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Null dust is matter with a stress-energy tensor $T_{ab}=\epsilon k_a{k}_b$ for a future-directed null vector $k$. For every future-directed timelike vector $t$, the energy current is $-T^a{}_b{t}^b=\epsilon (-k\cdot t)k^a$. Therefore its dominant energy condition holds exactly when $\epsilon \ge 0$, allowing the zero current. Rescaling $k$ rescales $\epsilon$ inversely by the square and leaves the stress-energy tensor unchanged. Ingoing Vaidya metric matter has $k=-{\partial }_r$, $k_a=-(dv{)}_a$, and $\epsilon =\dot{m}/(4\pi r^2)$.

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.