Starting with the Schwarzschild metric: $$ A = 1-\frac{2m}{r} $$ $$ \mathrm{d}\tau^2 = A\mathrm{d}t^2 - \mathrm{d}r^2/A -r^2\mathrm{d}\theta^2 - r^2\sin^2{\theta}\mathrm{d}\phi^2 $$ I want to calculate the angular velocity of a circular orbit ($\mathrm{d}r=\mathrm{d}\theta=0$). I can calculate the Christoffel symbols, write down the geodesic equation for $r$, then assign $\mathrm{d}r=0$ and $\theta=\pi/2$, solve for $\mathrm{d}\phi/\mathrm{d}t$ and indeed get the familiar $\sqrt{m/r^3}$ I expect from Kepler's 3rd law. Fine so far.
However, I can also write down the geodesic equation for $t$, which is $$ \frac {\mathrm{d}^2t}{\mathrm{d}\lambda^2} + \frac{1}{A}\frac{\mathrm{d}A}{\mathrm{d}r}\frac{\mathrm{d}r}{\mathrm{d}\lambda}\frac{\mathrm{d}t}{\mathrm{d}\lambda} = 0 $$ Which solves to $$ \frac {\mathrm{d}t}{\mathrm{d}\lambda} = \frac {k}{A} $$ with $k$ being a constant of integration. Likewise the geodesic equation for $\phi$ solves for $$ \frac {\mathrm{d}\phi}{\mathrm{d}\lambda} = \frac {k'}{r^2\sin^2\theta} $$ dividing these two, I expect to get an identical expression for $\mathrm{d}\phi/\mathrm{d}t$. Yet I don't.
Where's my mistake?
