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I want to compute the $\Gamma^a\vphantom{K}_{bc}$ in the case with torsion and verify that is equal to

$${\widetilde{\Gamma}^a\vphantom{\Gamma}_{bc}}+S^a\vphantom{S}_{bc}+S_{bc}\vphantom{S}^a+S_{cb}\vphantom{S}^a\tag{1}$$

where $\widetilde{\Gamma}^a\vphantom{\Gamma}_{bc}$ are the Christoffel symbol, while $S^a\vphantom{S}_{bc}= \Gamma^a\vphantom{S}_{[bc]} $. I refer to this article: https://arxiv.org/abs/1611.07878.

I start by the metricity condition:

$$\nabla_dg_{bc}=\partial_dg_{bc}-{\Gamma}^f\vphantom{\Gamma}_{cd}g_{fb}-{\Gamma}^f\vphantom{\Gamma}_{cb}g_{df}=0,$$

$$\Gamma_{dcb}+\Gamma_{bcd}=\partial_cg_{db}$$ $$\Gamma_{cdb}+\Gamma_{bdc}=\partial_dg_{cb}$$ $$\Gamma_{cbd}+\Gamma_{dbc}=\partial_bg_{dc}$$

I add the first and the third and subtract the second and I rais the indice with the metric obtaining:

$$ \frac{1}{2}g^{ad}(\partial_cg_{db}+\partial_bg_{dc}-\partial_dg_{cb})=\frac{1}{2}(\Gamma^a\vphantom{\Gamma}_{cb}+\Gamma_{bc}\vphantom{\Gamma}^a- \Gamma_{c}\vphantom{\Gamma}^a\vphantom{\Gamma}_{b} - \Gamma_{b}\vphantom{\Gamma}^a\vphantom{\Gamma}_{c} + \Gamma_{cb}\vphantom{\Gamma}^a + \Gamma^a\vphantom{\Gamma}_{bc})$$

and from this I obtain:

$$ \Gamma^a\vphantom{\Gamma}_{bc} = \widetilde{\Gamma}^a\vphantom{\Gamma}_{bc}+S^a\vphantom{S}_{bc}+S_{b}\vphantom{S}^a\vphantom{S}_c+S_{c}\vphantom{S}^a\vphantom{S}_b$$

that's different from the equation above. Where am I wrong?

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Try $$ \nabla_dg_{bc}=\partial_dg_{bc}-{\Gamma}^f\vphantom{\Gamma}_{cd}g_{fb}-{\Gamma}^f\vphantom{\Gamma}_{bd}g_{cf}=0, $$

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  • $\begingroup$ Ok, now it works. The difference between my definition of covariant derivative and yours is only a convention? $\endgroup$ May 16 '17 at 21:52
  • $\begingroup$ @raskolnikov No, it is not a convention, it is conceptual because the equation is not a definition, it follows from the definition of covariant derivative and Christoffel symbols. The subscript index $d$ should be present everywhere in the Christoffel symbols, and not as an index of the metric tensor, like you have in the last term. $\endgroup$ May 17 '17 at 3:07
  • $\begingroup$ Ok, what i mean is the following: I mistyped the covariant differentiation above, but i've in wald book (general relativity) a different definition of christoffel symbols, where the "d" is not the right indice but the central ones. There is a difference between wald definition of christoffel symbols and the above definition? $\endgroup$ May 17 '17 at 6:21

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