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[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 Rab[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.)
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 Rab[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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