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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.

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    $\begingroup$ Why do you think so? $\endgroup$ – Nihar Karve Jun 6 at 12:43
  • $\begingroup$ @Nihar Karve because $\nabla u$ is vector valued one form. One down index for the form and one up for the vector $\endgroup$ – amilton moreira Jun 6 at 12:45
  • $\begingroup$ $u_\alpha$ is just $u^\alpha$ with index lowered, and the derivative commutes with the metric. The equation you wrote is probably simpler with the $\alpha$ downstairs, because you can define these two-forms. $\endgroup$ – Eric David Kramer Jun 6 at 12:46
  • $\begingroup$ @Eric David Kramer could you elaborate your comment as an answer? $\endgroup$ – amilton moreira Jun 6 at 12:48
  • $\begingroup$ $u_\alpha$ is a covector, not a vector. $\endgroup$ – mike stone Jun 6 at 12:52
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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).

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  • $\begingroup$ Isn't $g_{\alpha\beta;\gamma}=0$ a property of the Christoffel connection specifically? I haven't done any GR recently so forgive me if I'm wrong. $\endgroup$ – AccidentalTaylorExpansion Jun 7 at 12:34
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    $\begingroup$ @AccidentalTaylorExpansion In GR, the covariant gradient (unless specified otherwise) corresponds to a Levi-Civita connection. This all connects to the structure of GR as built from the Einstein equivalence principle and the metric being the approximately constant Minkowski metric in the Local Inertial Frame. $\endgroup$ – Void Jun 7 at 14:08
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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.

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