Calculating from the metric

Since is a tensor we can lower the index using the metric tensor:

But by linearity, we have:

So consistency requires . Since is arbitrary this implies that

Thus the covariant derivative of the metric is zero in every frame.

We next prove that [ i.e. symmetric  in and ]. In a general frame we have for a scalar field :

in a local inertial frame, this is just , which is symmetric in and . Thus it must also be symmetric in a general frame. Hence is symmetric in and :

We now use this to express in terms of the metric. Since , we have:

By writing different permutations of the indices and using the symmetry of , we get

Multiplying by and using gives

Note that is not a tensor since it is defined in terms of partial derivatives.

In a local inertial frame since . We will see later the significance of this result.