% Bernd Berg jun 29 2018.
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\centerline{\bf Special and General Relativity (PHZ 4601/5606) Fall 2018 
Solutions} \medskip

\centerline{\bf Set 3} 

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\begin{description} 
\item{\bf 7. Relativistic distance measurements by light signals.}
\end{description}
\bigskip

\noindent For 2,757,789,531,312 counts of the Cesium clock the 
elapsed time is 300~$[s]$.
\smallskip

\noindent
(1) The relation $x=(\triangle t/2)\,c$ gives for all three times
$$ 10^{-3}\,[s]\times 3\times 10^5\,[km/s]\ =\ 300\,[km]\,. $$
Therefore, $O_2$ is at rest with respect to $O_1$.
\smallskip

\noindent
(2) We find the following positions at the following times:
\begin{eqnarray} \nonumber
  x_1=300\,[km]\ &{\rm at}&\ t_1=1.001\,[s]\,, \\ \nonumber
  x_2=600\,[km]\ &{\rm at}&\ t_2=2.002\,[s]\,, \\ \nonumber
  x_3=900\,[km]\ &{\rm at}&\ t_3=3.003\,[s]\,.
\end{eqnarray} 
This gives the velocities
\begin{eqnarray} \nonumber
  v_{21} &=& \frac{x_2-x_1}{t_2-t_1}\ =\ \frac{300\,[km]}
  {(2.002-1.001\,[s]}\ =\ \frac{300\,[km]}{1.001\,[s]}
  \ \approx\ 299.7\,[km/s]\,, \\ \nonumber
  v_{32} &=& \frac{x_3-x_2}{t_3-t_2}\ =\ \frac{300\,[km]}
  {(3.003-2.002)\,[s]}\ =\ \frac{300\,[km]}{1.001\,[s]}
  \ \approx\ 299.7\,[km/s]\,. 
\end{eqnarray} 
So, the results are consistent with the idea that $O_2$ is at rest
in an inertial frame, which moves with about 299.7$\,[km/s]$ with 
respect to the inertial frame of $O_1$. The position of $O_2$
with respect to $O_1$ is then given by
$$ x(t)\ =\ x_0 + v\,t\ =\ \frac{300\,[km]}{1.001\,[s]}\,t\,,$$
where $x_0=0$ followed from $x(t_1)=300\,[km]$ for $t_1=1.001\,[s]$.
\smallskip

\noindent
(3) We find again the positions $x_1=300\,[km]$ at $t_1=1.001\,[s]$
and $x_2=600\,[km]$ at $t_2=2.002\,[s]$, which give $v_{21}=(300/1.001)
\,[km/s]$, but now
\begin{eqnarray} \nonumber
  x_3=1200\,[km]\ &{\rm at}&\ t_3=3.004\,[s]\,,
\end{eqnarray} 
which gives 
\begin{eqnarray} \nonumber
  v_{32} &=& \frac{x_3-x_2}{t_3-t_2}\ =\ \frac{600\,[km]}{1.002\,[s]}
  \ \approx\ 598.2\,[km/s]\,. 
\end{eqnarray} 
As $v_{21}$ and $v_{32}$ disagree, $O_2$ cannot be at rest in an
inertial frame.

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