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In an intro to GR book the Ricci tensor is given as:

$$R_{\mu\nu}=\partial_{\lambda}\Gamma_{\mu \nu}^{\lambda}-\Gamma_{\lambda \sigma}^{\lambda}\Gamma_{\mu \nu}^{\sigma}-[\partial_{\nu}\Gamma_{\mu \lambda}^{\lambda}+\Gamma_{\nu \sigma}^{\lambda}\Gamma_{\mu \lambda}^{\sigma}]$$

I have gotten to the point where I can work out a given Christoffel symbol, but I am still having trouble working out the above tensor as a whole (just algebraically speaking). If I'm not mistaken, $R_{\mu\nu}$ should end up a $\mu$x$\nu$ (i.e. 4x4) matrix just like the energy-momentum tensor on the other side of the field equations. In the above rendering $\sigma$ is clearly a dummy index to be summed over, and I can see how $\lambda$ is also a dummy index in the first term. But the $\lambda$s in the other terms seem to be free indices, which would then introduce incompatible dimensions in the matrix operations. I appreciate it if someone can point out the error of my ways.

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  • $\begingroup$ Why did you think that the $\lambda$s are free? $\endgroup$
    – Ron Maimon
    Jul 24, 2012 at 2:17
  • $\begingroup$ Because I thought dummy indices had to be repeated in separate symbols. Now it's clear. $\endgroup$
    – ben
    Jul 24, 2012 at 4:18

1 Answer 1

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Every term contains one $\lambda$ in the superscript and one in the subscript, so you sum over those. The only indices which don't appear in both superscript and subscript in the same term are $\mu$ and $\nu$.

Example:

$$\Gamma_{\lambda\sigma}^\lambda\Gamma_{\mu\nu}^\sigma = \Gamma_{00}^0\Gamma_{\mu\nu}^0 + \Gamma_{01}^0\Gamma_{\mu\nu}^1 + \cdots + \Gamma_{10}^1\Gamma_{\mu\nu}^0 + \cdots$$

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    $\begingroup$ Note, all contraviariant/covariant indices should be written using LaTeX '\phantom' keyword $\Gamma^{\xi}_{\phantom{\lambda}\lambda\sigma}$; they should not be directly over eachother strictly speaking. Although it is a pain to write in this context... :] $\endgroup$
    – MoonKnight
    Jul 24, 2012 at 17:42
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    $\begingroup$ Yep, given that the point I want to make is clear without explicit positioning of the indices, I didn't bother with that. It's unfortunate that we don't have a tensor notation package for MathJax. $\endgroup$
    – David Z
    Jul 24, 2012 at 19:08
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    $\begingroup$ @Killercam +1 Thanks. That's good to know because the books I have looked at all position the Christoffel indices on top of each other, and I was wondering if they shouldn't rather be spaced in their own columns in keeping with the logic of, for eg., the Riemann tensor indices. $\endgroup$
    – ben
    Jul 25, 2012 at 0:41
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    $\begingroup$ @Killercam It's worth noting that plenty of authors do not offset the indices in the Christoffel symbols, as they are not actually tensors but rather just collections of coefficients. Any ambiguity in the lowered forms can be easily defined away, e.g. $\Gamma_{\lambda\mu\nu} = g_{\lambda\sigma} \Gamma^\sigma_{\mu\nu}$. In any event, the much more convenient way to TeX offset indices is to attach the offset ones to blank structures, as in \Gamma^\lambda{}_{\mu\nu}. Typesetting GR became significantly easier once I stopped using \phantom. ;) $\endgroup$
    – user10851
    Jul 31, 2012 at 10:20

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