Tuesday, 6 August 2019

Exercise 3.03 The Christoffel symbols with a diagonal metric

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.

Answer

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.

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