# I Confusion on Bianchi Identity proof

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1. Sep 21, 2016

### Dazed&Confused

This is from a general relativity book but I think this is the appropriate location.

The proof that $$\nabla_{[a} {R_{bc]d}}^e=0$$
is as follows:

Choose coordinates such that $\Gamma^a_{bc}=0$ at an event. We have $$\nabla_a {R_{bcd}}^e = \partial_a \partial_b \Gamma^e_{cd} - \partial_a \partial_c \Gamma^e_{bd} + \textrm{ terms in } \Gamma \partial \Gamma \textrm{ and } \Gamma \Gamma \Gamma.$$

Because the first term on the right-hand side is symmetric in $ab$ and the second in $ac$, and because the other terms vanish at the event, we have $$\nabla_{[a}{R_{bc]d}}^e=0$$
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.

2. Sep 21, 2016

### Orodruin

Staff Emeritus
"The other" terms have the Christoffel symbols as a factor and the Christoffel symbols are all zero because of the choice of coordinates. (You can always find such coordinates for any given event.)

3. Sep 22, 2016

### Dazed&Confused

Thanks I think I had a mental block because of the index notation.

Last edited: Sep 22, 2016