Hey guys! I am considering a space with a diagonal metric, which is maximally symmetric.(adsbygoogle = window.adsbygoogle || []).push({});

It can be proven that in that case of a diagonal metric the following equations for the Christoffel symbols hold:

[tex] \Gamma^{\gamma}_{\alpha \beta} = 0 [/tex]

[tex] \Gamma^{\beta}_{\alpha \alpha} = -(1/g_{\beta\beta})\partial_{\beta}g_{\alpha\alpha} [/tex]

[tex] \Gamma^{\beta}_{\alpha \beta} = \partial_{\alpha}\ln(\sqrt{|g_{\beta\beta}|}) [/tex]

[tex] \Gamma^{\alpha}_{\alpha \alpha} = \partial_{\alpha}\ln(\sqrt{|g_{\alpha\alpha}|}) [/tex]

Furthermore: for a maximally symmetric space we have for the Riemann tensor:

[tex] R_{\rho\sigma\mu\nu} = R/12(g_{\rho\mu}g_{\sigma\nu}-g_{\rho\nu}g_{\sigma\mu})[/tex], where R is the Ricci scalar.

Given these equations, I come accross a contradiction. From the above equation for the Riemann tensor we easily see that if it has three different indices, it must be zero (IF the metric is diagonal). However, if I plug in the Christoffelsymbols into the definition of the Riemanntensor expressed in Christoffel symbols and their derivatives, I do not find that this is zero in general. Does anyone know what is going wrong here?

EDIT: For a maximally symmetric space, three indices cannot be unequal, but for a diagonal space this might not be the case. How is this all related?

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# Space with a diagonal metric

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