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\begin{document}

\centerline{\bf Special and General Relativity PHZ 4601
Midterm October 17 Fall 2018.} % \bigskip \vskip -20pt

\begin{enumerate}

\item{\bf An astrophysical observation (40\%):}

For light from some galaxy the spectrum 
$$ \lambda=(729.2\,[nm])\, m^2/(m^2-4)\,,~~~m=3,\,4,\,5,\,\dots $$ 
is observed. Find the speed at which the galaxy moves away or towards 
us (ignore the possibility of transverse motion and the expansion of 
space). Note: In quantum mechanics books you find for the Balmer 
spectrum of the hydrogen atom $\lambda=(364.56\,[nm])\,m^2/(m^2-4)$,
and the relativistic Doppler equation reads $\omega'/\omega=\sqrt{
(1-\beta)/(1+\beta)}$.
\bigskip 

\item {\bf Light signals and travel in two inertial frames (40\%):}

\begin{figure}[th]
\centerline{\hbox{ \psfig{figure=figures/1mink,width=10cm} }}
Figure: Minkowski space in which observer~A is at rest and flashes a 
light signal at observer~B, who moves with speed $4/5$ and flashes 
the signal back. 
\end{figure}

In the above figure Minkowski space is parametrized by the coordinates 
of the rest frame $S$ of an observer~A. We use units of seconds $[s]$ 
and $c=1$ for the speed of light.
While observer A stays at rest, observer~B moves with speed $\beta=4/5$ 
along the positive $x$ axis. At their common origin both, A and B, have 
set their clocks to zero. After 15~$[s]$ observer~A emits at position 
A$_1$, i.e.\ at $(15\,[s],0)$, a light signal which reaches observer~B 
at position B$_0$, who flashes it back at observer~A, who receives it 
at position A$_2$ as drawn in the figure. 

\begin{enumerate}

\item Find the coordinates of $A_1$ in the rest frame $S'$ of B,
      where the translational freedom is used so that the origins 
      of the restframes of A and B agree at time $t=t'=0$.

\item Find the coordinates of $B_0$ in the rest frames
      $S$ of A and $S'$ of B.

\item Find the coordinates of $A_2$ in the rest frames
      $S$ of A and $S'$ of B.

\item Transform the above figure into the rest frame $S'$ of B
      (draw the resulting figure).

\end{enumerate}
\bigskip

\item{\bf Space travel (20\%):}

A spaceship travels with uniform speed to a star 8~light-years away, 
in a time the crew considers to be 8~years. Find the speed of the 
spaceship.

\end{enumerate}

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% \begin{figure}[th] % From EMscript/figures_plt/minkowski_space_ex.
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% \centerline{\hbox{ \psfig{figure=e3mink,width=10cm} }}
% \caption{Minkowski \label{fig_exmink3} } \end{figure}

\end{document}
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