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

,

hus .

Consider for example the Schwarzschild metric, given by

.

where .

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

,

so

,

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 .

Have fun!

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