I have a covector $\eta_\mu$. Then I have some notation which says $$\eta_{\alpha;\beta\gamma}$$ What does this mean? I understand that given a vector $A^\alpha$, that $$A^\alpha_{;\beta}=\nabla_\beta A^\alpha=(\nabla A)^\alpha_\beta$$This makes sense to me. However I do not understand what happens when two indices follow the semi-colon. If I had to guess, I would go with $$\eta_{\alpha;\beta\gamma}=\nabla_\beta\nabla_\gamma\eta_\alpha$$But I am not sure of this.
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4$\begingroup$ @JohnDoe your guess is almost correct. Actually, $\eta_{\alpha;\beta \gamma} = \eta_{\alpha;\beta;\gamma} = \nabla_{\gamma} \nabla_{\beta} \eta_{\alpha}$. Your result is only the same in the flat spacetime, where the Riemann tensor (which is related to the commutator of covariant derivatives) vanishes. $\endgroup$– Prof. LegolasovFeb 21, 2018 at 8:56
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2$\begingroup$ @SolenodonParadoxus So you're saying that I can think of it like this: after a semi-colon is introduced in one of the indices, everything that follows also has an invisible semi-colon in front of it. Is this correct? $\endgroup$– John DoeFeb 21, 2018 at 18:09
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6$\begingroup$ @SolenodonParadoxus That should be posted as an answer. $\endgroup$– Emilio PisantyFeb 21, 2018 at 18:19
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1$\begingroup$ @EmilioPisanty this question was put on hold earlier, which is when that comment was made (But yes, I do agree with you) $\endgroup$– John DoeFeb 22, 2018 at 0:31
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$\begingroup$ @JohnDoe yes, that is correct. $\endgroup$– Prof. LegolasovFeb 22, 2018 at 3:49
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Actually, $$\eta_{\alpha;\beta \gamma} = \eta_{\alpha;\beta;\gamma} = \nabla_{\gamma} \nabla_{\beta} \eta_{\alpha}$$ Your result is only the same in the flat spacetime, where the Riemann tensor (which is related to the commutator of covariant derivatives) vanishes.