I've been studying the Einstein field equations. I learned that the Ricci curvature tensor was expressed as the following commutator:(adsbygoogle = window.adsbygoogle || []).push({});

[∇_{[itex]\nu[/itex]}, ∇_{[itex]\mu[/itex]}]

I know that these covariant derivatives are being applied to some vector(s).

What I don't know however, is whether or not both covariant derivatives are being applied to the same vector and in the same frame of reference. Is this the case or not? Also, does the evaluation of both covariant derivatives need to use the same summation variables or can each covariant derivative involve different summation terms? If you do have to use the same summation terms for each covariant derivative, do all indices have to switch places when you do the other covariant derivative (ie. If one covariant derivative involves ∂V_{[itex]\mu[/itex]}/∂y^{[itex]\nu[/itex]}, then does the other have to have ∂V_{[itex]\nu[/itex]}/∂y^{[itex]\mu[/itex]})? Sorry if the notation is off or if the equation looks weird. This is my first time making a thread on this forum (as I am new here) so it is my first time typing an equation like this.

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# Ricci Curvature Tensor

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