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I am taking a course on General Relativity which is more mathematically oriented than physically, and in one of the lecture notes (discussing Riemann tensor, curvature and all that) this identity is stated without proof, and I just can't see how it could be proven. The Identity:

$\Delta \operatorname{grad} f-\operatorname{grad} \Delta f=-(\operatorname{Ric}(\operatorname{grad} f, \cdot))^{\#}$.

Any pointers or resources are highly appreciated. Thanks.

Joel
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  • I know this is not beautiful, but direct calculation in a free-falling frame (Christoffel symbols vanish) should work. – Paweł Czyż May 11 '20 at 08:00
  • @PawełCzyż I was trying to do that, but it got ugly real fast so I thought maybe there is a better or less messy approach. – Joel May 11 '20 at 08:27
  • @PawełCzyż why is it correct to assume that the Christoffel symbols vanish? – Joel May 11 '20 at 08:36
  • I haven't read in detail, but this looks promising. – Paweł Czyż May 11 '20 at 08:38
  • Considering your question about Christoffel symbols: https://math.stackexchange.com/questions/762270/christoffel-symbols-vanish-in-a-system-of-normal-coordinates (Note that Christoffel symbols vanish only at a given point, in particular their derivatives usually will not be zero). – Paweł Czyż May 11 '20 at 08:39
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    In abstract index notation this is very straightforward - just start with the Ricci identity applied to $\nabla^i f$ and contract with the metric. In other notations the core point will be the same, but complicated by the technicalities of dealing with coordinates or frames. – Anthony Carapetis May 11 '20 at 08:40
  • @PawełCzyż I will check that out, thanks. – Joel May 11 '20 at 08:52
  • @AnthonyCarapetis I think I see what you mean, I just need to do the calculations, Thanks for the suggestion. – Joel May 11 '20 at 08:53

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