Thread:  Christoffel symbol as tensor 		 		View Single Post

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Here's a quote from Wald. I have highlighted what I think are the important parts.

From Wald, p. 33. Thus, we have shown that $\tilde\nabla_a - \nabla_a$ defines a map of dual vectors at p (as opposed to dual vector fields defined in a neighborhood of p) to tensors of type (0, 2) at p. By property (1), this map is linear. Consequently $\tilde\nabla_a - \nabla_a$ defines a tensor of type (1,2) at p, which we will denote as $C^c{}_{ab}$. Thus, we have shown that given any two derivative operators $\tilde\nabla_a$ and $\nabla_a$ there exists a tensor field $C^c{}_{ab}$ such that  $\nabla_a \omega_b= \tilde\nabla_a \omega_b - C^c{}_{ab}\omega_c$ (3.1.7)  ...snip...  Continuing in a similar manner, we can derive the general formula for the action of $\nabla_a$ on an arbitrary tensor field in terms of $\tilde\nabla_a$ and $C^c{}_{ab}$. For $T \in {\cal T}(k,l)$ we find  $\nabla_a T^{b_1 \cdots b_k}{}_{c_1 \cdots c_l}= \tilde\nabla_a T^{b_1 \cdots b_k}{}_{c_1 \cdots c_l} + \sum_i C^{b_i}{}_{ad} T^{b_1 \cdots d \cdots b_k}{}_{c_1 \cdots c_l} -\sum_j C^{d}{}_{ac_j} T^{b_1 \cdots b_k}{}_{c_1 \cdots d \cdots c_l}$ (3.1.14)  Thus, the difference between the two derivative operators $\textcolor{red}{\nabla_a}$ and $\textcolor{red}{\tilde\nabla_a}$ is completely characterized by the tensor field $\textcolor{red}{C^c{}_{ab}}$.

From Wald, p. 34. The most important application of equation (3.1.14) arises from the case where $\tilde\nabla_a$ is an ordinary derivative operator $\partial_a$. In this case, the tensor field $C^c{}_{ab}$ is denoted $\Gamma^c{}_{ab}$ and called a Christoffel symbol. Thus, for example, we write  $\nabla_a t^b = \partial_a t^b + \Gamma^b{}_{ac} t^c$ (3.1.15)  Since we know how to compute the ordinary derivative associated with a given coordinate system, equation (3.1.15) (and, more generally, eq. [3.1.14] with $\partial_a$ and $\Gamma^b{}_{ac}$ replacing $\tilde\nabla_a$ and $C^b{}_{ac}$) tells us how to compute the derivative $\nabla_a$ if we know $\Gamma^b{}_{ac}$. Note that, as defined here, a Christoffel symbol is a tensor field associated with the derivative operator $\nabla_a$ and the coordinate system used to define $\textcolor{red}{\partial_a}$. However, if we change coordinates, we also change our ordinary derivative operator from $\partial_a$  to $\partial'_a$ and thus we change our tensor $\Gamma^c{}_{ab}$, to a new tensor $\Gamma'^c{}_{ab}$. Hence the coordinate components of $\textcolor{red}{\Gamma^c{}_{ab}}$, in the unprimed coordinates will not be related to the components of $\textcolor{red}{\Gamma'^c{}_{ab}}$ in the primed coordinates by the tensor transformation law, equation (2.3.8), since we change tensors as well as coordinates.
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