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- - **Quick question to clear up some confusion on Riemann tensor and contraction**
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Quick question to clear up some confusion on Riemann tensor and contractionLet's say I want to calculate the Ricci tensor, [tex]R_{bd}[/tex], in terms of the contractions of the Riemann tensor, [tex]{R^a}_{bcd}[/tex]. There are two definitions of the Riemann tensor I have, one where the [tex]a[/tex] is lowered and one where it is not, as above.
To change between the two all that I have ever seen written is 'we lower the indices' but I don't think I fully understand this. Does this mean... [tex]R_{abcd} = g_{aa} {R^a}_{bcd}[/tex] So the answer to my original question of finding the Ricci tensor is... [tex]R_{bd} = g^{ac} g_{aa} {R^a}_{bcd}[/tex] Following this, I also have written before me that... [tex]{R^b}_{bcd} = 0[/tex] since [tex]R_{abcd}[/tex] is symmetric on a and b. Shouldn't this be antisymmetric on a and b? Sorry if these are basic questions but I'm finding the vagueness of 'lowering the indices' a bit confusing... Cheers. |

Re: Quick question to clear up some confusion on Riemann tensor and contractionQuote:
[tex]R_{abcd} = g_{ae} {R^e}_{bcd}[/tex] |

Re: Quick question to clear up some confusion on Riemann tensor and contractionThe Riemann tensor
[tex]R^{\alpha}_{\;\beta\gamma\delta}=g^{\alpha\lambda}R_{\lambda\beta\gamma \delta}[/tex]. The raising and lowing of indices is done using the fundamental metric tensor [itex]g_{\alpha\beta}[/itex] and the inverse metric [itex]g^{\alpha\beta}[/itex]. The Ricci tensor [tex]R^{\alpha}_{\;\beta\gamma\delta}\rightarrow R^{\alpha}_{\;\beta\alpha\delta}=R_{\beta\delta}[/tex] is a contraction of the first and third indices. |

Re: Quick question to clear up some confusion on Riemann tensor and contractionI would argue that [tex]
R^{\alpha}_{\;\beta\gamma\delta} [/tex] is the more fundamental expression for the Riemann tensor since this is defined from the derivative operator [without reference to a metric]. Its trace yields the Ricci tensor [as jfy4 wrote]... and this too doesn't make use of a metric. To lower the upper index of Riemann or to get the scalar-curvature from Ricci now requires a metric. |

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