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\begin{document}

\centerline{\bf Special and General Relativity (PHZ 4601/560)} 
% \bigskip \vskip -20pt

\centerline{\bf Final (two pages) December 13, 10-12 am.} 
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\begin{enumerate}

\item{\bf Special Relativity: Electromagnetic invariants (30\%).}
\smallskip

In the notation of Rindler the components of the electromagnetic
field tensor are given by
$$ \left(E^{\alpha\beta}\right)=\left(\matrix{
    0  & -b^3&  b^2&  e^1\cr
    b^3&  0  & -b^1&  e^2\cr
   -b^2&  b^1&  0  &  e^3\cr
   -e^1& -e^2& -e^3&  0  }\right)\,.$$ \medskip
\begin{enumerate}
\item Calculate $E_{\alpha\beta}\,E^{\alpha\beta}$ as an expression of 
      $\vec{b}$ and $\vec{e}$.
\item Write down the matrix of the dual tensor $$ ^*\!E_{\alpha\beta} 
= \frac{1}{2}\,\epsilon_{\alpha\beta\gamma\delta}\,E^{\gamma\delta} $$ 
in terms of the components of $\vec{b}$ and $\vec{e}$. Here $\epsilon_{
\alpha\beta\gamma\delta}$ is the totally anti-symmetric Levi-Civita
tensor, $\epsilon^{1234}=+1$. 
\item Calculate $^*\!E_{\alpha\beta}\,E^{\alpha\beta}$ as an expression
       of $\vec{b}$ and $\vec{e}$.
\end{enumerate}
\bigskip

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

The Schwarzschild effective potential is given by
$$V_{GR}(r)=\frac{h^2}{r^2}-\frac{2m_s}{r}-\frac{2m_s\,h^2}{r^3}\,,$$
where $h$ is the specific angular momentum (i.e., angular momentum 
per rest mass).

In gravitational units $c=G=1$ the average distance of the earth to the 
sun becomes approximately $10^8$ solar masses $m_s$, $r_e=10^8\,[m_s]$, 
and a second $s$ becomes approximately $s=2\times 10^5\,[m_s]$.

\begin{enumerate}
\item For the orbit of the earth about the sun: Give a rough estimate 
of $h$ in units of the solar mass $m_s$.

\item Continue with $h=10^4\,[m_s]$ and find the extrema of the 
effective Schwarzschild potential. Give the numerical values of 
$V_{\min}=V_{GR}(r_+)$, $V_{\max}=V_{GR}(r_-)$, and~$r_{\pm}$. 

\item Sketch $V_{GR}(r)$ in the neighbourhood of its minimum value 
$V_{\min}=V_{GR}(r_+)$. Label the axis and indicate a few tics 
($\ge 3$) that have to be quantitatively correct.

\item Sketch $V_{GR}(r)$ in the neighbourhood of its maximum 
value $V_{\max}=V_{GR}(r_-)$. Label the axis as in~(c).

\end{enumerate}


\item{\bf Schwarzschild radius and density (10\%).} 

Consider a sphere of radius $R$ and uniform mass density $\rho$.
Find the smallest radius $R_0$ for which you can proof that 
$r_S>R$ holds for $R>R_0$, where $r_S$ is the Schwarzschild 
radius. Proof it by calculating $r_s-R$ explicitely. \bigskip


\item{\bf Einstein's field 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 an equation for $R_{\mu\nu}$ as function of $T_{\mu\nu}$.
% $$R_{\mu\nu}=-\kappa\,\left(T_{\mu\nu}-\frac{1}{2}\,g_{\mu\nu}\,
% T\right)\,.$$

\item Consider the Einstein field equation with a cosmological constant in
4~dimensions (3~space, 1~time):
$$G_{\mu\nu}+g_{\mu\nu}\,\Lambda = -\kappa\,T_{\mu\nu}\,.$$
Show that $R=n\,\Lambda$ holds in vacuum and find $n$.

\end{enumerate} 

\end{enumerate} 

\end{document}

