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Curvature 1-forms in NP formalism

  1. Jul 15, 2014 #1
    Hey guys, I'm working on a summer research project right now in diff. geo. I'm at the point where I have to define the spin coefficients for my spacetime. I'm following an appendix in another paper related to my problem (the equivalence problem for 3D Lorentzian spacetimes).

    In the appendix I am using, it defines the spin coefficients in terms of the Christoffel symbols; however, it expresses the Christoffel symbols as curvature 1-forms, rather than connection coefficients, as we usually see in GR, Riemannian geo., etc.

    The equations I am using are of the form:
    [itex] \Gamma_{12} = \kappa \omega^1 + \sigma \omega^2 + \tau \omega^3 [/itex].

    Now, I have already calculated the connection coefficients (the [itex] \Gamma^a_{bc} [/itex]'s). As I understand it, this would mean that [itex] \Gamma^1_{12}=\kappa[/itex], [itex] \Gamma^2_{12} = \sigma [/itex], etc. (this is what my supervisor has told me). Is this true/does this make sense to anyone? I find it hard to believe that I can just move the triad vector over to the other side of the equation, combine its index with the [itex] \Gamma [/itex], and then magically have a Christoffel symbol of the second kind.

    Any help/guidance would be greatly appreciated! I think my problem lies in something I am missing involving algebra of tensor equations...?
  2. jcsd
  3. Jul 23, 2014 #2
    I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?
  4. Sep 9, 2014 #3


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    I think you might want to define your quantities a little better. E.g. what is ##\kappa##, ##\sigma## etc? I have never seen a direct relationship between Christoffel symbols and connection one forms. As the Christoffel symbols are symmetric in the lower 2 indices, and the connection one-forms are anti-symmetric in the non-tetrad indices, they don't contain the same number of independent components, so I don't know how this would work.
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