# Confusion about convention for curvature tensor

I am a little bit confused about the convention of the curvature tensor. The books of Wald and Misner/Deser/Wheeler seem to have the same conventions, i.e. the indices of the Riemann curvature tensor are defined by

$${R_{\alpha\beta\gamma}}^{\delta}v_{\delta}=(\nabla_{\alpha}\nabla_{\beta}-\nabla_{\beta}\nabla_{\alpha})v_{\gamma}$$

and the Ricci tensor is defined by the contraction

$$R_{\alpha\beta}:={R_{\alpha\gamma\beta}}^{\gamma}={R^{\gamma}}_{\alpha\gamma\beta}.$$

However, the expression of the Ricci tensor in coordinates seems to be different:

$$R_{\alpha\beta}=\partial_{\lambda}\Gamma^{\lambda}_{\alpha\beta}-\partial_{\alpha}\Gamma^{\lambda}_{\beta\lambda}+\Gamma^{\lambda}_{\alpha\beta}\Gamma^{\mu}_{\lambda\mu}-\Gamma^{\lambda}_{\alpha\mu}\Gamma^{\mu}_{\beta\lambda}$$ $$R_{\alpha\beta}=\partial_{\lambda}\Gamma^{\lambda}_{\alpha\beta}-\partial_{\beta}\Gamma^{\lambda}_{\alpha\lambda}+\Gamma^{\lambda}_{\alpha\beta}\Gamma^{\mu}_{\lambda\mu}-\Gamma^{\lambda}_{\alpha\mu}\Gamma^{\mu}_{\beta\lambda}.$$

The first one is taken from Wald and the second one from Misner/Deser/Wheeler. The only difference is in the ordering of $$\alpha$$ and $$\beta$$ in the second term. Does anyone know why? Am I missing something?

• The second term is actually symmetric in $\alpha$ and $\beta$ so the ordering is irrelevant. Nov 29, 2022 at 8:49

The expressions are same, since $$\Gamma^{\lambda}_{\alpha\lambda}=\partial_{\alpha}\log\sqrt{|g|}$$ and that the ordering of $$\partial_{\alpha}\partial_{\beta}$$ acting on something doesn't matter as long as they are continuous.