## Tuesday, 6 August 2019

### Question

 A diagonal metric in 4-space
Imagine we had a diagonal metric $g_{\mu\nu}$. Show that the Christoffel symbols are given by
\begin{align}
\Gamma_{\mu\nu}^\lambda&=0&\phantom {10000}(1)\nonumber\\
\Gamma_{\mu\mu}^\lambda&=-\frac{1}{2}\left(g_{\lambda\lambda}\right)^{-1}\partial_\lambda g_{\mu\mu}&\phantom {10000}(2)\nonumber\\
\Gamma_{\mu\lambda}^\lambda&=\partial_\mu\left(\ln{\sqrt{\left|g_{\lambda\lambda}\right|}}\right)&\phantom {10000}(3)\nonumber\\
\Gamma_{\lambda\lambda}^\lambda&=\partial_\lambda\left(\ln{\sqrt{\left|g_{\lambda\lambda}\right|}}\right)&\phantom {10000}(4)\nonumber
\end{align}In these expressions, $\mu\neq\nu\neq\lambda$ and repeated indices are not summed over.

At last I have recovered from the grind of that geodesic on a sphere and completed this question. It involved a bit of index manipulation, some knowledge about the inverse of a diagonal matrix and the chain rule (taken slowly).

All three pages in Ex 3.03 Diagonal metric.pdf.