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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:
[tex]\frac{d}{ds}\left[A \dot t \right] = 0 [/tex]
[tex]0 = -Ac^2 \dot {t}^2 + \frac{1}{A} \dot {r}^2 + r^2 \dot {\phi}^2 [/tex]
[tex]\frac{d}{ds} \left[ r^2 \dot \phi \right] = 0 [/tex]
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:
[tex]\dot p^\mu + \frac{1}{\hbar} \Gamma^\mu_{\alpha \beta} p^\alpha p^\beta = 0 [/tex]
For the temporal component we have exactly what the question wants:
[tex]\dot p^0 = - \frac{A'}{A}p^0 \dot r [/tex]
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##?