Notation for covariant derivative in the book Gravitation I am not familiar with semicolon notation for covariant derivatives that is why I am asking this question.
In the book Gravitation  on page 566 they have the following exercise

Show that $\nabla u$ can be decomposed in the following manner:
$$
u_{\alpha ; \beta}=\omega_{\alpha \beta}+\sigma_{\alpha \beta}+\frac{1}{3} \theta P_{\alpha \beta}-a_{\alpha} u_{\beta}
$$

I am not understanding the notation $u_{\alpha ; \beta}$ for  the components of $\nabla u$.
Shouldn't we have $u^\alpha_{ ; \beta}$ instead of $u_{\alpha ; \beta}$? I think so because $\nabla u$ is a vector valued one form and so it should have one lower index for the form and one upper index for the vector.
 A: In the MTW book Gravitation as well as most relativity literature, index raising and lowering is taken as self-evident and/or implicit. This means that when we refer to four-velocity, we may mean the object with components $u_\alpha$ or $u^\alpha$.
These are related by index lowering/raising:
$$u_\alpha = u^\beta g_{\beta \alpha}$$
Since the metric has $g_{\alpha\beta;\gamma} = 0$ for all choices of $\alpha\beta\gamma$, index lowering and raising commutes with the covariant gradient:
$$u_{\alpha;\beta} = (u^{\gamma} g_{\gamma \alpha})_{;\beta} = u^\gamma{}_{;\beta} g_{\gamma \alpha} $$
That is, you may treat the covariant gradient as another tensor index behind which (and in front of which) indices can be raised and lowered freely.
You should understand the notation of MTW accordingly. The expression for the decomposition of the congruence gradient will hold no matter whether you put the $\alpha$ index "up" or "down", so you might as well put the index down. The reason why MTW do it is that then you can more directly examine the symmetries of the tensors you are decomposing into (e.g. shear $\sigma_{\alpha\beta}$ is symmetric in $\alpha\beta$, $\sigma_{\alpha \beta} = \sigma_{\beta \alpha}$, but vorticity $\omega_{\alpha \beta}$ is antisymmetric).
A: There are a couple of notations that are sometimes seen in tensor calculus: $u_{\alpha,\beta}$ and $u_{\alpha;\beta}$.
$u_{\alpha,\beta}$ (with a comma) stands for the partial derivative of $u_\alpha$ with respect to $x^\beta.$ This is often also written $\partial_\beta u_\alpha.$
$u_{\alpha;\beta}$ (with a semicolon) stands for the covariant derivative of $u_\alpha$ with respect to $x^\beta.$ This is often also written $D_\beta u_\alpha.$ Its relation to the partial derivative is
$$u_{\alpha;\beta} = u_{\alpha,\beta} - \Gamma_{\alpha\beta}^{\gamma}u_{\gamma},$$
where $\Gamma_{\alpha\beta}^{\gamma}$ are Christoffel symbols.
