Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 8; Problem 8.4.
The geodesic equation is
We can write this in terms of the four-velocity if we expand the first derivative:
Therefore, the geodesic equation is
We can express this in a different form by multiplying by and summing:
The geodesic equation thus confirms that along a geodesic, which we knew beforehand, since is an invariant.