Lorentz transformation

A Lorentz transformation preserves the Minkowski metric. For a Lorentz boost, let $S^{\prime }$ move at constant velocity $+v$ along the $x$-axis of $S$, with parallel spatial axes and coincident origins at $t=t^{\prime }=0$. With $\beta =v/c$ and the Lorentz factor $\gamma$, the coordinates of the same spacetime event obey

The inverse Lorentz transformation replaces $v$ by $-v$. In particular, $x^{\prime }=0$ traces the moving origin $x=vt$, fixing the sign convention. The time formula is $t^{\prime }=\gamma (t-vx/c^2)$; dropping its position term would lose the relativity of simultaneity. ^[ref].

For the moving-clock example in time dilation, the second tick has $(t,x)=(5\mu \mathrm{s},3c\mu \mathrm{s})$. At $v=0.6c$, substitution gives $t^{\prime }=1.25(5-0.6\times 3)\mu \mathrm{s}=4\mu \mathrm{s}$ and $x^{\prime }=1.25(3-0.6\times 5)c\mu \mathrm{s}=0$. The clock stays at its own origin, and its coordinate time there equals its proper time.

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.

The Lorentz transformation is a set of equations in the theory of special relativity that relate the space and time coordinates of two observers moving at constant velocity relative to each other. Named after the Dutch physicist Hendrik Lorentz, these transformations are essential for understanding how measurements of time and space change for observers in different inertial frames of reference, particularly when approaching the speed of light.

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:where:

Note that if $\frac{v}{c}$ tends towards zero, then this reduces to the usual Galilean transformations which our intuition expects:

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.