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Lorentz transformation

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This is a section about Lorentz transformation!

Lorentz transformation is a very important subject about which there is a lot to say.

For example, this sentence. And then another one.


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= Lorentz transform
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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:
* $(t, x, y, z)$ the observation of the standing observer
* $(t', x', y', z')$ the observation of the ending observer on a car
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:
$$
\begin{aligned}
t' & = \gamma \left( t - \frac{v x}{c^2} \right) \\
x' & = \gamma \left( x - v t \right) \\
y' & = y \\
z' & = z
\end{aligned}
$$
where:
$$
\gamma = \frac{1}{\sqrt{1 - \left(\frac{v}{c}\right)^2}}
$$

Note that if $\frac{v}{c}$ tends towards zero, then this reduces to the usual <Galilean transformations> which our intuition expects:
$$
\begin{aligned}
t' & = t
x' & = x - v t \\
y' & = y \\
z' & = z
\end{aligned}
$$

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


Lorentz transformation

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