After having worked for a while with the Schwarzschild geometry, I have realized something I hadn't seen before and that I found slightly disturbing. Consider the 4-dimensional Schwarzschild metric:
$$ ds^2 = -\left(1- \frac{2M}{r}\right)dt^2 + \left(1- \frac{2M}{r}\right)^{-1}dr^2 + r^2 d\Omega^2 ~, $$
and suppose two observers are held at fixed spatial positions $(R_1, 0, 0)$ and $(R_2,0,0)$, with $R_2 > R_1 > 2M$ (using some non-gravitational system, say rockets burning fuel). This is a typical situation to study gravitational redshift when the inner observer sends signals separated by $\Delta \tau_1$ (as measured by him, that is, $\tau_1$ is the proper time along his worldline). The famous result is:
$$ \Delta \tau_2 = \sqrt{\frac{1-\frac{2M}{R_2}}{1-\frac{2M}{R_1}}} \Delta \tau_1 ~~, \hspace{1cm} \nu_2 = \sqrt{\frac{1-\frac{2M}{R_1}}{1-\frac{2M}{R_2}}} \nu_1 ~, $$
where we think in terms of period of these signals in the last expression. If the inner observer approaches $R_1 \to 2M$, we get an infinite redshift from the outer observer's viewpoint. My question is the following: what happens if insted of the previous situation the outer observer sends signal to the inner one? Obviously, we obtain a blueshift (the analysis is the same), and that blueshift becomes arbitrarily large when $R_1 \to 2M$, since it is still true that:
$$ \nu_1 = \sqrt{\frac{1-\frac{2M}{R_2}}{1-\frac{2M}{R_1}}} \nu_2 ~, $$
and we are keeping $\nu_2$ constant now. This seems puzzling to me for several reasons:
- It seems that low energy excitations (say photons for example) sent by the outer observer to the inner one can become arbitrarily energetic when received by the inner observer. So in some sense we need to know the details of the high-energy physics to work in a neighbourhood of the horizon.
- At first I thought that keeping an observer at fixed $r$ close to the horizon without moving in space was unphysical in some sense (maybe the gravitational tidal forces become so large that it is not possible to do that). But, in principle, for large mass the horizon at $r = 2 M$ is not a place where gravitational forces become that strong (in a Newtonian approximation, $F \sim M / r^2 \sim 1/M $, something that I think I remember seeing somewhere justified more rigorously in the framework of GR by noting that the Riemann tensor components in a local inertial frame do not diverge at $r = 2M$).
I thought that maybe the above definition of frequency is tricky, so I also tried to compute the energy of a photon with four-momentum $k$ with the formula $E_{obs} = - k \cdot U_{obs}$, $U_{obs}$ being the observer's four-velocity. This also diverges for a photon sent with a finite energy from infinity (actually, you obtain the same expression as before), so nothing new. Is this a real problem or am I overestimating the consequences of this infinite blueshift? [For the sake of completeness, my fear of an infinite blueshift comes from some discussions I've read about stability of horizons. In particular, when considering rotating or charged black holes one has an inner horizon which spoils some nice features one expect physical solutions to have, like predictability. I found convincing in that case the fact that perturbations sent from outside the black hole become arbitrarily blueshifted when they reach the inner horizon, so maybe this horizon is unstable and it is only appearing as a consequence of the high degree of symmetry of our exact solutions - thus they do not violate strong cosmic censorship]. Any ideas or thoughts on this issue?
