Frequency of Photon in Schwarzschild Metric?

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Homework Statement

The schwarzschild metric is given by $ds^2 = -Ac^2 dt^2 + \frac{1}{A} dr^2 + r^2\left( d\theta^2 + sin^2\theta d\phi^2 \right)$. A particle is orbiting in circular motion at radius $r$.

(a) Find the frequency of photon at infinity $\omega_{\infty}$ in terms of when it is at $r$, $\omega_r$.
(b) Find the range of frequencies.

Homework Equations

The Attempt at a Solution

I have found the geodesic equations:
\frac{d}{ds}\left[A \dot t \right] = 0
0 = -Ac^2 \dot {t}^2 + \frac{1}{A} \dot {r}^2 + r^2 \dot {\phi}^2
\frac{d}{ds} \left[ r^2 \dot \phi \right] = 0

I have also found the christoffel symbols: $\Gamma^r_{rr}=-\frac{A'}{2A}$ and $\Gamma^r_{tt}=-\frac{1}{2}A A'c^2$ and $\Gamma^r_{\theta \theta} = -Ar$ and $\Gamma^r_{\phi\phi}=-Ar$ and $\Gamma^t_{rt}=\Gamma^r_{tr}=\frac{A'}{2A}$.

Part(a)
The differential equation is simply the geodesic equation for a photon:
\dot p^\mu + \frac{1}{\hbar} \Gamma^\mu_{\alpha \beta} p^\alpha p^\beta = 0
For the temporal component we have exactly what the question wants:
\dot p^0 = - \frac{A'}{A}p^0 \dot r

The energy of the photon is given by $E = p_\mu U^\mu =p_0 U^0+p_1 U^1+p_2 U^2+p_3 U^3 = \hbar \omega$.

How do I find $p^\mu$ and $U^\mu$?

 

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