% Bernd Berg sep 12 2018.
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\centerline{\bf Special and General Relativity (PHZ 4601/5606) Fall 2018 
Solutions} \medskip

\centerline{\bf Set 4} 

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% \end{figure} \centerline{Potential as function of $z$.}

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\item{\bf 8. Euclidean and hyperbolic rotations. }
\end{description}
\smallskip

Transformation of the 2D Euclidean rotation.
$$ \begin{pmatrix} x'^{\,1}\cr i\,x'^{\,0}\cr \end{pmatrix} = 
   \begin{pmatrix} \cosh(\zeta)& i\,\sinh(\zeta)\cr 
   -i\,\sinh(\zeta)&    \cosh(\zeta)\cr \end{pmatrix}\, 
   \begin{pmatrix} x^1\cr i\,x^0\cr \end{pmatrix} $$
and in components
\begin{eqnarray} \nonumber
     x'^{\,1} &=& \cosh(\zeta)\,x^1 - \sinh(\zeta)\,x^0\,,\\ \nonumber
  i\,x'^{\,0} &=& -i\,\sinh(\zeta)\,x^1 + i\,\cosh(\zeta)\,x^0\,,
\end{eqnarray}
or, equivalently,
\begin{eqnarray} \nonumber
  x'^{\,0} &=& +\cosh(\zeta)\,x^0 - \sinh(\zeta)\,x^1\,,\\ \nonumber
  x'^{\,1} &=& -\sinh(\zeta)\,x^0 + \cosh(\zeta)\,x^1\,,\\ \nonumber 
  \\ \nonumber
  \begin{pmatrix} x'^{\,0}\cr x'^{\,1}\cr \end{pmatrix} &=&
  \begin{pmatrix} \cosh(\zeta)& - \sinh(\zeta)\cr
                - \sinh(\zeta)&   \cosh(\zeta)\cr \end{pmatrix}\, 
  \begin{pmatrix} x^0\cr x^1\cr \end{pmatrix} 
\end{eqnarray} 

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