Coordinate rotation in 1+1 Minskowski space

Autor/Urheber:

Franzl aus tirol

Shortlink:

https://dewiki.de/b/22fab7

Quelle:

Wikimedia Commons

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410 x 410 Pixel (122213 Bytes)

Beschreibung:

Lorentz transformation:

${\displaystyle {\begin{pmatrix}u_{0}\\u_{1}\end{pmatrix}}={\begin{pmatrix}\cosh \theta &-\sinh \theta \\-\sinh \theta &\cosh \theta \end{pmatrix}}{\begin{pmatrix}x_{0}\\x_{1}\end{pmatrix}}}$

${\displaystyle {\begin{pmatrix}x_{0}\\x_{1}\end{pmatrix}}={\begin{pmatrix}\cosh \theta &\sinh \theta \\\sinh \theta &\cosh \theta \end{pmatrix}}{\begin{pmatrix}u_{0}\\u_{1}\end{pmatrix}}}$

with ${\displaystyle \tanh \theta =\beta }$. Here: ${\displaystyle \beta =0.5}$.

Basis and dual basis vectors:

${\displaystyle {\vec {e}}_{u_{0}}={\begin{pmatrix}\cosh \theta \\\sinh \theta \end{pmatrix}}\,,\ {\vec {e}}_{u_{1}}={\begin{pmatrix}\sinh \theta \\\cosh \theta \end{pmatrix}}}$

${\displaystyle {\vec {e}}_{u_{0}}^{\,*}={\begin{pmatrix}\cosh \theta \\-\sinh \theta \end{pmatrix}}\,,\ {\vec {e}}_{u_{1}}^{\,*}={\begin{pmatrix}-\sinh \theta \\\cosh \theta \end{pmatrix}}}$

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