After my studies of metric tensors and Cristoffel symbols, I decided to move on to the Riemann tensor and the Ricci curvature tensor. Now I noticed that the Einstein Field Equations contain the Ricci curvature tensor (R(adsbygoogle = window.adsbygoogle || []).push({}); _{[itex]\mu\nu[/itex]}).

Some sources say that you can derive this tensor by simply deriving the Riemann tensor using the commutator:

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

However, it seems to me (and to some other sources) that this would derive R^{a}_{b[itex]\nu[/itex][itex]\mu[/itex]}which in turn could contract to R_{[itex]\nu[/itex][itex]\mu[/itex]}rather than R_{[itex]\mu[/itex][itex]\nu[/itex]}.

The Einstein field equations involve R_{[itex]\mu[/itex][itex]\nu[/itex]}rather than R_{[itex]\nu[/itex][itex]\mu[/itex]}.

If you are trying to work with Einstein's equations, then wouldn't you have to do the commutator:

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

instead of the previous one that I mentioned and derive R_{[itex]\mu[/itex][itex]\nu[/itex]}instead? (Especially since every other tensor in the equations involve the indicies in the order [itex]\mu[/itex][itex]\nu[/itex] instead of the other way around.)

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# Question about Riemann and Ricci Curvature Tensors

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