Kepler's Law in Schwarzchild metric

[Fek]

Homework Statement

Show Kepler's Third Law holds for circular Schwarzschild orbits.

Homework Equations

The Attempt at a Solution

Setting $r' = 0 , \theta' = 0$ and $\theta = \pi / 2$ , where the derivatives are with respect to the variable $\lambda$ and setting c = 1 the Lagrangian is:

$L = (1 - r_s / r) \dot t^2 - r^2\dot\phi^2$

E-L equation for r:

$$\frac{r_s} {r^2} \dot t^2 - 2r \dot\phi^2 = 0$$

Therefore $$d\phi / dt = GM/r $$ , the solution.

However I'm struggling to understand why t here corresponds to the proper time at infinity, but r corresponds to the radius of the ciruclar orbit?

 

[BigJDubs]

Also, once I have this solution, how do I show that Kepler's Third Law holds?Any help would be greatly appreciated.