Observations of events seen from different reference systems always confuse me: A good example is the free fall of a particle into a -- for simplicity -- non-rotating black hole.
In the reference system of the falling particle the free fall from a distance $r_0$ (radial distance to the center of mass of the BH) to the Schwarzschild radius $r_g$ is finite and can be computed from the integral (Landau/Lifschitz "Classical field theory" )(102,8)
$$ \Delta\tau = \frac{1}{c}\int_{r_0}^{r_g} \left(\frac{r_g}{r} - \frac{r_g}{r_0}\right) dr$$
which is finite whereas the observation from a fixed point $r_0$ in the Schwarzschild reference system it takes an infinite amount of time according to the formula of Landau/Lifschitz (102,6).
$$r-r_g = const\cdot \exp(-ct/r_g)$$
Therefore for $r\rightarrow r_g$ $t\rightarrow \infty$.
I have a couple of questions with respect to this result. What actually meant with the time $t$? I mean, how do I measure it, in other words, how can the formula (102,6) be checked?
1) If I wait at a fixed position $r_0>r_g$ for the passage for the particle of the event horizon, I need to receive a (for instance light signal) from the falling particle, therefore I have to correct my time measurement at $r_0$ by the time span the signal needs to get from the event horizon to $r_0$ which already would amount to infinity, so the measurement, so I guess, the check of the formula could not really be done.
2) But I can also imagine "physical" Schwarzschild coordinate system with a long ruler reaching from the event horizon to $r_0$ which has a clock installed a each mark of the ruler which should be synchronized with the clock at $r_0$. Such a synchronization is apparently possible in the environment of a non-rotating black hole since the metric components $g_{0\alpha}$ with $\alpha = 1,2,3$ are zero in that case (SSM). First the clock at $r_0$ could send a signal to the clock close to the event horizon to inform about the start of the free fall (the measurement should be corrected by the time span the light needs to get there, but that span should be finite). Second the clock close to the event horizon could measure when the falling particle comes by. My question would be now: The time span I would measure like this would be the one that enters into the formula (102,6) (one could choose a $r$ which is close to $r_g$, but still $r>r_g$ to make the result finite)? What would be appropriate set-up for such a measurement? The first or the second?
I admit, the second set-up seems to be very unrealistic, but I always wonder about reference systems in GR which extend over large areas and how to perform measurements in such systems (and still relying on the same reference). That is actually also a question which could be answered by using this example.
Thank for any help.
