% Bernd Berg oct 16 2017.
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\centerline{\bf Special and General Relativity (PHZ 4601/5606) Fall 2017 
Solutions} \medskip

\centerline{\bf Set 6} 

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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 18. Raising and lowering; contractions.}
\end{description}
\smallskip

(a) Circle the correct equations in the following.
\begin{eqnarray} \nonumber
  T^{11} &=& + T_{11}\,,\\ \nonumber
  T^{12} &=& + T_{12}\,,\\ \nonumber
  T^{13} &=& + T_{13}\,,\\ \nonumber
  T^{14} &=& - T_{14}\,,\\ \nonumber
  T^{41} &=& - T_{41}\,,\\ \nonumber
  T^{42} &=& - T_{42}\,,\\ \nonumber
  T^{43} &=& - T_{43}\,,\\ \nonumber
  T^{44} &=& + T_{44}
\end{eqnarray}  % \smallskip

(b) $T^{\alpha\beta}\,T_{\alpha\beta}\ =$
\begin{eqnarray} \nonumber
  &+&T^{11}\,T_{11}+T^{12}\,T_{12}+T^{13}\,T_{13}+T^{14}\,T_{14}
  \\ \nonumber
  &+&T^{21}\,T_{21}+T^{22}\,T_{22}+T^{23}\,T_{23}+T^{24}\,T_{24}
  \\ \nonumber
  &+&T^{31}\,T_{31}+T^{32}\,T_{32}+T^{33}\,T_{33}+T^{34}\,T_{34}
  \\ \nonumber
  &+&T^{41}\,T_{41}+T^{42}\,T_{42}+T^{43}\,T_{43}+T^{44}\,T_{44}\,.
\end{eqnarray}  % \smallskip
\bigskip % \vfil\eject

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