= Christoffel symbols of the Vaidya metric
{c}
{title2=$\Gamma^\mu{}_{\nu\rho}$}
Writing $f=1-2M(v)/r$, the radial <Christoffel symbols> are $\Gamma^v{}_{vv}=M/r^2$, $\Gamma^r{}_{vv}=fM/r^2+M'/r$ 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 $\phi\phi$ terms, $\Gamma^\theta{}_{r\theta}=\Gamma^\phi{}_{r\phi}=1/r$, $\Gamma^\theta{}_{\phi\phi}=-\sin\theta\cos\theta$ and $\Gamma^\phi{}_{\theta\phi}=\cot\theta$. The lower indices are symmetric. The plus sign of $M'/r$ follows from the ingoing $+2dv\,dr$ cross term.
Back to article page
