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\begin{document}

\centerline{\bf Special and General Relativity (PHZ 4601/560 Fall 2017)}

\centerline{\bf Solutions Test on Homework November 20.} \bigskip 

% \begin{figure}[th] % From EMscript/figures_plt/minkowski_space_sol.
% \centerline{\hbox{ \psfig{figure=figures/1msol,width=12cm} }}
% \caption{Minkowski space in which observer~B is at rest and
% \label{fig_mid1mink} } \end{figure}

\begin{enumerate}

\item{Proper time under rotation.}

$$ c^2\,d\tau^2 = c^2\,dt^2 - \omega^2 R^2\,dt^2 \Rightarrow
   \frac{d\tau}{dt} = \sqrt{1-\frac{\omega^2 R^2}{c^2}}
   \approx 1 - \frac{1}{2}\,\frac{\omega^2\,R^2}{c^2}\ .$$
Now $g=v^2/R=\omega^2\,R$, where $g=9.81\,[m/s^2]$ and $v$ is the 
velocity of the satellite.
For $R=(6,378+160)\,[km]$ (equatorial radius plus $160\,[km]$) we get 
$\omega=0.0012249\,[s^{-1}]$, which correpsonds to a period of $T=2\pi
/\omega=85.5\,[minutes]$. For the time dilation we find
$$ 1-\frac{d\tau}{dt} = \frac{1}{2}\,\left(\frac{\omega\,R}{c}\right)^2 
   = 0.356321\times 10^{-9}\,. $$
Compared to this the time dilation of a clock on the equator is 
negligible:
$$ 1-\frac{d\tau}{dt} = \frac{1}{2}\, \left(
   \frac{\omega\,R}{c}\right)^2 = 0.119517\times 10^{-11}\,.$$
\bigskip


\item{See sol26.pdf of the Homework, Set 8.}
\bigskip

\item{Are $t=var$, $x^i=const$ lines geodesics?}

\noindent
The metric is $d\vec{s}\,^2=dt^2-dl^2$ and $dx^i=0$ holds for $t=var$.
Hence, $ds^2=dt^2 \Rightarrow ds=dt$. So, $\dot{t}=1$, $\ddot{t}=0$ and
$\dot{x}^i=0$, $\ddot{x}^i=0$.

\noindent
Now, let us check the geodesic equations of $L=\dot{t}^2-g_{ij}\,
\dot{x}^i\,\dot{x}^j$:
$$ \frac{d~}{dt}\,\frac{\partial L}{\partial\,\dot{t}} - 
   \frac{\partial L}{\partial\,t} = 0~~\Rightarrow~~ 
   2\,\ddot{t} - g_{ij,4}\,\dot{x}^i\,\dot{x}^j = 0 $$
which is solved by $\ddot{t}=0$, $\dot{x}^i=0$. Next,
$$ \frac{d~}{dt}\,\frac{\partial L}{\partial\,\dot{x}^i} - 
   \frac{\partial L}{\partial\,x^i} = 0~~\Rightarrow~~ 
   -2\,\frac{d~}{dt}\,\left(g_{ij}\,\dot{x}^j\right) - 
    g_{jk,i}\,\dot{x}^j\,\dot{x}^k = 0\,.$$
The geodesic equations are satisfied.

\end{enumerate} 
\end{document}
