Curvature 1-forms in NP formalism

[Quality Cheese]

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:
$\Gamma_{12} = \kappa \omega^1 + \sigma \omega^2 + \tau \omega^3$.

Now, I have already calculated the connection coefficients (the $\Gamma^a_{bc}$'s). As I understand it, this would mean that $\Gamma^1_{12}=\kappa$, $\Gamma^2_{12} = \sigma$, 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 $\Gamma$, 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...?

 

[Greg Bernhardt]

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?

 

[Matterwave]

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.