In Carroll, the author states:(adsbygoogle = window.adsbygoogle || []).push({});

[tex] \nabla^{\mu}R_{\rho\mu}=\frac{1}{2} \nabla_{\rho}R [/tex]

and he says "notice that, unlike the partial derivative, it makes sense to raise an index on the covariant derivative, due to metric compatibility."

I'm not seeing this very clearly :s

What's the reasoning behind this statement? And what does it even mean to raise an index on the covariant derivative? I'm thinking this is strange since the partial derivative is included in the covariant derivative, so raising an index on the covariant derivative would force you to raise an index on the partial as well.

(I'm not sure why this raising of an index on the partial derivative is so bad; maybe it's because it's non-tensorial. However, my initial response is to think it's bad cause the partial is [itex] \partial/\partial x^{\mu} [/itex], so raising an index on that doesn't really have such a clear meaning. But it seems like this would persist for the covariant derivative?)

I'm guessing this is simple, but clearly I'm lost haha, so any help would be appreciated :)

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# I Raising index on covariant derivative operator?

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