Christoffel symbol derivation in book by Wald

Question

In chapter 3 of Wald's General Relativity he starts by defining a covariant derivative $\nabla$ as a map on a manifold M from tensor fields $\mathscr{T}(k,l) \to \mathscr{T}(k,l+1)$ plus some required properties (linearity, Leibniz rule, etc.).

He then goes on to show that for any two derivatives $\nabla, \tilde{\nabla}$, their difference (applied to a one-form) can be expressed by a tensor as $$ \nabla_a \omega_b - \tilde{\nabla}_a \omega_b = C^c_{ab} \omega_c. $$ What I don't understand is that he says we choose $\tilde{\nabla}$ as the usual partial derivative $\partial$ and call the tensors $C^c_{ab} = \Gamma^c_{ab} $ the Christoffel symbols. I thought the partial derivative does not satisfy the required transformation properties of the covariant derivative hence I can't substitute it for $\tilde{\nabla}$.

Another minor issue is that he calls $C^c_{ab}$ a tensor field while he also says it doesn't transform according to the tensor transformation law. What does he then mean by that? That it is a multilinear map?

Answer 1

Under general coordinate transformation (GCT), a tensor transforms like, $$ T_{a'b'\cdots}^{c'd'\cdots}=\frac{\partial x^{c'}}{\partial x^c}\frac{\partial x^{d'}} {\partial x^d}\cdots\frac{\partial x^a}{\partial x^{a'}}\frac{\partial x^b}{\partial x^{b'}}\cdots T_{ab}^{cd} $$ It is easy to see from the above equation that partial derivative of a vector under GCT does not transform as a tensor. So we introduce an object called covariant derivative $$ D_\mu v^\nu =\partial_\mu v^\nu + \Gamma^{\nu}_{\mu\sigma} v^\sigma $$ and demand that the covariant derivative of a vector transforms tensorially, that is $$ D_{\mu'} v^{\nu'}= \partial_{\mu'} x^{\mu} \partial_{\nu} x^{\nu'} D_\mu v^\nu. $$ Note that this requires the Christoffel symbol to transform non-tensorially, $$ \Gamma^{\mu'}_{\nu'\lambda'}=\partial_\mu x^{\mu'} \partial_{\nu'} x^\nu \partial_{\lambda'} x^{\lambda} \Gamma^{\mu}_{\nu\lambda} - \partial_{\nu'} x^\nu \partial_{\lambda'} x^{\lambda} \partial_\mu\partial_\lambda x^{\mu'}. $$

However, it can be seen using the transformation law of the tensors that the difference of two Christoffel symbols transforms as a tensor. Furthermore, $$ D_\mu v^\nu - \hat{D}_\mu v^\nu = (\Gamma^{\nu}_{\mu\lambda}-\hat{\Gamma}^{\nu}_{\mu\lambda}) v^\lambda $$ We can define the differences of the Christoffel symbol as a (1,2) tensor, say $C^{\nu}_{\mu\lambda}$. And this indeed transforms as a tensor. I do not understand why you are saying that it does not transform according to the tensor transformation law.

Answer 2

Wald states, in eq. 3.1.14, that the difference between two distinct derivative operators is characterized exactly by the tensor field $C^c_{ab}$.

Schematically, he is saying that $$ \nabla T = \tilde{\nabla}T + CT $$ Where $\nabla$ and $\tilde{\nabla}$ are distinct derivative operators. He now chooses that one of the derivative operators is the regular partial derivative, i.e. he demands that $\tilde{\nabla} =\partial$, in order to find out how the regular partial derivative differs from the covariant derivative. Note that locally in some coordinate patch, $\partial$ does fulfill all of his 5 requirements.

As for the additional question on how Christoffel symbols transform, you can find many places on SE that have answered it, i.e. Under what representation do the Christoffel symbols transform?