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Shirish

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I'm reading Carroll's GR notes and I'm having trouble deciphering a particular expression for the Riemann curvature tensor. The coordinate-free definition is (eq. 3.71 in the notes): $$R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$$ An index-based expression is also given in (eq. 3.66): $$R^{\rho}_{\ \ \sigma\mu\nu}V^{\sigma}=[\nabla_{\mu},\nabla_{\nu}]V^{\rho}+T_{\mu\nu}^{\ \ \ \ \lambda}\nabla_{\lambda}V^{\rho}$$ How do I reconcile these two equations? My attempt so far is as follows: in the first equation (eq. 3.71), I can replace ##X,Y## by fields ##\partial_{\mu},\partial_{\nu}## respectively and ##Z## by ##V##, then I can get the local coordinates for both sides by acting them on ##x^{\rho}##, i.e. (AFAIK to make notation shorter, expressions like ##\nabla_{\partial_{\mu}}## are written as ##\nabla_{\mu}##): $$R(\partial_{\mu},\partial_{\nu})(V)(x^{\rho})=[\nabla_{\mu},\nabla_{\nu}]V(x^{\rho})-\nabla_{[\partial_{\mu},\partial_{\nu}]}V(x^{\rho})$$ So now we can compare this to (eq. 3.66) - the LHS term is the coordinate free version of LHS in eq. 3.66, and the first RHS term is the coordinate free version of the first RHS term in eq. 3.66. Although I have doubts about the 1st RHS term since it's ##([\nabla_{\mu},\nabla_{\nu}]V)(x^{\rho})##, which I guess may not be the same as ##[\nabla_{\mu},\nabla_{\nu}](V(x^{\rho}))=[\nabla_{\mu},\nabla_{\nu}]V^{\rho}##.

The main trouble is with the second term on the RHS. I've tried to reconcile 2nd RHS terms in both the coordinate-free and index-based versions, but no luck. I'd appreciate any help or corrections!

The main trouble is with the second term on the RHS. I've tried to reconcile 2nd RHS terms in both the coordinate-free and index-based versions, but no luck. I'd appreciate any help or corrections!

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