Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 17; Problem P17.10.
Here’s an application of the fact that the covariant derivative of any metric tensor is always zero. Suppose we define a coordinate transformation in which:
where is the Christoffel symbol in the primed system evaluated at a particular point (and therefore they are constants). (In Moore’s problem P17.10, he states that this is in the unprimed system, but the problem makes no sense in that case, since we’re summing over an index which refers to the unprimed coordinate system in one term and the primed system in the other.) The quantity represents a displacement from the point , as measured in the primed system.
Using the usual transformation equation for a tensor, we get for the metric tensor:
When , , so at point .
Now consider the second term. By renaming the dummy indices and , we have
Now because the covariant derivative (with respect to the primed coordinates) of the metric is zero, we have
Therefore, we can write
If we take the derivative of this with respect to a particular primed coordinate we can use
and then evaluate the result at so that all terms containing vanish. We get
Thus all the first derivatives of are zero in the primed coordinate system.