Lorentz transformation

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.

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

t^{'} x^{'} y^{'} z^{'} = γ (t - \frac{v x}{c ^{2}}) = γ (x - v t) = y = z

where:

γ = \frac{1}{1 - ( \frac{v}{c} ) ^{2}}

Note that if

\frac{v}{c}

tends towards zero, then this reduces to the usual Galilean transformations which our intuition expects:

t^{'} y^{'} z^{'} = t x^{'} = y = z = x - v t

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

\frac{v}{c}

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