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%----------------------------------------------------%
%%%%%%%%%%%%%%%%% CLASSICAL MECHANICS %%%%%%%%%%%%%%%%
%----------------------------------------------------%
\begin{center}
{\bf ADVANCED DYNAMICS --- PHY--4241/5227}\\\vspace {2mm} 
{\bf HOMEWORK 9}\\%
\vspace {3mm} (March 1, 2004) \\
\vspace {1mm}{Due Monday, March 14, 2004 (after springbreak, 
late afternoon)} \\ 
\end{center}
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\bigskip

%%%%%%%%%%%%%%%%%%%%%%%%%%
\centerline{\bf PROBLEM 20 (continuation of 18)}
\medskip

How far away from earth did the spaceship travel? Express the result in
light years.

\begin{enumerate}
\item After which time (on its own clock) did the spaceship reach 90\% 
      of the speed of light with respect to earth?

\item When does a message sent from earth in 2025 (precisely five years 
after take off) reach the spaceship occupants? Answer this question 

  \begin{enumerate}
  \item with respect to the time on earth and 
  \item with respect to the clock on the spaceship.          
  \end{enumerate}
\end{enumerate}

\bigskip
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%
\centerline{\bf PROBLEM 21 (continuation of 20)}
\medskip

Assume that the spaceship moves by exhausting particles at the 
speed $v=0.66c$.
\begin{enumerate}
\item Derive an expression for $m(\tau )$, the (remaining) mass
      of the spaceship at proper time $\tau$. 
\item Which fraction of the original mass is left, after the 
      spacetrip has been performed? 
\item Repeat the consideration of (1) and (2) under the assumption that, 
      instead of exhausting particles, the spaceship moves now by
      radiating mass into light.
\end{enumerate}
\smallskip


\bigskip
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%
\centerline{\bf PROBLEM 22}
\medskip

An electron and a positron, each with mass equal to $0.511~MeV$,
annihilate at rest into two photons. Chose the rest frame of the
positronium, such that one of the photons moves in positive
$x=x^1$ direction.
\begin{enumerate}
\item For each photon find the momentum 4-vector.

\item Find the 4-momentum in a frame that moves at a velocity 
       $c\beta\hat{x}$ with respect to the rest frame of the
       positronium.

\item Suppose that the annihilation took place $10^9$ years ago in a 
      galaxy that is receding from us at $\beta = {4\over 5}$. 
      What is the energy of the photon that we observe?
\end{enumerate}
\smallskip
\medskip
\centerline{\bf Turn Over}

\vfill\eject
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\medskip
%%%%%%%%%%%%%%%%%%%%%%%%%%
\centerline{\bf PROBLEM 23}
\medskip

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 $-g\, L$.
\item Write down the transpose matrix $\tilde{L}{\dagger}$.
\item Calculate $\tilde{L}\, g$.
\item Compare (a) and (c) to find the general form
      of $L$ ({\it i.e.} use $\tilde{L}\, g = -g\, L$).
\item Obtain the same result by discussing the elements of the
     equation $g^{\alpha\beta} \tilde{l}_{\beta}^{\ \gamma} 
      g_{\gamma\delta} = - l^{\alpha}_{\ \delta}$. (Hint:
      do the contractions first.)
\end{enumerate}
\smallskip


\end{document}
