# Christoffel symbol derivation in book by Wald

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?

• In a fixed coordinate patch $\partial_a$ satisfies all requirements Wald mentions. Changing coordinates that notion of covariant derivative transforms as required and it does not coincide with the standard derivative any more. Commented Apr 21, 2018 at 8:55

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.
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.