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For a quadratic derivative source $A^{\alpha \beta }{\partial }_{\alpha }u{\partial }_{\beta }u$, the classical null condition is $A^{\alpha \beta }{\ell }_{\alpha }{\ell }_{\beta }=0$ whenever $-{\ell }_0^2+|{\ell }_{\mathrm{sp}}{|}^2=0$. It removes the leading interaction of parallel lightlike derivatives. In three dimensions this structure gives global smooth solutions for sufficiently small, localized data through the vector field method for wave equations. This PDE condition is distinct from the null condition asserting that a single vector is lightlike.

For the round sphere, every normal has only $v,r$ components because the Reissner-Nordstrom metric has no mixed angular terms. The vector $X=-{\partial }_r$ is normal and null: $g_{rr}=0$. Write the second null vector normal as $Y=A{\partial }_v+B{\partial }_r$. The normalization gives $X\cdot Y=-A=-1$, so $A=1$. Its null condition then reads $-f+2B=0$. ThusThe specified future orientation of $X$, together with $X\cdot Y<0$, makes $Y$ future-directed as well. This choice fixes the reciprocal scaling freedom of the two null vectors on the sphere.

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.

An outgoing radial null geodesic of the Vaidya metric prescribed as $r=r_0(v)$ determines the mass along that curve by $M(v)=r_0(v)(1-2r_0^{\prime }(v))/2$. This inverse relation comes from the radial null condition. It determines a function of advanced time wherever the prescribed curve is differentiable and has positive radius.