Past exam of the relativity course of the University of Cambridge / 2017 / 1 / a / Solution

Use the Christoffel symbol formula with its first index raised:The diagonal metric tensor has $g_{rr}=e^{2\chi }$ and $g_{\theta \theta }=r^2{e}^{2\chi }$, with inverse components $g^{rr}=e^{-2\chi }$ and $g^{\theta \theta }=r^{-2}{e}^{-2\chi }$. Since ${\partial }_r\chi ={\chi }^{\prime }$ and all angular derivatives vanish,For example, ${\Gamma }^r{}_{\theta \theta }=-\frac{1}{2}{e}^{-2\chi }{\partial }_r(r^2{e}^{2\chi })$. The other components vanish. The formula supplied in the paper gives the first-kind coefficients ${\Gamma }_{ijk}$; multiplication by the inverse metric tensor is needed for the requested second-kind coefficients. These polar coordinates are understood away from $r=0$.