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For radial motion, take an affine parameter $\lambda$ and use dots for $d/d\lambda$. The radial null geodesics of the Vaidya metric satisfy the two radial geodesic equationsThe radial squared norm is $-f{\dot{v}}^2+2\dot{v}\dot{r}$. If $v$ is constant it vanishes, and the geodesic equation reduces to $\ddot{r}=0$. Thus constant-$v$ radial curves are null geodesics, with $r=a\lambda +b$ an affine parametrization. Future ingoing motion has $a<0$.

For the other family let $dr/dv=f/2$. Its tangent $k={\partial }_v+(f/2){\partial }_r$ is a null vector. Using the Christoffel symbols of the Vaidya metric, direct differentiation givesHence these are also null geodesics, but $v$ is generally not affine. For completeness, put $p=dv/d\lambda$ and chooseThis makes $\ddot{v}=-(M/r^2){\dot{v}}^2$. Differentiating $\dot{r}=(f/2)\dot{v}$ then gives $\ddot{r}=(f_v/2){\dot{v}}^2=-M^{\prime }{\dot{v}}^2/r$; substitution verifies the radial $r$ equation as well. Thus the null condition and both radial geodesic equations hold, not just the null condition alone.

Rearranging the outgoing radial null geodesics of the Vaidya metric equation along $r=r_0(v)$ givesFor the prescribed parametrization by $w$, the chain rule yieldsSince $n>1$, $v/w=1+2\beta w^{n-1}\to 1$ as $w\to 0^{+}$. The asymptotic equivalence $w\sim v$ therefore impliesMore precisely $w=v+O(v^n)$ and $M(v)=(\beta /2)v^n[1+O(v^{n-1})]$, using Big O notation. The positivity of $dv/dw$ justifies treating $w$ as a single-valued function of $v$ locally.