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advanced_tools:group_theory:lorentz_group [2018/04/06 15:56] jakobadmin [Concrete] |
advanced_tools:group_theory:lorentz_group [2018/04/08 17:13] georgefarr ↷ Links adapted because of a move operation |
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**Definition of the Lorentz transformations** | **Definition of the Lorentz transformations** | ||
- | It follows from the postulates of [[theories:special_relativity|special relativity]] that | + | It follows from the postulates of [[models:special_relativity|special relativity]] that |
$d s^2 = \eta^{\mu \nu} dx_\mu dx_\nu$ stays exactly the same in all inertial frames of reference: | $d s^2 = \eta^{\mu \nu} dx_\mu dx_\nu$ stays exactly the same in all inertial frames of reference: | ||
\begin{equation} ds'^2 = dx'_\mu dx'_\nu \eta^{\mu\nu} = ds^2 = dx_\mu dx_\nu \eta^{\mu\nu} \, ,\end{equation} | \begin{equation} ds'^2 = dx'_\mu dx'_\nu \eta^{\mu\nu} = ds^2 = dx_\mu dx_\nu \eta^{\mu\nu} \, ,\end{equation} | ||
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\[ \Lambda_{\mathrm{rot }}= \begin{pmatrix} 1 & \\ & R_{3 \times 3} \end{pmatrix} | \[ \Lambda_{\mathrm{rot }}= \begin{pmatrix} 1 & \\ & R_{3 \times 3} \end{pmatrix} | ||
\] | \] | ||
- | with the usual rotation matrices $R_{3 \times 3}$. | + | with the usual rotation matrices $R_{3 \times 3}$: |
+ | |||
+ | \begin{eqnarray} | ||
+ | & & R_x(\phi) = | ||
+ | \begin{pmatrix} | ||
+ | 1 & 0 & 0 \\ 0 & \cos\phi & \sin\phi \\ 0 & -\sin\phi & \cos\phi | ||
+ | \end{pmatrix} \label{eq:rotx} \\ | ||
+ | & & R_y(\psi) = | ||
+ | \begin{pmatrix} | ||
+ | \cos \psi & 0 & -\sin\psi \\ 0 & 1 & 0 \\ \sin\psi & 0 & \cos\psi | ||
+ | \end{pmatrix} \label{eq:roty} \\ | ||
+ | & & R_z(\theta) = | ||
+ | \begin{pmatrix} | ||
+ | \cos \theta & \sin \theta & 0 \\-\sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 | ||
+ | \end{pmatrix} \label{eq:rotz} | ||
+ | \end{eqnarray} | ||
__Boosts__ | __Boosts__ |