Lorentz transform

The Lorentz transform is a mathematical tool for converting vectors between different inertial reference frames. Given some vector $a$ in a frame $\O$ , the same vector in frame $\O'$ (i.e. the vector representing the same geometric object, but with different components) is

\mat{c a'_t \\ a'_x \\ a'_y \\ a'_z} = \Lambda \mat{c a_t \\ a_x \\ a_y \\ a_z}.

Let us assume that $\O'$ moves in the $+x$ direction with speed $v$ relative to $\O$ . Then the Lorentz transform matrix $\Lambda$ is

\begin{align*} \Lambda &= \mat{\gamma & -\gamma \frac vc & 0 & 0 \\ -\gamma \frac vc & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1} \\ \gamma &= \frac{1}{\sqrt{1 - \frac{v^2}{c^2} } }. \end{align*}

If the relative velocity was in the $y$ direction, we would instead get

\begin{align*} \Lambda &= \mat{\gamma & 0 & -\gamma \frac vc & 0 \\ 0 & 1 & 0 & 0 \\ -\gamma \frac vc & 0 & \gamma & 0 \\ 0 & 0 & 0 & 1}. \end{align*}

The Lorentz transform can be applied to any 4-vector to transform it beween frames.