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\centerline{\bf Special nd General Relativity (PHZ 4601) Fall 2017 }
\smallskip

\centerline{\bf Classwork and Homework} \bigskip

\begin{enumerate}

\item Take as Schwarzschild radius $r_s$ the approximate extension of the 
visible universe, $r_s = 46\times 10^9\,[ly]$.

\begin{enumerate}

\item Calculate calculate the corresponding density in units of 
      $[kg/m^3]$.

\item How many protons per $[m^3]$ would this be (take the proton weight
      from Wikipedia)? 

\item Compare with the actual estimated density of the universe (take 
      the estimate of the mass of the universe from Wikipedia, but keep
      $r_s$ as used above)? 

\item {\bf Solution:} sden.f.

\end{enumerate}

\item Homework, due F ???/2017 before class. 

\begin{enumerate}

\item Find the Shapiro time delay in the approximation of Rindler 
p.237 for $X_1$ the distance from Earth to Sun, $X_2$ the distance 
from Mercury to Sun and $R$ the radius of the Sun. State the result
in seconds. You may take the parameter values from Wikipedia. 
Calculate to at least three significant digits.

\item Estimate the distance light travels in the time period which
you find.

\item Repeat the above estimates with Mars replaced by Venus.

\item {\bf Solution:} shapiro.f.

\end{enumerate}

\item Homework, due F ???/2017 before class. Assume a circular orbit of 
radius $r = 149.6\times 10^9\,[m]$ for the earth about the sun  and use 
Eq.~(11.32) of Rindler to caculate $\omega$ in the units $[rad/s]$. 
Subsequently, calculate the angle covered in one year in the units
of $[deg]$. Give at least three significant digits and take the mass
of the sun from Google.

\item Schwarzschild effective potential. 

\begin{enumerate}

\item Find the extrema of
the Schwarzschild effective potential 
$$V_{GR}(r)=\frac{h^2}{r^2}-\frac{2m}{r}-\frac{2m\,h^2}{r^3}$$
in the form 
$$r_{1,2}=\frac{p}{2}\pm\sqrt{\left(\frac{p}{2}\right)^2-q}\,$$
where you have to calclate $p$ and $q$ in terms of the central 
mass $m$ and the angular momentum $h$. 

\item Express $h$ in terms of $m$ for the solution 
$$ r_{1,2} = \frac{p}{2} \pm \frac{p}{4}\,. $$

\item Sketch $V_{GR}(r)$ for the parameters above in the range 
$0\le r\le 30m$, $-0.08<V_{GR}<0.02$ and compare with Newton's
effective potential 
$$V_N(r)=\frac{h^2}{r^2}-\frac{2m}{r}$$
in the same range. What happens when you increase the values of 
$h$ in units of $m$ by about 10\%?
\end{enumerate}
\end{enumerate}

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% \begin{figure}[tb] % From EMscript/figures_plt/minkowski_space_ex.
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% \centerline{\hbox{ \psfig{figure=figures/e1mink.eps,width=10cm} }}
% \caption{Minkowski space in which observer~A is at rest and
% flashes a light signal at observer~B, who moves with speed $4c/5$
% and flashes the signal back. \label{fig_exmink1} } \end{figure}
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