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I know the covariant derivative of a tensor is $$\nabla_{\mu} V_{\nu}=\partial_\mu V_\nu-\Gamma_{\mu\nu}^{\lambda}V_{\lambda}$$ Now I want to obtain $\nabla_{\mu}\nabla_{\nu}\Phi(x)$ where $\Phi(x)$ is a scalar field. I consider $ V_\nu=\nabla_{\nu}\Phi(x)$ and then $$\nabla_{\mu}\nabla_{\nu}\Phi(x)=\partial_\mu \partial_\nu \Phi(x)-\Gamma_{\mu\nu}^{\lambda}\partial_\lambda \Phi(x)$$

Is this correct? If so, how can I interpret $\nabla_{\mu}\nabla_{\nu}\Phi(x)$?

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  • $\begingroup$ yes, it is correct. $\endgroup$
    – Prahar
    Commented Jun 12, 2022 at 15:51
  • $\begingroup$ I can not consider$∇μ∇νΦ(x)=∂μ∂νΦ(x)$?can I? $\endgroup$
    – Alice
    Commented Jun 12, 2022 at 16:00
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    $\begingroup$ No. The first derivative $\nabla_\nu$ is acting on a scalar so we can replace $\nabla_\nu \to \partial_\nu$, but the second one is acting on a tensor. $\endgroup$
    – Prahar
    Commented Jun 12, 2022 at 16:02
  • $\begingroup$ thank you very much. $\endgroup$
    – Alice
    Commented Jun 12, 2022 at 16:03
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    $\begingroup$ The question in your title is different from the one you're asking. Do you want to know how to interpret $\nabla_\mu\nabla_\nu\phi(r)$ or do you want to if your last statement is correct? $\endgroup$
    – AccidentalTaylorExpansion
    Commented Jun 12, 2022 at 16:03

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