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?

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Space with a diagonal metric

Loading...

Similar Threads - Space diagonal metric | Date |
---|---|

I How can General Relativity explain the Moon drifting apart from Earth | Mar 3, 2018 |

A Lense-Thirring effect - General Relativity | Feb 27, 2018 |

I Space contraction? | Feb 18, 2018 |

B 2D space embedded in 3D space | Feb 10, 2018 |

I Non-diagonal metric for testing elliptic PDE solver | Jun 22, 2017 |

**Physics Forums - The Fusion of Science and Community**