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Those darn symbols and their affine connections!http://mathworld.wolfram.com/ChristoffelSymboloftheSecondKind.html
The first term is the first cyclic form of the Riemann tensor iswhich has been well-established now it has antisymmetric indices. It also has to be true that the following commutation properties holdThat is the time deriative anti-commutes with the rest of the space derivatives. Is a form of the Bianchi identity. It may be a good question to ask, whether the first and last Riemann terms have commutation between the metric term at the end. We'll work that out in time. We seem to have ''partially'' created a quantum Bianchi identity by understanding the commutation written in the formSo question is, what notation do we use to express commutation on the first and last cyclic Riemann terms? This serves as a lead forward.
Quote from: Dubbelosix on 06/09/2017 23:02:10It seems I have done this last Bianchi identity incorrectly ... when you do it right (I mixed up the last indice wrong) then you do not get commutation with and - another good reason why double checking your indices is a good idea. The form this Bianchi identity takes isSo be mindful of this mistake and that it has been fixed.
Now we can move on to the three important identities we looked at and they will give a quantized look at the identities. You can write these three relationships out in the Bianchi identity, we can write the commutation again, on the indicesThe order of the commutation with respect to indices we have looked at have revealed their commutation in the last two indices.