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I am trying to make sense of this simple case in my book, but I am still baffled by the notation that is used in the indices with the commas and semicolons; I also do not understand how these are equivalent.

$$A_{\alpha;\beta}-A_{\beta;\alpha}=A_{\alpha,\beta}-A_{\beta,\alpha}$$

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    $\begingroup$ Please explain better the notation, or add a reference in which this notation is used, some (like me) may be not familiar with this notation $\endgroup$ – Fabio Di Nocera Mar 9 '20 at 14:03
  • $\begingroup$ Which book? Which page? $\endgroup$ – Qmechanic Mar 9 '20 at 14:08
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    $\begingroup$ The connection is presumably torsionfree. $\endgroup$ – Qmechanic Mar 9 '20 at 14:10
  • $\begingroup$ Sean Carroll-Spacetime and Geometry: An Introduction to General Relativity, 2019 edition, Chapter 3, Page 152. $\endgroup$ – UF6 Mar 9 '20 at 14:43
  • $\begingroup$ I think that the book probably explains the comma and semicolon notations somewhere, before page 152. $\endgroup$ – G. Smith Mar 9 '20 at 17:17
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By definition: \begin{align}\tag{1} A_{\mu,\nu} &\equiv \frac{\partial}{\partial x^{\nu}} \, A_{\mu} \equiv \partial_{\nu} \, A_{\mu}, \\[1ex] \tag{2} A_{\mu;\nu} &\equiv \frac{\partial}{\partial x^{\nu}} \, A_{\mu} - \Gamma_{\nu \mu}^{\lambda} \, A_{\lambda} \equiv \nabla_{\nu} \, A_{\mu},\end{align} If the connection is symetric: $\Gamma_{\mu \nu}^{\lambda} = \Gamma_{\nu \mu}^{\lambda}$ (Levi-Civita connection AKA Christoffel symbols, which means no torsion), then the antisymetric expression $A_{\mu;\nu} - A_{\nu;\mu}$ gives the same as $A_{\mu,\nu} - A_{\nu,\mu}$.

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    $\begingroup$ Just to be more precise, please don't take it as a critic, I completely understand the spirit of your answer. But the connection and the Christoffel symbols are two completely different objects, what you meant is that if you choose the Levi-Civita connection (as it is usually done in GR) then the Christoffel symbols are symmetric in the two "down" indices. $\endgroup$ – Fabio Di Nocera Mar 9 '20 at 14:17
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The semikolon indicates a covariant derivative, in General Relativity only covariant derivatives transform like a tensor, therefore it is so important.

The simple comma in difference indicates a simple derivative.

the difference between covariant derivative and simple derivative is:

$$A_{a;b} = A_{a,b} - \Gamma^c_{ba} A_c$$

where $\Gamma$ stands for the Christoffel-symbols. See a.o.: https://en.wikipedia.org/wiki/Christoffel_symbols

On the basis of this formula it can be easily shown that

$$A_{a;b} -A_{b;a} = A_{a,b} - A_{b,a}$$

because the Christoffel-symbols are symmetric on the 2. and 3. index (if the torsion is zero). Actually, in order to define a covariant derivative on a (pseudo)-Riemannian manifold an additional structure is required, the "connection". The Christoffel-symbols represent here the connection. However, for the computation of an exterior derivative of a one-form $A$ a connection is not required. Actually $ A_{a,b} - A_{b,a}$ represent the result of the exterior derivative of a one-form $A$ in components. The equality $A_{a;b} -A_{b;a} = A_{a,b} - A_{b,a}$ shows that an exterior derivative can be defined on a (pseudo)-Riemannian manifold without further structure introduced.

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