I was doing an exercise in Schutz (A First course in General Relativity). The exercise wanted the double covariant derivative calculated for a vector $V^\mu$ i.e. $\nabla_\alpha \nabla_\beta V^\mu$ i.e. $(V^\mu_{;\alpha})_{;\beta}$. This basically amounts to calculate the covariant derivative of a mixed (1,1) tensor $T^\alpha_\beta$. I was able to calculate the covariant derivative of this mixed tensor by converting it into $T^\alpha_\beta = A^\alpha B_\beta$ and it worked out.
But I originally tried to calculate it by lowering the index on a (2,0) tensor $T^{\alpha\beta}$ using a metric tensor and then calculating the covariant derivative, since the covariant derivative of metric tensor is zero. But it did not work out. Here is how I did it.
Given that,
$$
T^{\alpha \beta}_{;\gamma}= T^{\alpha \beta}_{,\gamma} + \Gamma^\alpha_{\mu\gamma}T^{\mu\beta}+\Gamma^\beta_{\mu\gamma}T^{\alpha\mu}
$$
I lowered the index,
$$
(T^{\alpha}_\lambda g^{\lambda\beta})_{;\gamma} =(T^{\alpha}_\lambda )_{;\gamma}g^{\lambda\beta}= (T^{\alpha \beta}_{,\gamma} + \Gamma^\alpha_{\mu\gamma}T^{\mu\beta}+\Gamma^\beta_{\mu\gamma}T^{\alpha\mu})g^{\lambda\beta}
$$
$$
(T^{\alpha}_\lambda)_{;\gamma} = g_{\lambda\beta}T^{\alpha \beta}_{,\gamma} + g_{\lambda\beta}\Gamma^\alpha_{\mu\gamma}T^{\mu\beta}+g_{\lambda\beta}\Gamma^\beta_{\mu\gamma}T^{\alpha\mu}
$$
The confusion is that $g_{\lambda\beta}\Gamma^\beta_{\mu\gamma}$ in the last term, after the contraction on $\beta$ becomes, $\Gamma_{\lambda\mu\gamma}$ which is not the correct term for the covariant derivative of $T^{\alpha}_\lambda$.
So, I am missing something in the contraction of $\Gamma^\beta_{\mu\gamma}$ or may be that is not a valid operation(?). I have a hunch that the reason has something to do with $\Gamma^\beta_{\mu\gamma}$ not being a valid tensor but I cant place what it is mathematically/physically(?). Or may be there is an identity of $\Gamma^\beta_{\mu\gamma}$ that I am missing here(?).
Best Answer
Your hunch is basically correct. Raising and lowering indices really means that you are exploiting a canonical isomorphism between tangent and contangent spaces in the presence of a metric: $v^\mu=v_\nu g^{\nu\mu}$ is just the map $$\begin{align*} V &\longrightarrow V^*\\v&\longmapsto g(v,\cdot)\end{align*}$$ expressed in a basis. The same applies to tensors with more indices, i.e. elements of tensor products of $V$ and $V^*$.
This implies that raising and lowering is only sensible for proper tensors (elements of $V\otimes\dotsm \otimes V\otimes V^*\otimes\dots \otimes V^*$). Partial derivatives and Christoffel symbols are not such tensors, and so you should not raise/lower the indices here. (Of course, the covariant derivative combines $\partial_\mu$ and $\Gamma_{\mu\nu}^\rho$ in the right way to be a tensor, hence the above iosomrphism applies, and you can freely raise/lower indices her.)