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How does one properly evaluate something of the form $$ g_{a}^{\, [b} R_{c] b}~? $$ when I try to expand using the definition of anti-symmetrization brackets the Einstein summation seems to break: $$ g_{a}^{\, [b} R_{c] b} = \frac{1}{2} \bigg[ g_{a}^{\, b} R_{c b} - g_a^{\,c} R_{bb}\bigg] $$ the second term should be summing over the $b$ index except now I have two lower indices which breaks convention. Is there some implied index raising that's built into the anti-symmetrization brackets? Because I don't see any reason why I should have introduced a metric factor into the second term. It also seems a little fishy since $b$ is a dummy index

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  • $\begingroup$ Is this from a book? $\endgroup$ – Qmechanic Mar 22 at 4:14
  • $\begingroup$ @Qmechanic I'm trying to compute the contraction of the Weyl tensor $C_{abc}^{\quad b}$ where $C_{abcd} = R_{abcd} + A(g_{a[d} R_{c] b} + g_{b[c} R_{d] a} ) + B R g_{a [ c} g_{d ] b}$, it's from my course notes $\endgroup$ – David Feng Mar 22 at 4:16
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When contracting $b$ with $d$, you should have raised the index that was not being covariantly antisymmetrized. For example, instead of the term you showed, write it as $g_{a[b}{R_{c]}}^b$.

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    $\begingroup$ Ah thank you, that makes sense. In general can we only antisymmetrize indices that are of the same type and is there any deeper mathematical reason behind this? $\endgroup$ – David Feng Mar 22 at 5:29
  • $\begingroup$ Yes, you can only symmetrize or antisymmetrize a set of covariant indices or a set of contravariant indices. Otherwise you get the kind of nonsense that you encountered where free indices don’t have matching placement and contractions aren’t between an upper and a lower index as they must be to maintain Lorentz covariance. $\endgroup$ – G. Smith Mar 22 at 5:45
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You cannot (anti) symmetrize indices that belong to different spaces. For example, if you try to symmetrize a mixed tensor like $A^\mu_{\ \nu}$ you have to swap $\mu$ and $\nu$ somehow, but that produces a new tensor with the wrong index placements. The original tensor $A^\mu_{\ \nu}$ had $\mu$ on top and $\nu$ at bottom; trying to symmetrize these generates a new tensor with opposite index placements.

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