This is from a general relativity book but I think this is the appropriate location.(adsbygoogle = window.adsbygoogle || []).push({});

The proof that [tex]

\nabla_{[a} {R_{bc]d}}^e=0[/tex]

is as follows:

Choose coordinates such that [itex] \Gamma^a_{bc}=0[/itex] at an event. We have [tex]

\nabla_a {R_{bcd}}^e = \partial_a \partial_b \Gamma^e_{cd} - \partial_a \partial_c \Gamma^e_{bd} + \textrm{ terms in } \Gamma \partial \Gamma \textrm{ and } \Gamma \Gamma \Gamma.[/tex]

Because the first term on the right-hand side is symmetric in [itex] ab[/itex] and the second in [itex] ac [/itex], and because the other terms vanish at the event, we have [tex]

\nabla_{[a}{R_{bc]d}}^e=0[/tex]

at the event in this coordinate system. However, this is a tensor equation, so it is valid in every coordinate system.

The bit I'm unsure of is how ''the other terms'' vanish at the event, so help would be appreciated.

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# I Confusion on Bianchi Identity proof

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