March 10, 2009
As we know since long Evans has some problems with the derivation of the first Bianchi identity. See p.4, no.4 of his papernote 12/118. His problem: He has forgotten all about the Poincaré ''Lemma'' d Ù d = 0. So he accuses S.M. Carroll of not having given a proof of the first Bianchi identity Eq. (3.140) in his Lecture Notes on GR. (And besides, in addition he completely denies the valitity of the second Bianchi identity.) The (straight forward) proofs missing in Carroll's L.N. were left to the reader.
Therefore last year I wrote a note where both (straight forward) proofs were given using Cartan's exterior calculus exclusively as known from literature.