Christoffel symbols of the Vaidya metric (${\Gamma }^{\mu }{}_{\nu \rho }$)

Writing $f=1-2M(v)/r$, the radial Christoffel symbols are ${\Gamma }^v{}_{vv}=M/r^2$, ${\Gamma }^r{}_{vv}=fM/r^2+M^{\prime }/r$ and ${\Gamma }^r{}_{vr}=-M/r^2$. The angular couplings are ${\Gamma }^v{}_{\theta \theta }=-r$, ${\Gamma }^r{}_{\theta \theta }=-fr$, with ${\sin }^2\theta$ factors for the corresponding $\varphi \varphi$ terms, ${\Gamma }^{\theta }{}_{r\theta }={\Gamma }^{\varphi }{}_{r\varphi }=1/r$, ${\Gamma }^{\theta }{}_{\varphi \varphi }=-\sin \theta \cos \theta$ and ${\Gamma }^{\varphi }{}_{\theta \varphi }=\cot \theta$. The lower indices are symmetric. The plus sign of $M^{\prime }/r$ follows from the ingoing $+2dvdr$ cross term.