% Bernd Berg Nov 22 2017.
\documentclass{ws-book9x6}  

\begin{document}

\centerline{\bf Special and General Relativity (PHZ 4601/5606) Fall 2017 }
\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 9/1/2017 before class. In 2D Euclidean space a
straight line goes in $K$ from 
$\left(\begin{matrix} 1 \cr 1 \end{matrix}\right)$ to
$\left(\begin{matrix} 2 \cr 2 \end{matrix}\right)$. From where to 
where does it go in $K'$ which is rotated by $\phi=\pi/8$? Is it
still a straight line? \bigskip

\item Homework, due F 9/1/2017 before class. In 2D Euclidean space find 
the matrix that takes $\left(\begin{matrix} x'\cr y'\end{matrix}\right)$ 
in $K'$ to $\left(\begin{matrix} x\cr y\end{matrix}\right)$ in $K$ along 
the lines used in the lecture from $K$ to $K'$ (orientation of the frames
the same as there). 

\end{enumerate} \bigskip 

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

\item Homework, due M 9/18/2017 before class. Exercise 1.6 of Rindler.
\bigskip 

\item Homework, due M 9/18/2017 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 Classwork, due W 9/5/2017 in class (late acceptance 9/18/2017). 
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.
\bigskip 

\item Homework, due M 9/18/2017 before class. Exercise 1.9 of Rindler.
By ``source'' a light source is meant.
\bigskip 
\end{enumerate} \bigskip 
\vfil\eject

{\bf Set 3:} \begin{enumerate} \setcounter{enumi}{6}

\item Homework, due F 9/22/2017 before class: Spherical coordinates.
\medskip

(1) Given are the spherical coordinates
$$ \theta = {3\, \pi /4}~[rad],\ \phi =  {4\, \pi \over 3}~[rad]
~~{\rm and}~~ r=5~[ly]\, .$$
Calculate the corresponding (contravariant) Cartesian coordinates
$x^1$, $x^2$ and $x^3$.
\smallskip

\noindent (2) Given are
$$ x^1 = 2~[ly],\ x^2= -1~[ly] ~~{\rm and}~~ x^3 = -3~[ly]\, .$$
Find the corresponding polar coordinates $\theta$, $\phi$ and $r$
(angles in $[rad]$). 
% \noindent Solution: EM97/8 exercises, p.4 (\# 2/97).
\bigskip 

\item Homework, due F 9/22/2017 before class: Euclidean and hyperbolic 
rotations.

Consider the 2D Euclidean rotation
$$ \begin{pmatrix} x'^{\,1}\cr x'^{\,4}\cr \end{pmatrix} = 
  \begin{pmatrix} \cos(\phi)& \sin(\phi)\cr - 
                  \sin(\phi)&\cos(\phi)\cr \end{pmatrix}\,
  \begin{pmatrix} x^1\cr x^4\cr \end{pmatrix}               $$
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 Classwork, due M 9/18/2017 in class: Time and relativistic 
      distance measurements (synchronization of clocks).

A Cesium clock counts 2,757,789,531,312 cycles (starting at 0). Find
the elapsed time in seconds and round to the nearest integer number.

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: 

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

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

$ (3)~1.002,\,2.004,\,3.008\,\dots\, [s]$. \\
Determine for each case whether the data are consistent with assuming 
that $O_2$ is also in 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 $O_1$. Approximate the speed of light by $300,000\,[km/s]$,
but perform all calculations with a precision of at least four digits.

\end{enumerate}

\begin{figure}[tb]
\centerline{\hbox{ \psfig{figure=figures/1mink,width=9cm} }}
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. 
% \end{figure}
% \begin{figure}[tb]
\centerline{\hbox{ \psfig{figure=figures/2mink,width=9cm} }}
Minkowski space in which observer~A is at rest and
observer~B moves with speed $4c/5$. Observer~B flashes a light
signal at observer~A, who flashes it back.
\end{figure}

{\bf Set 4:}

\begin{enumerate} \setcounter{enumi}{9}

\item Homework, due F 9/29/2017 before class. Time in Minkowski space.
\medskip

Use natural units with $c=1$ for this exercise.

In the figures the Minkowski space is parametrized by the coordinates 
of the rest frame of an observer~A. While observer A stays at rest, 
observer~B moves with speed $4c/5$ along the positive $x$ axis. At 
their common origin both, A and B, have set their clocks to zero. After 
15~$[s]$ observer~A emits at position A$_1$ a light signal which reaches 
observer~B at position B$_0$, who flashes it back at observer~A, who 
receives it at position A$_2$  as drawn in the first figure.

\begin{description}
\item{1.} What are the coordinates of the position B$_0$?
\item{2.} What is the time on the clock of observer~B at position $B_0$?
\item{3.} What is the time on the clock of observer~A when the back 
          signal is received at position A$_2$?
\end{description}

In the second figure observer~B flashes a light signal at A from 
position B$_1$ when the time on the clock of B shows 15~$[s]$.
It is received by A at position A$_0$ and reflected back at B,
who receives it at position B$_2$.

\begin{description}
\item{4.} What are the coordinates of the position B$_1$?
\item{5.} What time does the clock of A show at position A$_0$?
\item{6.} What are the coordinates of B$_2$ in the Minkowski space and
          what time does the clock of B show at this position?
\item{7.} Compare the time reported by A at position A$_2$ with the 
          time reported by B at position $B_2$. Without the actual 
          calculation are there reasons why these times should agree 
          or disagree? 
\end{description}
\bigskip

\item Homework, due F 9/29/2017 before class. Exercise 2.9 of Rindler.
\bigskip

\item Homework, due F 9/29/2017 before class. 
      Time dilation for a satellite.  \smallskip \vskip -10pt

The circular orbit of a satellite is described by its radius $R$
and angular velocity $\omega$. 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 (neglect gravitational effects).

Use $d\tau$ for a clock on the satellite and $dt$ for a clock fixed on
the equator of the earth. Calculate the numerical value $1-d\tau/dt$ 
for a satellite that orbits above the equator at an altitude of 160~[km].
Compare with the competing effect from gravity.
\bigskip \vfil\eject

\item Graduate students only: Homework, due F 9/29/2017 before class. 
      Addition theorem for transverse velocity components. 
\smallskip 

The addition theorem of velocities can also be derived as follows:
Assume that IF $K'$ moves with respect to $K$ with velocity $\vec{v}
= v\,\hat{x}^1$ and a particle moves with respect to $K'$ with 
velocity $\vec{u}^{\,'}$:
$$ x'^{\, i} = c^{-1} u'^{\, i} x'{\, ^0} . $$
The Lorentz boost equations 
\begin{eqnarray} \nonumber
  x'^{\,0} =  x^0 \cosh (\zeta ) - x^1 \sinh (\zeta )\,,\\
  \nonumber
  x'^{\,1} = -x^0 \sinh (\zeta ) + x^1 \cosh (\zeta )\,,\\
  x'^{\,i} = x^i,\ (i=2,3)\,. \nonumber
\end{eqnarray}
imply
$$ \gamma\, (x^1-\beta x^0) = 
   c^{-1} u'^{\, 1} \gamma\, (x^0-\beta x^1)\,. $$
Sorting with respect to $x^1$ and $x^0$ gives
$$ \gamma\, \left(1+{u'^{\, 1} v\over c^2}\right)\, x^1 = 
   c^{-1}\gamma\, (u'^1 + v)\, x^0\, . $$
Using the definition of the velocity in $K$, 
$\vec{x}=c^{-1} \vec{u}\, x^0$, gives
\begin{equation} \nonumber
 u^1 = c\, {x^1\over x^0} 
     = {u'^1 + v \over 1 + {u'^1 v/ c^2}}\, . 
\end{equation}
Derive along similar lines the equation for the two other components
$u^i, (i=2,3)$. 
\end{enumerate}
\bigskip

{\bf Set 5:}

\begin{enumerate} \setcounter{enumi}{13}

\item Homework, due F 10/6/2017 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 
$0\le M < 2m$.
\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

\item Homework, due F 10/6/2017 before class. Spacetrip and elapsed 
time on Earth.  \medskip

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? (This problem is adapted 
from Jackson, Electrodynamics.)

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 10/6/2017 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 Classwork, due M 10/2/2017 in class, 5 points. 

An astrophysical observation.  \smallskip

For light from some galaxy the spectrum 
$$ \lambda=(729.2\,[nm])\, m^2/(m^2-4)\,,~~~m=3,\,4,\,5,\,\dots $$ 
is observed. Find the speed at which the galaxy moves away or towards 
us (ignore the possibility of transverse motion and the expansion of 
space). Note: In quantum mechanics books you find for the Balmer 
spectrum of the hydrogen atom $\lambda=(364.56\,[nm]) \,m^2/(m^2-4)$.
\bigskip \vfil\eject

\end{enumerate} 
\bigskip \vfil\eject

{\bf Set 6:}

\begin{enumerate} \setcounter{enumi}{17}

\item Homework, due F 10/13/2017 before class. Relative velocity.
\smallskip

An observer in an inertial frame $S'$ measures the velocities of two 
particles and finds
$$ \vec{\beta'}_1=\left(\begin{matrix}\beta'^1_1\cr \beta'^2_1
\end{matrix} \right)=\left(\begin{matrix}1/2\cr 0\end{matrix}\right)$$ 
along the $x$ direction for the first particle and 
$$\vec{\beta'}_2=\left(\begin{matrix}\beta'^1_2\cr \beta'^2_2\end{matrix}
  \right)=\left(\begin{matrix}0\cr 1/2\end{matrix}\right)$$ 
along the y direction for the second particle.
\smallskip

Find the speed of the second particle in the rest frame of the first 
particle. You may use the formulas of problem~13.
\bigskip % \vfil\eject


\item Homework, due F 10/13/2017 before class. 

The four-dimensional Levi-Civita tensor $\epsilon^{\alpha\beta
\gamma\delta}$ is completely antisymmetric in its indices and let 
$\epsilon^{1234}=1$. 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

\item Graduate students only: Homework, due F 10/13/2017 before class.
\smallskip

The dual tensor is defined by 
$$ ^*E^{\alpha\beta}=\frac{1}{2}\,\epsilon^{\alpha\beta\gamma\delta}
     E_{\gamma\delta}\,.$$
Use Eq.~(7.32) of the book to express the dual tensor
in terms of $\vec{e}$ and $\vec{b}$. Subsequently, compare
$\left.^*E^{\alpha\beta}\right._{,\alpha}=0$ with the 
homogeneous Maxwell equations, Eq.~(7.26) and~(7.27) of
the book.
\bigskip

\item Classwork, due  10/6/2016 in class. Electromagnetic field tensor.
\smallskip

Let $E^{\alpha\beta}$ be an antisymmetric tensor of rank 2. Compare 
$$ \frac{\partial E^{\alpha\beta}}{\partial\,x^{\alpha}} =
   \partial_{\alpha}E^{\alpha\beta}=E^{\alpha\beta}_{~~,\alpha} =
   \frac{4\pi}{c}\,J^{\beta}$$
for $\beta=4$ with the inhomogeneous Maxwell equations 
$$ \nabla\cdot\vec{e} = 4\pi \rho = \frac{4\pi}{c}\,J^4
   ,~~~\nabla \times \vec{b} - {1\over c}{\partial\vec{e}
   \over\partial t}={4\pi\over c}{\vec J} $$
to 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 $\beta=3$ too 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

{\bf\large Prepare for the Midterm on F October 20!}

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

{\bf Set 7:}

\begin{enumerate} \setcounter{enumi}{21}

\item Classwork, due  10/16/2016. 
Twin travel in 2D Minkowski space.  

\begin{figure}[th] % From EMscript/figures_plt/minkowski_space_ex.
\centerline{\hbox{ \psfig{figure=figures/e3mink,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}

The figure above 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 $K$ of A with constant velocity $\beta_1$ away 
from A, turns at position $\begin{pmatrix}75\cr 60\end{pmatrix}$ and 
travels back with velocity $-\beta_1$ to meet A again.

\begin{enumerate}

\item What is the value of $\beta_1$?

\item Consider the inertial frame $K'$ in which B is for the first part 
of its travel at rest, which we call ``initial rest frame (IRF)''. The 
IRF is arranged so that its origin agrees at time zero with the origin 
of the rest frame of~A.  At what time leaves $B$ the $x=0$ axis of the
IRF frame?

\item Find the coordinates of the last position of B (where B meets A
again) in the IRF and sketch the entire travel in this frame.

\item Calculate the velocity for the second part of the travel of 
$B$ in the IRF $K'$. 

\end{enumerate} \medskip

\item Homework, due F 10/27/2017 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. 
You may want to use   

~~~~~http://www.wolframalpha.com/calculators/integral-calculator/

for the integration.
\bigskip

\item Homework, due F 10/27/2017 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

\item Graduate students only: Homework, due F 10/27/2017 before class.
\smallskip

Derive the Euler-Lagrange equations from the action principle
$$ \delta \int_{t_1}^{t_2} L(q_i,\dot{q}_i)\,dt = 0\,,~~~(i=1,\dots,N)\,.$$
Prepare to present you solution in class.

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

{\bf Set 8:}

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

\item Homework, due F 11/3/2017 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 for this system.
\item Using $$
\dot{x}^\lambda+\Gamma^\lambda_{\mu\nu}\dot{x}^\mu\dot{x}^\nu=0 $$
derive the two coupled, second-order differential equations
that describe the motion of a free particle in this system.
\end{enumerate}
\medskip

\item Homework, due F 11/3/2017 before class. Verify that the 
differential equations of the previous problem are fulfilled by 
$$ r=\sqrt{y_0^2+s^2}~~{\rm and}~~\tan(\theta)=\frac{y_0}{s}\,,
   ~~{\rm where}~~\cdot = \frac{d~}{ds} $$
and $y_0$ is a constant. Describe the motion in the $x$-$y$ plane.
\bigskip

\item Graduate students only: Homework, due F 11/3/2017 before class. 
In the following units with $c=1$ are used. Solve the spaceship 
problem~(15) by transforming $(d\beta)_S=g\,d\tau$ to $(\beta)_E$,
where $E$ is the earth frame, considered to be inertial, and $S$ 
is the instantaneous rest frame of the spaceship at velocity 
$\beta(\tau)$ seen from earth. You may use Mathematica for
the integral over $(d\beta)_E$.
\end{enumerate} 
\bigskip % \vfil\eject

{\bf Set 9:}

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

% After Rindler 10.2.
\item Homework, due W 11/8/2017 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 
$$ L = \dot{t}^{\,2}-g_{ij}\,\dot{x}^i\,\dot{x}^j $$
allow for the proposed solutions.
\bigskip % \vfil\eject

\item Homework, due W 11/8/2017 before class. Use the appendix of 
Rindler, p.419, 420, to show Eq.~(11.4) for the Ricci tensor element 
$R_{tt}$.
\end{enumerate} \bigskip 

{\bf Set 10: Three problems.}

\begin{enumerate} \setcounter{enumi}{30}

\item Homework, due F 11/17/2017 before class. Gravitational Units.
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 F 11/17/2017 before class. 

Solar system:
Schwarzschild radius, ruler distance and radar distance.
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 F 11/17/2017 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 11:}

\begin{enumerate} \setcounter{enumi}{33}

\item Homework, due W 11/29/2017 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 Estimate the distance light travels in the time period which
you find.

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

\end{enumerate}

\item Homework, due W 11/29/2017 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 11/27/2017, graduate students only: Prepare 
to present motion in a central field with potential $U(r)=\alpha/r$
to the class. In particular, derive 
$$ t_2-t_1 = \sqrt{\frac{m}{2}}\,\int_{r_1}^{r_2} dr\,\left(
   E+\frac{\alpha}{r}-\frac{L^2}{2\,m\,r^2}\right)^{-1/2} $$
from the Lagrangian 
$$ {\cal L} = \frac{1}{2}\,m\,\vec{r}\,^2 + \frac{\alpha}{r}\,,
   ~~\alpha >0\,. $$
Define and use the effective potential to  explain the physically 
allowed $r(t)$ functions for different values of the enery $E$
\end{enumerate} \bigskip 


{\bf Set 12:}

\begin{enumerate} \setcounter{enumi}{36}

\item Homework, due M 12/4/2017 before class. Radial coordinate
velocity of light in Schwarzschild spacetime.  Exercise 11.2 of 
Rindler.  \bigskip 

\item Homework, due M 12/4/2017 before class. Retarded potential 
and Lorenz gauge: Show that current conservation $\partial_{\mu} 
J^{\mu}(\{x^{\nu}\})=0$ implies that the retarded solution for the 
potential [Eq.~(7.50) of the book]
$$ \Phi^{\mu}(\{x^{\nu}\}) = \frac{1}{c} \int d^3y\,\frac{J^{\mu}
   (x^0-|\vec{x}-\vec{y}|,\vec{y})}{|\vec{x}-\vec{y}|}\,, $$
where $y^0=x^0-|\vec{x}-\vec{y}|$ is the retarded time, fulfills 
the Lorenz gauge $\partial_{\mu}\Phi^{\mu} (\{x^{\nu}\})=0$.
\bigskip 

\item Homework, due M 12/4/2017 before class. Lorenz gauge with 
covariant derivatives. Exercise 14.1 of Rindler.
\end{enumerate} \bigskip 

{\bf Set 13: Prepare for the Final Thursday, December 14, 3-5 pm.}

\vfil \end{document}
