I'm taking a course in GR and I want to determine for which radius a photon can move in a circular orbit in the gravitational field produced by a massive spherical body (Schwarzild's metric).
What I have done is to calculate the tangent vector for an arbitrary geodesic of this metric and make the norm equal to $0$ because a massless particle must move in a null-like geodesic. This gives me $r_0=2GM$, the Schwarzild radius.
Then the problem ask me to calculate the period of the photon in this orbit for the coordinate time and for an observer in a point of that orbit.
I was wondering how to do this. Physically, as the radius that I've got is the Schwarzild radius, I expect the solution for the coordinate time to be $\infty$ and some value for an observer in the orbit.
To calculate the period should I apply the classical formula for the angular velocity of a particle $\omega=v/r$ and then say that $T=2\pi/\omega$ or isn't this a valid approximation?
Is there any way to get this information from the null-like geodesic besides calculating the Cristoffel symbols and then the velocity in coordinates $(t,r,\theta, \phi)$?