Why don't Christoffel symbols appear in second partial derivatives of the metric?

[quasar_4]

Hi,

I am incredibly confused about second derivatives of the metric. I know that in general, the covariant derivative of a vector is given by

$$\nabla_a v^b = \partial_a v^b + \Gamma^b_{ac}v^c$$

and I think I understand how to generalize to higher rank tensors (just decompose into an outer product of vectors and covectors and use Leibniz, right?).

But, if we have the metric tensor, and we want to take the 2nd derivative

$$\partial_a \partial_b g_{cd} = g_{cd, ab}$$

do I just differentiate the components of g twice or are there Christoffel symbols hiding in there too?? They are just partial derivative symbols and not covariant derivatives, so I don't see how/why we'd do anything but differentiate components of the metric.

But I've been trying to compute a curvature tensor for hours now and I cannot for the life of me get the answer that Maple gets, so I must not understand how to compute 2nd derivatives of the metric...

 

[CompuChip]

Indeed, to calculate
$$\partial_a \partial_b g_{cd} = g_{cd, ab}$$
you simply take the second derivatives of the components of g. The Christoffel symbols only come in when you want to calculate the covariant derivative
$$\nabla_a \nabla_b g_{cd} = g_{cd; ab}.$$
Usually, in GR, we choose the Christoffel symbols such that
$$\nabla_a g_{bc} = g_{bc;a} = 0,$$
the so-called metric compatible connection.