<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">% Bernd Berg Nov 28 2018.
\documentclass{ws-book9x6}  

\begin{document}

\centerline{\bf Special and General Relativity (PHZ 4601/5606) Fall 2018 }
\smallskip

\centerline{\bf Classwork and Homework} \bigskip

Every exercise counts 10 points unless stated differently.
\bigskip

{\bf Set 1:}

\begin{enumerate}

\item Homework, due F 8/31/2018 before class. Consider the 2D
Euclidean vectors 
$\vec{r}_0=\left(\begin{matrix} x_0\cr y_0 \end{matrix}\right)$ and
$\vec{r}_1=\left(\begin{matrix} x_1\cr y_1 \end{matrix}\right)$. Prove
or disprove algebraically that their scalar product $\vec{r}_0\cdot
\vec{r}_1$ is invariant under a rotation by an angle $\phi$. \smallskip

\item Homework, due F 8/31/2018 before class, 5~points. Use $c=3\times 
10^5\ [km/s]$ and $g=9.81\ [m/s^2]$, and one year $=$ 365 days to 
calculate $g$ in units of light years $[ly]$ and years $[y]$ to three 
significant digits. \smallskip 

\item Homework, due F 8/31/2018 before class, 5~points. Consider 
the triangle of Fig.~8.4 (p.168) of the book for the special case 
$r=a\,\pi/2$ and show that equation (1.9) holds.
\end{enumerate} \bigskip 

{\bf Set 2:} \begin{enumerate} \setcounter{enumi}{3}

\item Homework, due F 9/7/2018 before class. Exercise 1.6 of Rindler.
\smallskip
\end{enumerate} \bigskip 

{\bf Set 3:} \begin{enumerate} \setcounter{enumi}{4}

\item Homework, due F 9/14/2018 before class. Use $c=3\times 10^5\ [km/s]$,
$g=9.8\ [m/s^2]$, $[ft]=0.3[m]$, and one year $=$ 365 days. Estimate the 
time difference encountered due to the gravitational frequency shift 
after running for one year identical atomic clocks positioned in 
Colorado at an altitude of $5400\,[ft]$ and at the Greenwich 
Observatory at an altitude of $80\,[ft]$. State the result in
microsecond and round to the first two digits. Assume that $g$ 
is constant in the range from $80\,[ft]$ to $5400\,[ft]$ and 
give a reason why that is a reasonable approximation.
\bigskip 


\item Homework, due F 9/14/2018 before class. Exercise 1.9 of Rindler.
By ``source'' a light source is meant.
\bigskip 

\item Classwork, due F 9/12/2018 in class: Time and relativistic 
      distance measurements (synchronization of clocks).  \smallskip

A Cesium clock counts 2,757,789,531,312 cycles. Find the elapsed time 
in seconds and round to the nearest integer number.
\smallskip

In the following approximate the speed of light by $300,000\,[km/s]$,
but perform all calculations with a precision of at least four digits.
\smallskip

An observer $O_1$ is located in an inertial system and flashes at
times $1,\,2,\,3\,\dots\,[s]$ light signals towards another observer
$O_2$, who reflects them back with a mirror. Assume that the returned
signals are received by $O_1$ at the following times: 
\smallskip

$ (1)~1.002,\,2.002,\,3.002\,\dots\, [s]$, 
\smallskip

$ (2)~1.002,\,2.004,\,3.006\,\dots\, [s]$, 
\smallskip

$ (3)~1.002,\,2.004,\,3.008\,\dots\, [s]$. 
\smallskip

Determine for each case whether the data are consistent with assuming 
that a frame attached to $O_2$ is also an inertial system. If this is 
the case, write down the equation for the distance of $O_2$ as function 
of the time as seen by $O_1$, $x(t)=x(0)+v\,t$. 

\end{enumerate} \bigskip 

{\bf Set 4:} \begin{enumerate} \setcounter{enumi}{7}

\item Homework, due F 9/21/2017 before class: Euclidean and hyperbolic 
rotations.

Consider the 2D Euclidean rotation
$$ \begin{pmatrix} x'^{\,1}\cr x'^{\,4}\cr \end{pmatrix} = 
  \begin{pmatrix} \cos(\phi)&amp; \sin(\phi)\cr - 
                  \sin(\phi)&amp;\cos(\phi)\cr \end{pmatrix}\,
  \begin{pmatrix} x^1\cr x^4\cr \end{pmatrix}               $$
and substitute $\phi = i\,\zeta$, $x^4=i\,x^0$, $x'^{\,4}=i\,x'^{\,0}$. 
Write out the equations for $x'^{\,1}$ and~$x'^{\,0}$ in terms of 
$\sinh$ and $\cosh$.
\bigskip

\item Homework, due  F 9/21/2018. 
Twin travel in 2D Minkowski space.  

\begin{figure}[th] % See subfolder figs08plt.
\centerline{\hbox{ \psfig{figure=figures/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 $\begin{pmatrix}t\cr 
x \end{pmatrix} = \begin{pmatrix} 0\cr 0\end{pmatrix}$. 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 
$\begin{pmatrix}5\cr 4 \end{pmatrix}$ 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 $K_2$.

(g) What is the velocity $\beta_B^2$ in $K_2$ of $B$ from $B_0$ to its 
final position, where $A$ meets $B$ again.

(h) What are the coordinates in $K_2$ of the position where $A$ 
meets $B$ again?

(i) Draw the travel in $K_2$ similarly as it is drawn for $K_1$
in the figure.
\bigskip

\item Homework, due F 9/21/2017 before class. 
      Time dilation for a satellite.  \smallskip \vskip -10pt

For this problem use $G=6.674\times 10^{-11}\,[m^3\,kg^{-1}\,s^{-2}]$
for Newton's gravitational constant, $m_e=5.972\times 10^{24}\,[kg]$
for the mass of the earth and $r_e=6371\times 10^3\,[m]$ for the radius
of the earth. The earth is approximated by a sphere.
\smallskip

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.
\smallskip

A stationary satellite orbits on a circle about the center of the earth. 
Find the radius of this circle and compare with the radius of the earth
(given above).
\smallskip

Use $d\tau_s$ for a clock on the stationary satellite and $d\tau_e$ for 
a clock fixed on the equator of the earth. Calculate the numerical value 
$1-d\tau_s/d\tau_e$ to order $c^{-2}$.
\smallskip

Calculate the the competing effect from gravity for the same satellite.
\bigskip 

\end{enumerate} \bigskip 


{\bf Set 5:} \begin{enumerate} \setcounter{enumi}{10}

\item Homework, due  F 9/28/2018. 
Twin travel in 2D Minkowski space. Continuation of the previous
problem.

Consider the inertial frame $K_3$ in which B is for the second part of 
its travel, i.e., from $B_0$ to the position where $A$ meets $B$ again,
at rest. Arrange $K_3$ so that its origin agrees at time zero with the 
origin of $K_1$. State again all results as integers or fractions of 
integers.
\medskip 

(a) Find the velocity $\beta_A^3$ of $A$ in $K_3$

(b) What is the velocity $\beta_B^3$ of $B$ in $K_3$ from the 
origin to $B_0$.

(c) Write down the coordinates of $B_0$ in $K_3$.

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

(e) What are the coordinates in $K_3$ of the position where $A$ 
meets $B$ again?

(f) Draw the travel in $K_3$ similarly as it is drawn for $K_1$
in the figure of the previous problem.

(g) Write down the translation that transforms the inertial 
frame $K_3$ into the inertial frame $K_4$ for which the $B_0$
position agrees with the $B_0$ position in inertial frame $K_2$.

(h) Draw the travel in $K_4$.

(i) Patch the figures for $K_2$ and $K_4$ together in the following
way: Take the worldlines of $K_2$ up to the time of $B_0$ in $K_2$.
For times larger than the time of $B_0$ in $K_2$ take the worldlines
of $K_4$. Plot the resulting figure.

(j) If you did everything right, the worldlines of $A$ from $K_2$
and $K_4$ meet at $t=3$, $x=-2.4$ in the last figure. Are these 
identical spacetime points? Find this space-time point from $K_2$, 
then from $K_4$, in the initial frame $K_1$.
\bigskip


\item Homework, due F 9/28/2017 before class. Spacetrip and elapsed 
time on Earth (adapted from Jackson, Electrodynamics).
\smallskip

Assume that the earth is in an inertial frame. A spaceship leaves the 
earth in the year zero. The spaceship is constructed so that it has 
an acceleration $g$ in its own frame (to make the occupants feel 
comfortable). By its own clock, it accelerates on a straight-line 
path for 5 years, decelerates at the same rate for 5 more years, 
turns around, accelerates for 5 years, decelerates for 5 years, and 
lands on earth. What is the time on earth? 
\smallskip

How far did the spaceship travel?
\smallskip

Instructions: Use $g=9.81~[m/s^2]$, one year = $365\times 24\times 
3600~[s]$, and for the speed of light $c=300,000~[km/s]$. Calculate 
three significant digits. Hint: Find first $\zeta(\tau)$, where $\tau$ 
is the proper time in the rocket and $\zeta(\tau)$ its rapidity with 
respect to Earth.
\bigskip


\item Homework, due F 9/29/2017 before class. 
\medskip

Two particles, each of mass $m$ and with negligible kinetic energy,
annihilate into a particle of mass $M$ and a photon. 

\noindent 1.
Use natural units with $c=1$ and calculate the photon energy 
$E_{\gamma}=|p_{\gamma}|$ as function of $M$ in the range 
$0\le M &lt; 2m$.
\smallskip 

\noindent 2.
Compute $E_{\gamma}(M)$ for $M\ =\ 0\,,\ 0.5\,m,\ 1\,m,\ 
\sqrt{2}\,m\,,\ \sqrt{3}\,m$. 
\smallskip 

\noindent 3.
Sketch $f(M)=2m-M$ and $E_{\gamma}(M)$ in one graph together. 
\bigskip

\end{enumerate} %\bigskip 


{\bf Set 6:} \begin{enumerate} \setcounter{enumi}{13}

\item Homework, due F 10/5/2018 before class. End of spacetrip.
\smallskip

Assume that the spaceship of the previous exercise moves by exhausting 
particles (photons) at the speed of light $c$.
\smallskip

\noindent 1. Derive an expression for $m(\tau )$, the (remaining) mass
of the spaceship at proper time $\tau$.  \smallskip

\noindent 2. Which fraction of the original mass is left, after the 
spacetrip has been performed?
\bigskip

\item Homework, due F 10/5/2018 before class. Ccontractions.

Let (in unspecified units)
\begin{eqnarray} \nonumber
&amp;(1)&amp; \ ct=5,\ x^1=1,\ x^2=2,\ x^3=3, \\ \nonumber
&amp;(2)&amp; \ ct=5,\ x_1=1,\ x^2=2,\ x^3=3, \\ \nonumber
&amp;(3)&amp; \ ct=5,\ x^1=1,\ x^2=2,\ x^3=-3,\\ \nonumber
&amp;(4)&amp; \ ct=5,\ x^1=0,\ x^2=3,\ x^3=4, \\ \nonumber
&amp;(5)&amp; \ ct=5,\ x_1=0,\ x_2=3,\ x_3=4, \\ \nonumber
&amp;(6)&amp; \ ct=5,\ x^1=2,\ x^2=3,\ x^3=4, \\ \nonumber
&amp;(7)&amp; \ ct=5,\ x^1=0,\ x^2=3,\ x^3=-4.
\end{eqnarray}
Calculate $x_{\alpha}x^{\alpha}$ for each case. Comment
on $x^{\alpha}\,x_{\alpha}$.
\bigskip 

\item Homework, due F 10/5/2017 before class. Levi-Civita tensor.

The four-dimensional Levi-Civita tensor $\epsilon^{\alpha\beta
\gamma\delta}$ is completely antisymmetric in its indices and 
$\epsilon^{1234}=1$. Show the following properties: \vskip 2pt

\noindent (1) Show $\epsilon_{\alpha\beta\gamma\delta}= - 
\epsilon^{\alpha\beta\gamma\delta}$. 
\smallskip

\noindent (2) Express $\epsilon_{\alpha\beta\gamma_1\delta_1} 
\epsilon^{\alpha\beta\gamma_2\delta_2}$ in terms of Kronecker delta.
\smallskip

\noindent (3) Express $\epsilon_{\alpha\beta_1\gamma_1\delta_1} 
\epsilon^{\alpha\beta_2\gamma_2\delta_2}$\ in terms of Kronecker delta.
\bigskip 
% \vfil\eject

\item Classwork, due  10/3/2018 in class. Electromagnetic field tensor.
\smallskip

Let $E^{\alpha\beta}$ be an antisymmetric tensor of rank 2. Compare 
$$ \frac{\partial E^{\alpha\beta}}{\partial\,x^{\alpha}} =
   E^{\alpha\beta}_{~~,\alpha} = \frac{4\pi}{c}\,J^{\beta} $$
for $\beta=4$ with the inhomogeneous Maxwell equations 
$$ \nabla\cdot\vec{e} = 4\pi \rho = \frac{4\pi}{c}\,J^4
   ,~~~\nabla \times \vec{b} - {1\over c}{\partial\vec{e}
   \over\partial t}={4\pi\over c}{\vec J} $$
and identify $E^{44}$, $E^{14}$, $E^{24}$ and $E^{34}$ in terms of
$\vec{e}$ and $\vec{b}$. Continue with $\beta=1$, then $\beta=2$. 
Do you need also $\beta=3$ to determine all components of the tensor
$E^{\alpha\beta}$ in terms of $\vec{e}$ and $\vec{b}$?
\smallskip

Write down the matrix of the finally obtained electromagnetic field 
tensor $\left(E^{\alpha\beta}\right)$ in terms of $\vec{e}$ and 
$\vec{b}$.
\bigskip % \vfil\eject


\item Classwork, 5 points, due F 10/5/2018 in class. 

Raising, lowering and contractions of indices.
\medskip

Let $T^{\alpha\beta}$ be an arbitrary SR rank~2 tensor 
($T$ for tensor) \medskip. 

(a) Circle the correct equations in the following.
\begin{eqnarray} \nonumber
  T^{11} = + T_{11}~~&amp;{\rm or}&amp;~~T^{11} = - T_{11}\,,\\ \nonumber
  T^{12} = + T_{12}~~&amp;{\rm or}&amp;~~T^{12} = - T_{12}\,,\\ \nonumber
  T^{13} = + T_{13}~~&amp;{\rm or}&amp;~~T^{13} = - T_{13}\,,\\ \nonumber
  T^{14} = + T_{14}~~&amp;{\rm or}&amp;~~T^{14} = - T_{14}\,,\\ \nonumber
  T^{41} = + T_{41}~~&amp;{\rm or}&amp;~~T^{41} = - T_{41}\,,\\ \nonumber
  T^{42} = + T_{42}~~&amp;{\rm or}&amp;~~T^{42} = - T_{42}\,,\\ \nonumber
  T^{43} = + T_{43}~~&amp;{\rm or}&amp;~~T^{43} = - T_{43}\,,\\ \nonumber
  T^{44} = + T_{44}~~&amp;{\rm or}&amp;~~T^{44} = - T_{44}\,,
\end{eqnarray}  \smallskip
(b) Write out the 16 terms of $T^{\alpha\beta}\,T_{\alpha\beta}$.
\bigskip % \vfil\eject

\end{enumerate} 


{\bf Set 7:}

\begin{enumerate} \setcounter{enumi}{18}

\item Homework, due M 10/15/2018 before class.
\smallskip

The dual tensor is defined by 
$$ ^*E^{\alpha\beta}=\frac{1}{2}\,\epsilon^{\alpha\beta\gamma\delta}
     E_{\gamma\delta}\,.$$
Write down the matrix $\left( ^*E^{\alpha\beta}\right)$ and express
it subsequently in terms of $E_{\alpha\beta}$ matrix elements. 

Then, use Eq.~(7.32) of the book to express the dual tensor in terms 
of $\vec{e}$ and $\vec{b}$. Finally, compare $\left.^*E^{\alpha\beta}
\right._{,\alpha}=0$ with the homogeneous Maxwell equations, Eq.~(7.26) 
and~(7.27) of the book.
\end{enumerate} 
\bigskip

{\bf\large Prepare for the Midterm on W October 17!} \bigskip
\vfil\eject

{\bf Set 8:}

\begin{enumerate} \setcounter{enumi}{19}

\item Homework, due M 10/22/2017 before class. Distance in a saddle.
\smallskip

Consider a saddle described by the function
$$ z = f(x,y) = x^2 - y^2 $$
in arbitrary units. A two-dimensional being walks a distance $r=10$ 
from the origin $x=y=0$ of the saddle into its $\hat{x}$ direction. 
At which value of $x$ is it? Give at least three significant digits. 
% For the integration you may want to use   
%~~~~http://www.wolframalpha.com/calculators/integral-calculator/
\bigskip % \vfil\eject

\end{enumerate} %\bigskip %\vfil\eject


{\bf Set 9:}

\begin{enumerate} \setcounter{enumi}{20}

\item Homework, due M 10/29/2018 before class. Radar distance.
\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 Homework, due M 10/29/2017 before class. Rindler 9.8.
\smallskip

Hints: First, complete the square for $dt$. Then, apply the 
transformation $t'=t+f(x,y)$ (called a ``gauge transformation''
in the context of gravity) and find a function $f(x,y)$ so that
the metric becomes static (i.e., has no cross terms $dx\,dt$ and
so on).
\bigskip % \vfil\eject

\item Classwork, due M 10/29/2018 in class. Partial Derivatives.
\smallskip

To limit the scope of this classwork, take only positive roots.
Let $x'=x^2\,y^2$ and $y'=x^2/y^2$.

\begin{enumerate} 
\item Find $x(x',y')$ and $y(x',y')$.
\item Calculate 
$$\frac{\partial~}{\partial x'}\,\frac{\partial x'}{\partial x}~~{\rm 
and}~~\frac{\partial~}{\partial x}\,\frac{\partial x'}{\partial x'}~.$$ 
Express the result in terms of $x$ and $y$.
\item Compare the previous results with 
$$\frac{\partial^2x'}{\partial x\,\partial x}\,\frac{\partial x}
  {\partial x'}+\frac{\partial^2x'}{\partial x\,\partial y}\,
  \frac{\partial y}{\partial x'} $$
in terms of $x$ and $y$. List the result for each of the two 
contributions separately before adding them up.
\item Explain
$$\frac{\partial~}{\partial x}\,\frac{\partial~}{\partial y} =
  \frac{\partial~}{\partial y}\,\frac{\partial~}{\partial x}~~
  {\rm versus}~~
  \frac{\partial~}{\partial x'}\,\frac{\partial~}{\partial x} \ne
  \frac{\partial~}{\partial x}\,\frac{\partial~}{\partial x'}~.$$
\item Consider 2D coordinates systems which are related by  
      $$ x^{1'}=(x^1)^2\,(x^2)^2~~{\rm and}~~x^{2'}=(x^1)^2/(x^2)^2\ .$$
      Calculate $p^{1'}_{1 i}\,p^i_{1'}$ in terms of $x^1$ and $x^2$. 
      The $p$ (partial derivatives) symbols are defined in chapter~7 
      of Rindler, p.132.
\end{enumerate} 
\end{enumerate} 
\bigskip % \vfil\eject

{\bf Set 10:}

\begin{enumerate} \setcounter{enumi}{23}

\item Homework, due F 11/2/2018 before class. Consider a flat 2D 
plane mapped by the standard polar coordinates.
\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 (affine connections) of second 
      kind $\Gamma^i_{jk}$ for this system.
\item Using 
      $$ \ddot{x}^i+\Gamma^i_{jk}\dot{x}^j\dot{x}^k=0 $$
      derive the two coupled, second-order differential equations
      that describe the motion of a free particle in this system.
\end{enumerate}
\bigskip

\item Homework, due F 11/2/2018 before class. Verify that the 
differential equations 
$$ \ddot{r} = r\,\dot{\theta}^2~~{\rm and}~~\ddot{\theta}=
  -\frac{2}{r}\,\dot{r}\,\dot{\theta}\,,
   ~~{\rm where}~~\cdot = \frac{d~}{ds}\,, $$
are fulfilled by 
$$ r=\sqrt{y_0^2+s^2}~~{\rm and}~~\tan(\theta)=\frac{y_0}{s}\,. $$
Here $y_0$ is a constant. Describe the motion in the $x$-$y$ plane.
\medskip

Find at least one more solution of these differential equations.
\end{enumerate} 
\bigskip

{\bf Set 11:} \medskip

\begin{enumerate} \setcounter{enumi}{25}

% Rindler 10.2 slightly modified.
\item Homework, due W 11/14/2018 before class. 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 
$$ L = \dot{t}^{\,2}-g_{ij}\,\dot{x}^i\,\dot{x}^j $$
allows for the proposed solutions.
\bigskip % \vfil\eject

\item Homework, due W 11/14/2018 before class. Exercise 10.6 of Rindler.
\bigskip

\item Homework, due W 11/14/2018 before class. Exercise 14.1 of Rindler.

\end{enumerate}
\bigskip

{\bf Set 12:} \medskip

\begin{enumerate} \setcounter{enumi}{28}

\item Homework, due F 13/14/2018 before class. Exercise 14.2 of Rindler.

\item Homework, due F 13/14/2018 before class. Use Gaussian units 
$(g\,cm\,s)$ and the following values for this problem:

Speed of light: $3\times 10^{10}\,[cm/s]$.

Newton's gravitational constant: 
$G=6.674\times 10^{-8}\,[cm^3/(s^2\,g)]$.

Radius of the visible universe: $r_u=4.4\times 10^{28}\,[cm]$.

Ordinary matter in the visible universe: $10^{56}\,[g]$.

The cosmological constant
$\Lambda$ leads to a modified Poisson equation in the Newtonian 
limit (Rindler, p.304). The observed value (Planck satellite in 
2015) is
$$ \Lambda = 1.11\times 10^{-56}\,[1/cm^2]\,.$$
Compare the absolute values of the implied negative energy density
(sometimes called ``dark energy'') with the energy density due to
ordinary matter. State your answer in percent of the two absolute 
values added up.
\bigskip

\item Classwork, due F 11/16/2018 in class. Schwarzschild metric.

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}\ =\ &amp;+&amp; \beta\,a_{22} + \gamma\,a_{33} + \delta\,a_{44}
  \\ \nonumber
             &amp;+&amp; \beta\,b_{11} + \gamma\,c_{11} + \delta\,d_{11}
  \\ \nonumber
             &amp;-&amp; \beta^2\,b_1^2 - \gamma^2\,c_1^2 - \delta^2\,d_1^2
  \\ \nonumber - \alpha\,a_1\ (
             &amp;+&amp; \beta\,b_1 + \gamma\,c_1 + \delta\,d_1~~)
  \\ \nonumber - \beta\,a_2\ (~~\alpha\,a_2
             &amp;+&amp; \beta\,b_2 - \gamma\,c_2 - \delta\,d_2~~)
  \\ \nonumber - \gamma\,a_3\ (~~\alpha\,a_3
             &amp;-&amp; \beta\,b_3 + \gamma\,c_3 - \delta\,d_3~~)
  \\ \nonumber
               - \delta\,a_4\ (~~\alpha\,a_4
             &amp;-&amp; \beta\,b_4 - \gamma\,c_4 + \delta\,d_4~~)
\end{eqnarray}
where we use the notation (Rindler Appendix)
\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}
We want to construct 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 use units with speed of light $=1$.
\begin{enumerate}
\item Let $r=x^1$, $\theta=x^2$, $\phi=x^3$ and $t=x^4$ and 
identify the non-zero contributions to $R_{11}=R_{rr}$.
\item Use the result of (a) to proof equation (11.3) of the 
book $$R_{rr}=\frac{A_{11}}{2} - \frac{A_1\,B_1}{4} + \frac{(A_1)^2}{4}
-\frac{B_1}{r}\,.$$
\end{enumerate}
\end{enumerate}
\bigskip

{\bf Set 13:}

\begin{enumerate} \setcounter{enumi}{31}
\item Homework, due M 11/19/2018 before class. Use the appendix of 
Rindler, p.419/420, or better the version posted on the Web, to show 
Eq.~(11.4) for the Ricci tensor element $R_{tt}$.
\end{enumerate} \bigskip 

{\bf Set 14:}

\begin{enumerate} \setcounter{enumi}{32}
\item Homework, due W 11/28/2018 before class. Gravitational Units.
\medskip

In the following use $G=6.674\times 10^{-11}\,[m^3\,kg^{-1}\,s^{-2}]$ 
for the gravitational constant, $m_e=5.972\times 10^{24}\,[kg]$ for 
the mass of the earth, which is considered to be a perfect sphere of 
radius $r_e=6371\times 10^3\,[m]$ of uniform mass density. Perform
calculations for this problem to the three leading digits (rounding 
the fourth digit).

\begin{enumerate}

\item Calculate the Scharzschild radius $r^s_e$ of the earth in
units of meter $[m]$ and, subsequently, the ratio $r^s_e/r_e$.

Continue using gravitational units defined by $c=G=1$. 

\item Write down the values for $m_e$ and $r_e$ in units of 
seconds [s]. Subsequently, calculate the ratio $2\,m_e/r_e$.

\item Write down the values for $m_e$ and $r_e$ in units of 
years [y]. Subsequently, calculate the ratio $2\,m_e/r_e$.

\item Write down the values for $m_e$ and $r_e$ in units of 
meters [m]. Subsequently, calculate the ratio $2\,m_e/r_e$.

\item Write down the values for $m_e$ and $r_e$ in units of 
light years [ly]. Subsequently, calculate the ratio $2\,m_e/r_e$.
\end{enumerate} \bigskip 

\item Homework, due W 11/28/2018 before class. Solar system: 

Schwarzschild radius, ruler distance and radar distance.
\medskip

In the following use $G=6.674\times 10^{-11}\,[m^3\,kg^{-1}\,s^{-2}]$ 
for the gravitational constant, $m_s=1.9886\times 10^{30}\,[kg]$
for the mass of the sun, which is considered to be a perfect sphere
of radius $r_s=6957\times 10^5\,[m]$ of uniform mass density. Assume
$r_1=149.6\times 10^9\,[m]$ for the coordinate distance of the earth to
the center of the sun. Neglect the mass of the earth (or assume 
it to be replaced by a satellite).  Perform calculations for this 
problem to the three leading digits (rounding the fourth digit).

\begin{enumerate}

\item Calculate the Scharzschild radius $r^s_s$ of the sun and, 
subsequently, the ratio $r^s_s/r_s$.

\item Calculate the ruler (geodesic) distance from the earth to
the surface of the sun and compare it with the corresponding 
coordinate distance.

\item Calculate the radar distance from the earth to the surface of 
the sun and compare it with the corresponding coordinate and ruler 
distances.
\end{enumerate} \bigskip 

\item Homework, due W 11/28/2018 before class. Isotropic form of
the Schwarzschild metric. Substitute 
$$ r =\left(1+\frac{m}{2\overline{r}}\right)^2\overline{r} $$
into the Scharzschild metric to derive its ``isotropic'' form
$$ d\vec{s}\,^2 = \frac{(1-m/2\overline{r})^2}{(1+m/2\overline{r})^2}
   \,dt^2 - \left(1+\frac{m}{2\overline{r}}\right)^4\,\left(
   d\overline{x}^2+d\overline{y}^2+d\overline{z}^2\right)\,.  $$

\end{enumerate} \bigskip 

{\bf Set 15:}

\begin{enumerate} \setcounter{enumi}{35}

\item Homework, due W 12/5/2018 before class. Shapiro time delay.

\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 Repeat the above estimates with Mercury replaced by Venus.
\end{enumerate}

\item Homework, due W 12/5/2018 before class. Mercury precession.

\begin{enumerate}

\item Estimate the Einsteinian advance of the perihelion of Mercury 
in radians from the equation $\triangle\approx 6\pi\,m^2/h^2$, where 
you may use the approximation $\omega=\sqrt{m/r^3}$ to find the 
specific angular momentum of Mercury from its frequency.

\item Find from the previous result the precession of Mercury over
100 earth years and express the result in arc seconds.

\end{enumerate}

\item Homework, due M 12/5/2018 before class. Radial coordinate
velocity of light in Schwarzschild spacetime.  Exercise 11.2 of 
Rindler.  \bigskip 

\end{enumerate}

\centerline{\bf Prepare for the Final} \medskip

\centerline{\bf Thursday, December 13, 2018, 10 am - 12 pm.}

\vfil \end{document}
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