Lorentz boost

Also called hyperbolic rotations. They are a subgroup of Lorentz group, important in special relativity.

Lorentz boosts in 1+1 dimensions are a fundamental concept when considering how measurements of time and space change for observers moving relative to one another. In a 1+1 dimensional space, we only consider one spatial dimension $x$ and one time dimension $t$ .

The Lorentz boost formula in 1+1 dimensions describes how the coordinates (time and space) of an event transform from one inertial frame to another moving at a constant velocity $v$ relative to the first. The transformation is given by:

\begin{aligned} t^{'} & = γ (t - \frac{v}{c^{2}} x) \\ x^{'} & = γ (x - v t) \end{aligned}

Where:

$t^{'}$ and $x^{'}$ are the time and space coordinates in the moving frame.
$t$ and $x$ are the time and space coordinates in the stationary frame.
$v$ is the relative velocity between the two frames.
$c$ is the speed of light.
$γ$ is the Lorentz factor, defined as $γ = \frac{1}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}$ .

Also written as

\begin{aligned} t^{'} & = (\cosh ϕ) t - (\sinh ϕ) x \\ x^{'} & = - (\sinh ϕ) t + (\cosh ϕ) x \\ y^{'} & = y \\ z^{'} & = z, \end{aligned}

where $ϕ$ is called the rapidity, and is defined so that $\tanh (ϕ) = v$ .
This transformation ensures that the speed of light is the same in all inertial frames, a cornerstone of special relativity. It also leads to effects such as time dilation and length contraction.

We will denote this Lorentz boost by $Λ^{x} (v)$ or $Λ^{x} (η)$ , depending on the context.