Suppose you have $$\nabla_\alpha\nabla_\beta\nabla_\mu\nabla^\mu f$$ with $f$ a scalar function. Suppose that you know the expression for $\nabla_\mu\nabla^\mu f$. Is it possible to do calculate the 4 covariant derivative expression as follows?

$$\nabla_\alpha\nabla_\beta\nabla_\mu\nabla^\mu f=(\partial_\alpha\partial_\beta-\Gamma^\eta_{\alpha\beta}\partial_\eta)\nabla_\mu\nabla^\mu f.$$


closed as off-topic by Kyle Kanos, heather, honeste_vivere, sammy gerbil, ACuriousMind Sep 10 '17 at 1:09

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  • 2
    $\begingroup$ Yes. That's correct, because $\nabla_\mu\nabla^\mu f$ is a scalar. $\endgroup$ – Immanuel Sep 3 '17 at 20:07
  • $\begingroup$ Welcome to Physics! Please note that homework-like questions and check-my-work questions are generally considered off-topic here. We intend our questions to be potentially useful to a broader set of users than just the one asking, and prefer conceptual questions over those just asking for a specific computation. $\endgroup$ – ACuriousMind Sep 10 '17 at 1:09