Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 17; 9.
One interesting and useful theorem is that the covariant derivative of any metric tensor is always zero. We can show this by using the expression for the covariant derivative of a general tensor to say:
We can combine this with the explicit expression for the Christoffel symbols:
Substituting, we get
Since , we get
Now we use the symmetry of the metric tensor: :