Lorentz Boost

up:: 032 MOC Relativity

A Lorentz Boost is a linear transformation of the type

$$\{\begin{array}{ll}c{t}^{\prime }& =\gamma (ct-\beta x)\\ x^{\prime }& =\gamma (x-\beta ct)\end{array}⟺\left(\begin{array}{c}c{t}^{\prime }\\ x^{\prime }\end{array}\right)=\left(\begin{array}{cc}\gamma & -\gamma \beta \\ -\gamma \beta & \gamma \end{array}\right)\left(\begin{array}{c}ct\\ x\end{array}\right)$$

where the boost can be seen as the (symmetric) matrix

$$\Lambda =\left(\begin{array}{cc}\gamma & -\gamma \beta \\ -\gamma \beta & \gamma \end{array}\right)$$

when only looking at a single-direction boost, in the $x$-direction.

Matrix form in Minkowski Space

In Minkowski space, this boost (only in the $x$ direction) would be the matrix

$$\Lambda =\left(\begin{array}{cccc}\gamma & -\gamma \beta & 0& 0\\ -\gamma \beta & \gamma & 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$$