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{\bf ADVANCED DYNAMICS --- PHY 4241/5227}\\\vspace {2mm} 
{\bf HOME AND CLASS WORK -- SET 9}\\%
\vspace {3mm} (March 29, 2010) \\
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\begin{description}

\item{(26)} Exercises with the rapidity variable.
\begin{enumerate}
\item Show $1/\sqrt{1-\beta^2} = \cosh(\zeta)$. Due in class (2~points). 
\item Show $\cosh(\zeta_2)\,\cosh(\zeta_1)
      +\sinh(\zeta_2)\,\sinh(\zeta_1)=\cosh(\zeta_1+\zeta_2)$. Due in 
      class (2~points). 
\item Velocities are given by $\beta_1=\tanh(\zeta_1)$ and $\beta_2=
      \tanh(\zeta_2)$. Use the addition theorem of rapidities to 
      express the velocity given by $\beta=\tanh(\zeta_1+\zeta_2)$
      in terms of $\beta_1$ and $\beta_2$. Due in class (4~points). 
\end{enumerate}

\item{(27)} D mesons have a (mean) lifetime of $1051\times 10^{-15}\,$s. 
Assume such a meson is created in a particle collission at Fermilab and 
moves at $0.995\,$c with respect to the Lab frame. Compute the (mean) 
distance travelled by such a D meson.
Due April~4 before class (6~points). 

\item{(28)} Assume that a rocket ship leaves the earth in the year 2020. 
The rocket ship is so constructed so that it has an acceleration $g$ in 
its own frame (to make the occupants feel comfortable). It accelerates 
on a straight-line path for 5 years (by its own clocks), decelerates at 
the same rate for 5 more years, turns around, accelerates for 5 years, 
decelerates for 5 years, and lands on earth. What year is on earth? 
% (This problem is 11.4a, p.564 from~\cite{Jackson}.)
Use $g=9.81~[m/s^2]$, one year = $365\times 24\times 3600~[s]$, and 
the speed of light $c$, and assume that earth defines an inertial
system. Due April~4 before class (10~points). 

\item{(29)} Exercises with tensors.
\begin{enumerate}

\item Let $S_{\alpha\beta}$ be a (covariant) symmetric tensor. Write 
$S_{\alpha\beta}\,dx^{\alpha}dx^{\beta}$ explicitly as a sum of ten
terms. Due in class (2~points). 

\item Let $F_{\alpha\beta}$ be a (covariant) antisymmetric tensor. 
Calculate $F_{\alpha\beta}\,dx^{\alpha}dx^{\beta}$. 
Due in class (2~points). 

\item Calculate $h^{\alpha}_{~\beta}=g^{\alpha\gamma}
g_{\gamma\beta}$ and $r_{\alpha}^{~\beta}=g_{\alpha\gamma}
g^{\gamma\beta}$. (A)~As $g$-tensor. (B)~The explicit matrix
and identify then with another well-known symbol. Due 
in class (2~points). 

\item Write down explicitly the matrix $(a^{\beta}_{~\alpha})$ and
its transpose $(\tilde{a}_{\beta}^{~\alpha})$,
$\tilde{a}_{\beta}^{~\alpha}=a^{\beta}_{~\alpha}$.
Due in class (2~points). 
\end{enumerate}

\item{(30)} Continuation of (28). Seen from earth, how far away did 
the spaceship travel? Express the result in light years. 
Due April~6 before class (6~points). 
 
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