Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

I Confusion on Bianchi Identity proof

  1. Sep 21, 2016 #1
    This is from a general relativity book but I think this is the appropriate location.

    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.
     
  2. jcsd
  3. Sep 21, 2016 #2

    Orodruin

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    "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.)
     
  4. Sep 22, 2016 #3
    Thanks I think I had a mental block because of the index notation.
     
    Last edited: Sep 22, 2016
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Confusion on Bianchi Identity proof
  1. Third Bianchi identity (Replies: 2)

Loading...