Gravitational redshift and doppler effect in Schwarzschild metric

Question

I am trying to calculate the change in frequency of a photon emitted by beacon that has a circular orbit with $r=r_1$. In $r_2 >r_1$ there is a static observer that observes the photon when he, the beacon and the BH are in a aligned. This means that the photon is not emitted in a radial trayectory.

I calculate it using an orthonormal basis. For the observer

$$\hat{e}_0=\bar{U}_{obs}=\frac{dt}{d\tau}\partial_t=\frac{1}{(1-\frac{r_s}{r_2})^{1/2}}\partial t$$

and for the beacon

$$\hat{e}_0=\frac{dt}{d\tau}(\partial_t+\Omega\;\partial_\varphi)=\frac{1}{(1-\frac{3r_s}{2r_1})^{1/2}}(\partial_t+\Omega\;\partial_\varphi)$$

where this follows from normalization of $\bar{U}^2=-1$, and $\Omega=d\varphi/dt$. Then

$$\frac{\omega_{obs}}{\omega}=\frac{-g(k,\hat{e}_{0,obs})}{-g(k,\hat{e}_{0,em})}=\left(\frac{1-\frac{3r_s}{2r_1}}{1-\frac{r_s}{r_2}}\right)^{1/2} \frac{1}{1+\Omega\;\frac{g(k,\partial\varphi)}{g(k,\partial_t)}}$$

I got stucked in this part. I know that both $\partial_t$ and $\partial_\varphi$ are Killing vectors, so the g( , ) are conserved quantities

$$\frac{g(k,\partial\varphi)}{g(k,\partial_t)}=-\frac{r^2d\varphi/d\tau}{(1-r_2/r)dt/d\tau}$$

but now, with respect to what observer should i evaluate this? Any help is appreciated.

Answer 1

Demtrio wrote: "With respect to what observer should i evaluate this? Any help is appreciated."

First you get the ratio of the gravitational time dilation between observer and emitter. Then you check the light ray's vector at emission and see how much of it was in direction of motion so you multiply the resulting special relativistic Doppler. The result should look like this:

The image above is in the frame of a stationary observer at r=1000GM/c², θ=85°. If the observer is moving (see here for an example with different directions of motion) as well just apply the special relativistic aberration for his local velocity relative to a stationary reference observer (see here for an example how it's done).

Demtrio wrote: "there is a static observer that observes the photon when he, the beacon and the BH are in a aligned. This means that the photon is not emitted in a radial trayectory."

If the photon is emitted when they are all aligned then that photon travelled radially. If they are aligned when the light reaches the observer just calculate the Δt between emission and absorption and apply that to the dφ/dt of the circular orbit, then you know how far the emitter travelled while the light was on its way.