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\begin{document}

\centerline{\bf Special and General Relativity (PHZ 4601/560 
Fall 2017) Final December 14, 3-5 pm.} % \bigskip \vskip -20pt

% \begin{figure}[tb]
% \centerline{\hbox{\psfig{figure=figures/1mink,width=10cm}}}
% Minkowski space.  \end{figure}

\begin{enumerate}

\item{\bf Schwarzschild effective potential (40\%).}

\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 Choose $q$ so that $(p/2)^2-q=(p/4)^2$ holds and express 
$h$ in terms of $m$ for the effective potential resulting from 
the solution 
$$ r_{1,2} = \frac{p}{2} \pm \frac{p}{4} $$
for its extrema.

\item Sketch $V_{GR}(r)$ for the parameters of (b) 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 value of $h$ in units of 
% mass $m$ by, say, about 10\%?
\end{enumerate}


\item{\bf Schwarzschild radius and density (30\%).} 

Consider a sphere of radius $R$ and uniform mass density $\rho$.

\begin{enumerate}

\item Find the radius $R_0$ so that $r_S>R$ holds for $R>R_0$,
where $r_S$ is the Schwarzschild radius.

\item Calculate $R_0$ for the density $\rho=10^{-27}\,[kg/m^3]$,
which is a rough approximation for the mass density of the universe.
Express the result in light years $[ly]$ an compare with the 
size  of the universe. Use $G=6.7\times 10^{-11}m^3\cdot kg^{-1}
\cdot s^{-2}$, $c=3\times 10^8\,m/s^2$, and $1~year=365\times 24
\times 60\times 60\,s$.

\end{enumerate} 
\rightline{\bf Turn over~~~~~~~~~~}
\vfil\eject


\item{\bf Einstein's equations (20\%).}

\begin{enumerate}

\item Transform the equation
$$G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}\,g_{\mu\nu}\,R=-\kappa\,T_{\mu\nu}$$
into the form
$$R_{\mu\nu}=-\kappa\,\left(T_{\mu\nu}-\frac{1}{2}\,g_{\mu\nu}\,
  T\right)\,.$$

\item Consider the Einstein equation with a cosmological constant,
$$G_{\mu\nu}+g_{\mu\nu}\,\Lambda = -\kappa\,T_{\mu\nu}\,.$$
Show that $R=n\,\Lambda$ holds in vacuum and find $n$.

\end{enumerate} 

\item{\bf Covariant Lorenz gauge (10\%).}

Prove that $\Phi^{\mu}_{~;\rho\mu}=-\Phi^{\alpha}R_{\alpha\rho}$
holds in the Lorenz gauge.

\end{enumerate} 

\end{document}
