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.