% Bernd Berg jun 29 2018.
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\centerline{\bf Special and General Relativity (PHZ 4601/5606) Fall 2018 
Solutions} \medskip

\centerline{\bf Set 1} 

% \begin{figure}[tb] % \vspace{5pc}
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% \caption{for assignment~14.} \label{fig_lader} \vskip -10pt
% \end{figure} \centerline{Potential as function of $z$.}

\begin{description} 
\item{\bf 1. Invariance of 2D scalar product $\vec{r}_0\cdot\vec{r}_1$ 
under rotation.} 
\end{description}

Consider the 2D Euclidean rotation
$$ \begin{pmatrix} x'^{\,1}\cr x'^{\,2}\cr \end{pmatrix} = 
  \begin{pmatrix} \cos(\phi)& \sin(\phi)\cr - 
                  \sin(\phi)&\cos(\phi)\cr \end{pmatrix}\,
  \begin{pmatrix} x^1\cr x^2\cr \end{pmatrix} = 
  \begin{pmatrix} \cos(\phi)\,x^1+\sin(\phi)\,x^2\cr 
                 -\sin(\phi)\,x^1+\cos(\phi)\,x^2\cr \end{pmatrix}\,.$$
Therefore, we have
\begin{eqnarray} \nonumber
 \vec{r}_0\!'\cdot\vec{r}_1\!' &=&
 \begin{pmatrix} \cos(\phi)\,x^1_0+\sin(\phi)\,x^2_0\cr 
                -\sin(\phi)\,x^1_0+\cos(\phi)\,x^2_0\cr \end{pmatrix}
 \cdot
 \begin{pmatrix} \cos(\phi)\,x^1_1+\sin(\phi)\,x^2_1\cr 
                -\sin(\phi)\,x^1_1+\cos(\phi)\,x^2_1\cr \end{pmatrix}\\
  \nonumber &=& 
      \cos(\phi)^2\,x^1_0\,x^1_1+\cos(\phi)\,\sin(\phi)\,x^1_0\,x^2_1
     +\sin(\phi)\,\cos(\phi)\,x^2_0\,x^1_1+\sin(\phi)^2\,x^2_0\,x^2_1\\
  \nonumber &+&
      \sin(\phi)^2\,x^1_0\,x^1_1-\sin(\phi)\,\cos(\phi)\,x^1_0\,x^2_1
     -\cos(\phi)\,\sin\phi)\,x^2_0\,x^1_1+\cos(\phi)^2\,x^2_0\,x^2_1\\
  \nonumber &=& 
      x^1_0\,x^1_1+x^2_0\,x^2_1\,.
\end{eqnarray}

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