% 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 coordinate 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. The times observed with standard clocks are 
$$ \triangle\tau_0 = \sqrt{1-\frac{2m}{r_0}}\,\triangle t~~
   \to~\infty~{\rm for}~~r_1\to 2m $$
and
$$ \triangle\tau_1 = \sqrt{1-\frac{2m}{r_1}}\,\triangle t~~
   \to~0~{\rm for}~~r_1\to 2m\,. $$
By Eq.~(11.25) of Rindler the redshift of the light received by us 
approaches infinity:
$$ \frac{\lambda_0}{\lambda_1} = \left(\frac{1-2m/r_0}{1-2m/r_1}
\right)~\to~\infty~~{\rm for}~~r_1\to 2m\,.$$
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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