Lorentz Boost

Theorem: Lorentz Boost

Suppose $S$ and $S^{\prime }$ are two inertial frames of reference and $S^{\prime }$ is moving relative to $S$ with velocity $\mathbf{u}$.

If an event happens at the point $(x,y,z)$ and time $t$ in $S$, then in $S^{\prime }$ it happens at the point $(x^{\prime },y^{\prime },z^{\prime })$ and time $t^{\prime }$, where the relationship between $x,y,z,t$ and $x^{\prime },y^{\prime },z^{\prime },t^{\prime }$ is given by the Lorentz transformation

$$\left[\begin{array}{c}c{t}^{\prime }\\ x^{\prime }\\ y^{\prime }\\ z^{\prime }\end{array}\right]=\left[\begin{array}{cccc}\gamma & -\gamma {\beta }_x& -\gamma {\beta }_y& -\gamma {\beta }_z\\ -\gamma {\beta }_x& 1+\frac{{\gamma }^2}{1+\gamma }{\beta }_x^2& \frac{{\gamma }^2}{1+\gamma }{\beta }_x{\beta }_y& \frac{{\gamma }^2}{1+\gamma }{\beta }_x{\beta }_z\\ -\gamma {\beta }_y& \frac{{\gamma }^2}{1+\gamma }{\beta }_x{\beta }_y& 1+\frac{{\gamma }^2}{1+\gamma }{\beta }_y^2& \frac{{\gamma }^2}{1+\gamma }{\beta }_y{\beta }_z\\ -\gamma {\beta }_z& \frac{{\gamma }^2}{1+\gamma }{\beta }_x{\beta }_z& \frac{{\gamma }^2}{1+\gamma }{\beta }_y{\beta }_z& 1+\frac{{\gamma }^2}{1+\gamma }{\beta }_z^2\end{array}\right]\left[\begin{array}{c}ct\\ x\\ y\\ z\end{array}\right],$$

where $\gamma$ is the Lorentz factor for the two inertial frames of reference, $c$ is the speed of light in a vacuum and $\mathbf{\beta }=\frac{1}{c}\mathbf{u}$.

PROOF

TODO