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\begin{center}
{\bf ADVANCED DYNAMICS --- PHY 4241/5227}\\\vspace {2mm} 
{\bf HOME AND CLASS WORK -- SET 10}\\%
\vspace {3mm} (April 11, 2011) \\
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\begin{description}

\item{(31)} The matrix $L$ is defined by
$$ L\ =\ \left( \matrix{ 
  l^0_{\ 0} & l^0_{\ 1} & l^0_{\ 2} & l^0_{\ 3} \cr
  l^1_{\ 0} & l^1_{\ 1} & l^1_{\ 2} & l^1_{\ 3} \cr
  l^2_{\ 0} & l^2_{\ 1} & l^2_{\ 2} & l^2_{\ 3} \cr
  l^3_{\ 0} & l^3_{\ 1} & l^3_{\ 2} & l^3_{\ 3} \cr} \right) $$
\begin{enumerate}
\item Calculate $-g\, L$ (2 points).
\item Write down the transpose matrix $\tilde{L}$ (2 points).
\item Calculate $\tilde{L}\,g$ (2 points).
\item Compare 1.\ and 3.\ to find the general form of $L$ 
      ({\it i.e.} after using $\tilde{L}\, g = -g\, L$,
      4 points).
\item Obtain the same result by discussing the elements of the
      equation $g^{\alpha\beta} \tilde{l}_{\beta}^{\ \gamma} 
      g_{\gamma\delta} = - l^{\alpha}_{\ \delta}$ (4 extra 
      points. Hint: Do the contractions first.)
\end{enumerate}
Due Wednesday, April 13, before class. 
\medskip

\item{(32)} In the following $K_1$ and $S_3$ are generators as 
defined in class.
\begin{enumerate}
\item Calculate $\exp(-\zeta\,K_1)$ and explain its physical
meaning (3 points).
\item Calculate $\exp(-\phi\,S_3)$ and explain its physical
meaning (3 points).
\end{enumerate}
Due in class. 
\medskip

\item{(33)} Consider the spaceship journey again and plot versus the 
proper time $\tau$ the following quantities (NOT required solution
posted):
\begin{enumerate}
\item The time on earth at which the spaceship, seen from earth,
      is at its position at time $\tau$.
\item The distance from earth as seen from earth.
\item The date of news received from earth, which is transmitted
      at the speed of light.

\end{enumerate}
\end{description}
\end{document}

{\bf New items with no solutions yet:}

\item The distance of earth as seen at proper time $\tau$ from the 
      instantaneous rest rest frame of the spaceship.

\item The time on earth as seen at proper time $\tau$ from the 
      instantaneous rest rest frame of the spaceship.

\item The apparent velocity from the distance in the instantaneous
      rest frame and the proper time.
\end{enumerate}

\item{(NEW1)} Plot the $K'$ frame $(x'^{\,0},\,x'^{\,1}$ with origin at
             $x^0_0,\,x^1_0$ into the $K$ frame $x^0,\,x^1$.

\item{(NEW2)} Discuss travel at constant speeds in 2D. 1. After first
              travel period: Junp from $K'$
              onto $K$ and compare with the jump from $K$ into $K'$.
              Then with the jumps into the $K''$ frame that goes back
              or catches up. The distances are always the same. As 
              they are spacelike, equal times for them differ largely.
              2. After back-travel period: Final 
              comparison of clocks. 
\end{description}
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