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

\centerline{\bf Test on Homework} % \bigskip \vskip -20pt

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\begin{enumerate}

\item Time dilation for a satellite from special relativity (neglect 
      gravitational effects) (20\%).  \smallskip \vskip -20pt

The circular orbit of a satellite is described by its radius $R$
and angular velocity $\omega$. Use special relativity to calculate 
the ratio of the proper time interval $d\tau$ of a clock on the 
satellite over the time interval $dt$ of the inertial frame in which 
the center of the orbit is at rest.

Use $d\tau$ for a clock on the satellite and $dt$ for a clock fixed on
the equator of the earth. Calculate the numerical value $1-d\tau/dt$ 
for a satellite that orbits above the equator at an altitude of 160~[km].
Use $g=9.81\,[m/s^2]$ for the gravitational acceleration and $R=
6378\,[km]$ for the radius of the earth.
% Compare with the competing effect from gravity.
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\item Flat 2D plane mapped by the standard polar coordinates,
	      $x=r\,\cos\theta$, $y=r\,\sin\theta$ (60\%). \vskip -10pt
\begin{enumerate}
\item Write down $ds^2$.
\item What are the coefficients $g_{ij}$?
\item What are the coefficients $g^{ij}$?
\item Calculate the Christoffel symbols $\Gamma^\lambda_{\mu\nu} =
\frac{1}{2}\,g^{\lambda\gamma}\,(g_{\gamma\mu,\nu}+g_{\gamma\nu,\mu}
-g_{\mu\nu,\gamma})$ for this system.
\item Using 
$$\ddot{x}^\lambda+\Gamma^\lambda_{\mu\nu}\dot{x}^\mu\dot{x}^\nu=0$$
derive the two coupled, second-order differential equations
that describe the motion of a free particle in this system.
\end{enumerate}
\medskip

% After Rindler 10.2.
\item Is $t=var$, $x_i$ fixed, geodesic? (20\%).

Consider the spacetime metric $d\vec{s}\,^2=dt^2-dl^2$, where 
$dl^2$ is an arbitrary and possibly time-dependent metric. Are 
coordinate lines with $t$ variable and the other coordinates 
fixed at constant values geodesic? Hint: Investigate whether 
the geodesic equations of the Lagrangian 
$$ L = \dot{t}^{\,2}-g_{ij}\,\dot{x}^i\,\dot{x}^j $$
allow for the proposed solutions.
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\end{enumerate} % \bigskip

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