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.
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.