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Source: Thomas Moore's A General Relativity Workbook

In Moore's "diagonal metric worksheet" he doesn't explain his process of determining the "only possible non zero components" of the Ricco tensor, i.e why there are zeros included a priori. I'm assuming one could prove that certain components are always zero by taking a metric, finding the Christoffel symbols, using the definition of the Riemann tensor which takes the Christoffel symbols into account, and then determining which components of the Ricci tensor are always zero? Does that follow?

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  • $\begingroup$ The scan doesn’t include the “only possible non-zero components” quote that you are asking about. $\endgroup$
    – G. Smith
    Commented Jul 18, 2019 at 17:51
  • $\begingroup$ "only possible non-zero components" is implied from the zeros included in the computation of the Ricci component $\endgroup$
    – Lopey Tall
    Commented Jul 18, 2019 at 19:38
  • $\begingroup$ You seem to be confusing certain terms in a Ricci component being zero with the Ricci component itself (the sum of those terms) being zero. The various terms come from derivatives and products of Christoffel symbols. For a diagonal metric, many Christoffel symbols are zero, so that fact that many of these terms are zero is not surprising. Their sum is not zero. Perhaps this is just confusion over terminology. You must not refer to the individual terms as “components of the Ricci tensor” or no one will understand what you are talking about. $\endgroup$
    – G. Smith
    Commented Jul 18, 2019 at 20:24
  • $\begingroup$ If you do not understand why some Christoffel symbols vanish for a diagonal metric, think about the symbols where the three indexes are all different. $\endgroup$
    – G. Smith
    Commented Jul 18, 2019 at 20:26
  • $\begingroup$ Ah, yes yes yes. Ok I believe you confirmed my suspicion. I just did some trial calculations and it is indeed the case that all diagonal metrics yield some Christoffel symbols that are zero. And hence, since Christoffel symbols are used to compute the Riemann tensor, and the Riemann tensor is used to compute the Ricci tensor, some "terms in the sum of each components of the Ricci tensor" will a priori be zero. $\endgroup$
    – Lopey Tall
    Commented Jul 19, 2019 at 0:19

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For a diagonal metric, every Christoffel symbol where all three indices are different vanishes. This means that some of the terms in each Ricci component vanish, in the case of a diagonal metric, since Riemann and Ricci are constructed from derivatives and products of Christoffels. This does not mean that any entire Ricci component vanishes... just some of the terms that contribute to it.

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  • $\begingroup$ Nice answer (+1) Do you know of any solution to the EFE where $R$ (Ricci scalar) is non-zero? (I have read that $R=0$ in Schwarzchild metric, gravitational wave, black hole other than centermost singularity.) $\endgroup$
    – James
    Commented Aug 5 at 8:28

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