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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.

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