In this post, I will write about the geodesic equation. I suggest a simplified form that is simpler to solve.
The geodesic equation is given by
where . We have used the convention that . Then we may write
Then it is clear that IF the metric does not depend on we obtain the equation
Consider for example the Schwarzschild metric, given by
Then for we obtain the equation
where is a solution. Next we consider the equation
Say we consider circular orbits – thus then we obtain
thus we obtain
which is know as the Third Law of Kepler.
We have here derived the Third law of Kepler for the Schwarzschild metric, without calculating the affine connection .