## The Lorentz Transformation

Suppose two inertial frames

$S, \: S'$
coide at
$t=0$
, with moving at constant speed
$v$
along the
$x$
axis of
$S'$
.

In the inertial frame

$S$
, an event
$E_1$
occurs at coordinates
$(x, y, z, t)$
according to an observer at the origin of
$S'$
.
An observer at the origin of
$S'$
will measure the coordinates of
$E_1$
to be
$(x'.y',z',t')=(\frac{x-vt}{\sqrt{1-v^2/c^2}}), y, z, \frac{t-vx/c^2}{\sqrt{1-v^2/c^2}})$
.
The transformation is symmetric, so
$(x.y,z,t)=(\frac{x'+vt'}{\sqrt{1-v^2/c^2}}), y', z', \frac{t'+vx'/c^2}{\sqrt{1-v^2/c^2}})$
.

The Lorentz Transformation supercedes the Galilean Transformation, which is only accurate in the limit of low speeds.