# How is semicolon derivative notation defined for multiple derivatives?

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.

• @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. Feb 21, 2018 at 8:56
• @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? Feb 21, 2018 at 18:09
• @SolenodonParadoxus That should be posted as an answer. Feb 21, 2018 at 18:19
• @EmilioPisanty this question was put on hold earlier, which is when that comment was made (But yes, I do agree with you) Feb 22, 2018 at 0:31
• @JohnDoe yes, that is correct. Feb 22, 2018 at 3:49

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.