I am calculating for many hours and I am really confused with this exercise.
Consider a comoving observer sitting at constant spatial coordinates$(r∗,θ∗,φ*)$, around a Schwarzschild black hole of mass $M$. The observer drops a beacon into the black hole (straight down, along a radial trajectory). The beacon emits radiation at a constant wavelength $λ_{\mathrm{em}}$ (in the beacon’s rest frame).
From Schutz's book, I think "comoving" might be wrong, because it is only true when the object flies on geodesic which has to be a circle and that is not guaranteed.
The Schwarzschild metric:
$$ds^2=-(1-2M/r)dt^2+(1-2M/r)^{-1}dr^2+r^2d\Omega$$
I tried to solve this in this way:
In the observers MCRF, he has the four velocity:
$$U_{\operatorname{obs}}=\vec{(1,0,0,0)}$$
Now we calculate the energy of the object by the observer:
$$E/m=\bar{E}=-U_{\mu}^PU^{\mu}_{\operatorname{obs}}=-U_{0}^P$$
with $U^p$ the objects velocity measured by the observer. Then we found
$$U_{0}^P=-g^{00}\bar{E}=\dot{t}$$
The last equating can you show with respect to the proper time.
The other component follows from:
$$U_{\mu}^PU^{\mu}_P=-1$$
From this equation you can compute:
$$(U^r)^2=-(1-2G/R)+\bar{E}$$
something like that…
But now I have a question.
I am in the MCRF of the observer hence a local initial system with a flat metric. But I used the in this point the Schwarzschild metric not the Minkowski one.
I thought you can do this, because the Schwarzschild metric is valid there in every point. And when you use the Minkowski one you get that the velocity gets infinity on $r =2M$ and this is wrong.
Best Answer
OK, I"m answering this:
"I am in the MCRF of the observer hence a local initial system with a flat metric. But I used the in this point the Schwarzschild metric not the Minkowski one. I thought you can do this, because the Schwarzschild metric is valid there in every point. And when you use the Minkowski one you get that the velocity gets infinity on r =2M and this is wrong."
The very key term here is LOCAL. The reference frame of a comoving observer is only Minkowski over a reference frame small enough that you can ignore curvature. And, in a frame such as your constant $r$ reference frame, you can't pretend that it is Minkowski at all--you have to exert a force to stay in the frame! So, for this setup, I'd recommend doing everything in the Schwarzschild geometry and forgetting about any reference to Minkowskian things.