## FANDOM

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The Lorentz boost transforms 4 vectors from one intertial reference frames to the other. The common form of this transformation is

$c t \rightarrow ct \gamma + \beta \gamma x$

$x \rightarrow \gamma x + \beta \gamma c t$

$y \rightarrow y$

$z \rightarrow z$

where $\gamma = \left(1-\beta^2\right)^{-1/2}$, $\beta = v/c$, $v$ is the relative velocity between the two reference frames and $c$ is the speed of light. We would like to generalise this transformation to a general velocity vector $\vec{\beta}$. To so this, we split the position vector into a component parallel and perpendicular to $\vec{\beta}$. The parallel component $x = \vec{\beta} \cdot \vec{r} / \beta$ transforms in the way described above, and the perpendicular component does not change. Hence

$c t \rightarrow c t \gamma + \gamma \vec{\beta} \cdot \vec{r}$

$\vec{r} \rightarrow \gamma \frac{\vec{\beta} \cdot \vec{r}}{\beta^2} \vec{\beta} + \vec{r} - \frac{\vec{\beta} \cdot \vec{r}}{\beta^2} \vec{\beta} + \gamma \vec{\beta} c t$

Hence the transformation matrix can be written in the following form

$\Lambda \left( \vec{\beta} \right) = \begin{pmatrix} \gamma & \gamma \vec{\beta}\\ \gamma \beta^T & 1 + \left(\gamma-1 \right )\frac{\vec{\beta}^T\vec{\beta}}{\beta^2} \end{pmatrix}$

where $\vec{\beta}^T \vec{\beta}$ is an outer an outer product (i.e. a tensor) rather than a scalar.