% Bernd Berg Oct 13 2017.
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\centerline{\bf Special and General Relativity (PHZ 4601/5606) Fall 2017 
Solutions} \medskip

\centerline{\bf Set 12} 

\begin{description} 
\item {\bf 37. Radial coordinate velocity of light in Schwarzschild
spacetime.  Exercise 11.2 of Rindler.}
\end{description}

\noindent As $\theta,\ \phi = const$, 
$$d \vec{s}\,^2=\alpha\,dt^2-\alpha^{-1}\,dr^2
    ~~{\rm with}~~\alpha=1-2m/r\,.$$
Light: $d\vec{s}\,^2=0\ \Rightarrow\ dt=\pm\alpha^{-1}\,dr$. So
the coordinate velocity is
$$ \frac{dr}{dt}=\pm\alpha\ \to 0~~{\rm for} ~~r\to 2m\,. $$
For the time to a mirror at $r_1$ from $r_0>r_1$ and back, we have
\begin{eqnarray} \nonumber
  \triangle t &=& 2 \int_{r_1}^{r_0} (1-2m/r)^{-1} dr =
  2 \int_{r_1}^{r_0} r/(r-2m)\,dr \\ \nonumber
  &=& 2 \int_{r_1}^{r_0} (r-2m+2m)/(r-2m)\,dr 
   =  2 \int_{r_1}^{r_0} \left[1+2m/(r-2m)\right]\,dr\\ \nonumber 
  &=& 2\,\left[r_0-r_1 + 2m\,\ln\left(\frac{r_0-2m}{r_1-2m}\right)
  \right]\ \to\ \infty~~{\rm for}~~ r_1\to 2m\,.
\end{eqnarray}
As we watch a source fall towards the horizon, it appears to slow down 
and to stop at the horizon. By Eq.~(11.25) of Rindler the redshift of 
the light received by us approaches infinity. So the signal becomes ever 
dimmer, since both the frequency of photons reaching us becomes smaller,
and the rate at which they arrive also decreases, each by the Doppler
factor.

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