Physlib.SpaceAndTime.SpaceTime.Boosts

Let $x$ be a point in spacetime with coordinates $(t, x, y, z)$, where $t$ is the temporal component and $x, y, z$ are the spatial components. For a velocity parameter $\beta \in \mathbb{R}$ such that $|\beta| < 1$ and the Lorentz factor $\gamma(\beta) = \frac{1}{\sqrt{1 - \beta^2}}$, the action of a Lorentz boost in the $x$-direction on $x$ transforms the coordinates according to: - $t' = \gamma(\beta)(t - \beta x)$ - $x' = \gamma(\beta)(x - \beta t)$ - $y' = y$ - $z' = z$

In a spacetime with $d+1$ spatial dimensions, consider a point with temporal component $t$ and spatial vector $\mathbf{x} = (x_0, x_1, \dots, x_d) \in \mathbb{R}^{d+1}$. Given a speed of light $c$ and a velocity parameter $\beta \in \mathbb{R}$ such that $|\beta| < 1$, the action of the inverse Lorentz boost $B^{-1}$ in the first spatial direction (the direction corresponding to index $0$) on the spacetime point $(t, \mathbf{x})$ results in a transformed point $(t', \mathbf{x}')$ defined by: - $t' = \gamma(\beta) \left( t + \frac{\beta}{c} x_0 \right)$ - $x'_0 = \gamma(\beta) (x_0 + c \beta t)$ - $x'_j = x_j$ for all $j \in \{1, \dots, d\}$ where $\gamma(\beta) = \frac{1}{\sqrt{1 - \beta^2}}$ is the Lorentz factor.