Zero null optics supplies the missing null Ricci condition

Fix any regular parameter ${\lambda }_0$ and work locally on its spacelike section $S_{{\lambda }_0}$, restricting to an embedded section patch if necessary. Rebase the same congruence at that section by using $s=\lambda -{\lambda }_0$ and relabeling each generator by its point on $S_{{\lambda }_0}$. Its tangent is still $\mathrm{\ell }$, its parameter is still affine, and its section directions are still orthogonal to $\mathrm{\ell }$. Under the identification of $T{S}_{{\lambda }_0}$ with the screen at ${\lambda }_0$, its screen Jacobi map is ${\tilde{A}}_s=A_{{\lambda }_0+s}{A}_{{\lambda }_0}^{-1}$. Since the right factor is fixed in $s$, ${\tilde{B}}_s={\tilde{A}}_s^{\mathrm{\prime }}{\tilde{A}}_s^{-1}=B_{{\lambda }_0+s}$. Apply the initial-section identification of [F1] to this rebased congruence: at $s=0$, its optical expansion and shear correspond to the trace and trace-free part of ${\chi }_{\mathrm{\ell }}$ on $S_{{\lambda }_0}$. These vanish by the assumed zero second fundamental form. As the regular point and section were arbitrary, $\theta =0$ and $\sigma =0$ throughout the open congruence neighborhood, with all constructions taken on the chosen parameter-domain branch.

The smooth function $\theta$ vanishes on this open neighborhood, so its affine derivative also vanishes there. Substitution of $\theta =0$, $\sigma =0$ and ${\theta }^{\mathrm{\prime }}=0$ into the identity in [F2] gives $\mathrm{Ric}(\mathrm{\ell },\mathrm{\ell })=0$ throughout that neighborhood. The open-neighborhood hypothesis is what justifies taking the derivative of the identically zero expansion; vanishing at one isolated section would not suffice.

At the specified section, pointwise bilinearity of the covariant two-tensor in [F3] gives $\mathrm{Ric}(f\mathrm{\ell },f\mathrm{\ell })=f^2\mathrm{Ric}(\mathrm{\ell },\mathrm{\ell })=0$ for every smooth nonzero rescaling $L=f\mathrm{\ell }$. This pointwise rescaling needs no affine extension of $L$.