 Radial null geodesics of the Vaidya metric

The two radial null geodesic families are $v=\text{constant}$ and $dr/dv=(1-2M(v)/r)/2$. The first has affine radius, while the second has tangent $k={\partial }_v+(f/2){\partial }_r$ obeying ${\nabla }_{k}k=(M/r^2)k$, so advanced time is generally a nonaffine parameter. An affine parameter $\lambda$ for the outgoing family satisfies $d\log (dv/d\lambda )/dv=-M/r^2$.