^2}}}$

and the rest of Lorentz transformations follows.

What does it mean when we use light speed this way?  $c$, instead of one unit?  It means, $v$ is in units of 299792458 ms^-1 and $x$ in units of 299792458 m.  But it is OK because,

$c.x^{'}=\cfrac{c.x-c.vt}{\sqrt{1-\cfrac{(c.v)^2}{c^2}}}$

where $x^{' }$, $x$ and $v$ are in units or unit meter and unit meter per unit time, respectively.

$c.x^{'}=c.\cfrac{x-vt}{\sqrt{1-v^2}}$

$x^{'}=\cfrac{x-vt}{\sqrt{1-v^2}}$

which is the same as before.  But! when $\gamma=\cfrac{1}{\sqrt{1-\cfrac{v^2}{c^2}}}$,

$\cfrac{x}{|c|}\rightarrow x$ 

$\cfrac{x^{'}}{|c|}\rightarrow x^{'}$

and

$\cfrac{v}{c}\rightarrow v$

we are using light speed as reference, everything is measured in 299792458 m or 299792458 ms^-1.

Have a nice day.