The equation that allows us to calculate stuff in special relativity!

Take two observers with identical rules and stopwatch, and aligned axes, but one is on a car moving at towards the $+x$ direction at speed $v$.

TODO image.

When both observe an event, if we denote:It is of course arbitrary who is standing and who is moving, we will just use the term "standing" for the one without primes.

- $(t,x,y,z)$ the observation of the standing observer
- $(t_{′},x_{′},y_{′},z_{′})$ the observation of the ending observer on a car

Then the coordinates of the event observed by the observer on the car are:
where:

$t_{′}x_{′}y_{′}z_{′} =γ(t−c_{2}vx )=γ(x−vt)=y=z $

$γ=1−(cv )_{2} 1 $

Note that if $cv $ tends towards zero, then this reduces to the usual Galilean transformations which our intuition expects:

$t_{′}y_{′}z_{′} =tx_{′}=y=z =x−vt$

This explains why we don't observe special relativity in our daily lives: macroscopic objects move too slowly compared to light, and $cv $ is almost zero.