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\centerline{\bf Special and General Relativity PHZ 4601}
\centerline{\bf Test on Homework November 30, 2018.} 
% \bigskip \vskip -20pt

\begin{enumerate}

\item{\bf Twin travel in 2D Minkowski space (36\%).}

\begin{figure}[th] % See subfolder figs08plt.
\centerline{\hbox{ \psfig{figure=18-1mink,width=10cm} }}
Figure: Minkowski space in which observer B travels away from observer 
A to the space-time point $B_0$ and back to A. Natural units with $c=1$ 
and arbitrary time units are used.
\end{figure}

We use dimensionless units with $c=1$ in this problem. The figure 
for this problem is drawn in the rest frame of observer~A. The initial 
position of observers A as well as B is $\left(\matrix{t\cr x}\right) 
= \left(\matrix{ 0\cr 0}\right)$. Observer B travels in the rest frame 
$K_1$ of A with constant velocity $\beta_B^1$ away from A, turns at 
position $B_0$ given by $\left(\matrix{5\cr 4}\right)$ and travels back 
with velocity $-\beta_B^1$ to meet A again. In the following state all 
results as integers or fractions of integers.
\bigskip

(a) What is the value of $\beta_B^1$?

(b) What is the elapsed time on the clock of $B$ at $B_0$?

(c) What is the elapsed time on the clock of $A$ when $A$ 
meets $B$ again?

(d) What is the elapsed time on the clock of $B$ when $A$ 
meets $B$ again?
\medskip

Consider the inertial frame $K_2$ in which B is at rest for the first 
part of its travel, i.e., from the origin to $B_0$. Arrange $K_2$ so 
that its origin agrees at time zero with the origin of $K_1$.
\medskip

(e) Find the velocity $\beta_A^2$ of $A$ in $K_2$

(f) Write down the coordinates of $B_0$ in $K_2$.
\bigskip


\item{\bf Radar distance (28\%).}
\smallskip

The radar distance is defined as $(c/2)\,\triangle\tau$ with 
$\triangle\tau$ being the proper time elapsed on a standard clock 
between emission and reception of a radio echo.\smallskip

Observers at two fixed points $A$ and $B$ in a stationary gravitational
field, respectively at $\Phi_A$ and $\Phi_B$, determine the radar 
distance between them by use of standard clocks. Let $L_A$ and $L_B$ 
be the determinations made at $A$ and $B$ respectively. Find 
$L_A/L_B$.
\bigskip


\item{\bf Schwarzschild metric (36\%).}.

Note that this problem is for $R_{tt}$ as in HW~32 and not for
$R_{rr}$ as in CW~31.

For the diagonal metric
\begin{eqnarray} \nonumber
  d\vec{s}^{\,2} = a\,(dx^1)^2+b\,(dx^2)^2+c\,(dx^3)^2+d\,(dx^4)^2\,,
\end{eqnarray}
where $a,\,b,\,c,\,d$ are functions of the coordinates, the $R_{11}$
compoment of the Ricci tensor is given by
\begin{eqnarray} \nonumber
  R_{11}\ =\ &+& \beta\,a_{22} + \gamma\,a_{33} + \delta\,a_{44}
  \\ \nonumber
             &+& \beta\,b_{11} + \gamma\,c_{11} + \delta\,d_{11}
  \\ \nonumber
             &-& \beta^2\,b_1^2 - \gamma^2\,c_1^2 - \delta^2\,d_1^2
  \\ \nonumber - \alpha\,a_1\ (
             &+& \beta\,b_1 + \gamma\,c_1 + \delta\,d_1~~)
  \\ \nonumber - \beta\,a_2\ (~~\alpha\,a_2
             &+& \beta\,b_2 - \gamma\,c_2 - \delta\,d_2~~)
  \\ \nonumber - \gamma\,a_3\ (~~\alpha\,a_3
             &-& \beta\,b_3 + \gamma\,c_3 - \delta\,d_3~~)
  \\ \nonumber
               - \delta\,a_4\ (~~\alpha\,a_4
             &-& \beta\,b_4 - \gamma\,c_4 + \delta\,d_4~~)
\end{eqnarray}
where we use the notation 
\begin{eqnarray} \nonumber
  \alpha = \frac{1}{2a}\,,~~~\beta=\frac{1}{2b}\,,~~~
  \gamma = \frac{1}{2c}\,,~~~\delta=\frac{1}{2d}\,,
\end{eqnarray}
and for $i,j=1,2,3,4$:
\begin{eqnarray} \nonumber
  a_{i} = \frac{\partial a}{\partial x^i}\,,~~~~a_{ij} 
  = \frac{\partial^2 a}{\partial x^i\,\partial x^j}\,,
  ~~{\rm and\ so\ on.}
\end{eqnarray}
Consider a stationary, static, spherically symmetric metric of the form
$$ d\vec{s}^{\,2} = e^{A(r)}\,dt^2 - e^{B(r)}\,dr^2 - r^2\,
   (d\theta^2+\sin^2\theta\,d\phi^2)\,, $$
where we use units with speed of light $=1$.
\begin{enumerate}
\item Let $x^1=t$, $x^2=r$, $x^3=\theta$, $x^4=\phi$ and identify the 
non-zero contributions to $R_{11}=R_{tt}$.
\item Use the result of (a) to proof equation (11.4) of the book 
$$ R_{tt}=-e^{A-B}\,\left(\frac{A_{22}}{2} - \frac{A_2\,B_2}{4} + 
   \frac{(A_2)^2}{4} +\frac{A_2}{r}\right)\,.$$
\end{enumerate}

\end{enumerate}
\end{document}
