Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 17; Box 17.3.
The Christoffel symbols are defined in terms of the basis vectors in a given coordinate system as:
Remember that the basis vectors are defined so that
In a locally flat frame using rectangular spatial coordinates, the basis vectors are all constants, so from 1, all the Christoffel symbols must be zero: .
Now let’s look at the second covariant derivative of a scalar field :
where in 6 we used rule 1 for the covariant derivate: the covariant derivative of a scalar is the same as the ordinary derivative.
In the locally flat frame, this equation reduces to
Since the covariant derivative is a tensor, this is a tensor equation, and since ordinary partial derivatives commute, this equation is the same if we swap the indices and . Tensor equations must have the same form in all coordinate systems, so this implies that 7 must also be invariant if we swap and . This means that the Christoffel symbols are symmetric under exchange of their two lower indices:
At first glance, this seems wrong, since from the definition 1 this symmetry implies that
In 2-D polar coordinates, if we take the usual unit vectors and then both these vectors are constants as we change and both of them change when we change , so it’s certainly not true that , for example. However, remember that the basis vectors we’re using are not the usual unit vectors; rather they are defined so that condition 4 is true. In polar coordinates, we have
For the derivatives, we have
Thus the condition 10 is actually satisfied here.