Spacetime and Geometry · Problem 3

Gravitation — Sean Carroll

Physics — General Relativity · christoffel-symbols-diagonal-metric

Problem

Imagine we have a diagonal metric

g_{\mu\nu}

. Show that the Christoffel symbols are given by

\begin{aligned} \Gamma^\lambda_{\mu\nu} &= 0 \tag{3.213}\\ \Gamma^\lambda_{\mu\mu} &= -\tfrac{1}{2}(g_{\lambda\lambda})^{-1}\partial_\lambda g_{\mu\mu} \\ \Gamma^\lambda_{\mu\lambda} &= \partial_\mu\left(\ln\sqrt{|g_{\lambda\lambda}|}\right) \\ \Gamma^\lambda_{\lambda\lambda} &= \partial_\lambda\left(\ln\sqrt{|g_{\lambda\lambda}|}\right) \end{aligned}

In these expressions,

\mu \neq \nu \neq \lambda

, and repeated indices are not summed over.