I am learning GR and the notation is killing me here. So my understanding is, the comma notation is used to indicate a derivative, such as:
\begin{equation}
V^{\alpha}_{\;\;,\gamma}=\partial_{\gamma}V_{\alpha}
\end{equation}
and a semicolon is used to represent a covariant derivative, such as:
\begin{equation}
V^{\alpha}_{\;\;;\gamma}= \partial_{\gamma}V^{\alpha}+\Gamma_{\gamma\mu}^{\alpha}V^{\mu} = V^{\alpha}_{\;\;,\gamma}+\Gamma_{\gamma\mu}^{\alpha}V^{\mu} = \nabla_{\gamma}V^{\alpha}
\end{equation}
However! In problem 7.7 in "The Problem Book of Relativity and Gravitation" they write (for the metric tensor g):
\begin{equation}
g_{\alpha \beta , \gamma} = \nabla_{\gamma}(\mathbf{e}_{\alpha}\cdot \mathbf{e}_{\beta}) = \Gamma^{\mu}_{\alpha\gamma}\mathbf{e}_{\mu}\cdot\mathbf{e}_{\beta}+\Gamma^{\mu}_{\beta \gamma}\mathbf{e}_{\mu}\cdot\mathbf{e}_{\alpha}
\end{equation}
Christoffel symbols?! How? I thought these only popped up when taking the COVARIANT derivative. Then later on, they write:
\begin{equation}
A^{\alpha}_{\;;\alpha} = A^{\alpha}_{\;,\alpha}+\Gamma^{\alpha}_{\;\beta\alpha}A^{\beta}
\end{equation}
Which makes sense given my definition above, but does not make sense with the notation used in the first example. Am I missing something? Is it just a typo??
Best Answer
All formulas you shown above are using abstract index notation except the third formula which is fully expressed is a basis. For a vector field, you can write for example $$V = V^\mu e_\mu\;,$$ where $V^\mu$ is a scalar while $e_\mu$ is a vector basis. This is a sort of confusion because in abstract index notation we view $V^\mu$ as a vector field.
When we take the covariant derivative, it reads $$\nabla_\mu V=\nabla_\mu (V^\nu e_\nu) = \nabla_\mu (V^\nu) e_\nu + V^\nu \nabla_\mu ( e_\nu)$$ \begin{eqnarray} &=& \partial_\mu (V^\nu) e_\nu + V^\nu \Gamma_\mu{}^\lambda{}_\nu e_\lambda\;,\\ &=&\big( \partial_\mu V^\nu +\Gamma_\mu{}^\nu{}_\lambda V^\lambda\big)e_\nu \end{eqnarray} If we define $$\nabla_\mu V =: (\nabla_\mu V^\nu) e_\nu $$ we will have the relation in the abstract index notation $$\nabla_\mu V^\nu = \partial_\mu V^\nu +\Gamma_\mu{}^\nu{}_\lambda V^\lambda\;.$$ (More general, you can start with $\nabla V$ and then define $\nabla V =:(\nabla_\mu V^\nu) e^\mu \otimes e_\nu$ ) Next, the metric $g$, it is (o,2) tensor so it has two slots for inserting 2 vectors if we insert the basis into these slots we will get a component of the metric tensor which is a scalar field $$g(e_\mu, e_\nu)=g_{\mu\nu}$$
($g= g_{\alpha\beta} e^\alpha \otimes e^\beta,\;g(e_\mu, e_\nu) =g_{\alpha\beta} e^\alpha(e_\mu) \otimes e^\beta(e_\nu) =g_{\alpha\beta}\delta^\alpha_\mu \delta^\beta_\nu= g_{\mu\nu} $)
It is also usually define that $\eta(A,B):= A\cdot B$, $\eta$ is a Minkowskian metric $A\cdot B$ is a scalar so invariants under coordinate transformations $$A\cdot B =\eta(A,B) \equiv \eta_{IJ} A^I B^J$$ $$= g(A,B) \equiv g_{\mu\nu} A^\mu B^\nu$$ where $A^I = e^I_\mu A^\mu$ for some scalar $e^I_\mu$ (a vierbein), and you can easily prove that $g_{\mu\nu} = \eta_{IJ} e^I_\mu e^J_\nu$.
So now we have $$e_\mu \cdot e_\nu =\eta_{IJ}e^I_\mu e^J_\nu= g_{\mu\nu}$$
In this step, we can view $\eta_{IJ},g_{\mu\nu}$ as the scalar fields $e^I_\mu$ as a vector field \begin{eqnarray} \nabla_\gamma g_{\alpha\beta} (= \partial _\gamma g_{\alpha\beta})&=& \nabla_\gamma (e_\alpha\cdot e_\beta) \\ &=&\eta(\nabla_\gamma e_\alpha,e_\beta) +\eta(e_\alpha,\nabla_\gamma e_\beta) \equiv \eta_{IJ}\nabla_\gamma(e^I_\alpha) e^J_\beta + \eta_{IJ}e^I_\alpha \nabla_\gamma(e^J_\beta) \\ &=&\eta(\Gamma_\gamma{}^\rho{}_\alpha e_\rho,e_\beta) +\eta(e_\alpha,\Gamma_\gamma{}^\sigma{}_\beta e_\sigma) \equiv \eta_{IJ}\Gamma_\gamma{}^\rho{}_\alpha e^I_\rho e^J_\beta + \eta_{IJ}e^I_\alpha \Gamma_\gamma{}^\sigma{}_\beta e^J_\sigma \\ &=&\Gamma_\gamma{}^\rho{}_\alpha \eta( e_\rho,e_\beta) + \Gamma_\gamma{}^\sigma{}_\beta\eta(e_\alpha, e_\sigma) \equiv \Gamma_\gamma{}^\rho{}_\alpha \eta_{IJ} e^I_\rho e^J_\beta + \Gamma_\gamma{}^\sigma{}_\beta \eta_{IJ}e^I_\alpha e^J_\sigma \\ &=& \Gamma_\gamma{}^\rho{}_\alpha e_\rho \cdot e_\beta + \Gamma_\gamma{}^\sigma{}_\beta e_\alpha \cdot e_\sigma \equiv \Gamma_\gamma{}^\rho{}_\alpha g_{\rho \beta} + \Gamma_\gamma{}^\sigma{}_\beta g_{\alpha \sigma} \end{eqnarray} Note: Not fully detailed as much as possible but may be helpful for you.