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 $γ = \frac{1}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}$ .

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.

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Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es

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