= Solution
For a curve with fixed $v,\theta,\phi$, 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:
$$
\Gamma^a{}_{rr}=\frac12g^{ab}(2\partial_rg_{br}-\partial_bg_{rr})=0,
$$
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 $-r$. \b[These are the ingoing radial null geodesics.]
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