Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 54 / 2 / a / Solution

For a curve with fixed $v,\theta ,\varphi$, its tangent is proportional to ${\partial }_r$. The Vaidya metric has $g_{rr}=0$, so the curve is a null curve. Moreover, the Christoffel symbols with two lower $r$ indices vanish:because $g_{vr}=1$ is constant and every other $g_{br}$ is zero. Thus the covariant derivative along this tangent gives ${\nabla }_{{\partial }_r}{\partial }_r=0$. The geodesic equation is satisfied with $r$ as an affine parameter; choosing future orientation gives the tangent $k=-{\partial }_r$ and an affine parameter increasing as $-r$. These are the ingoing radial null geodesics.