 Area-square-root optical focusing equation

For an affinely parametrized twist-free null geodesic congruence, let $A$ be local transported screen area and ${\sigma }^2={\hat{\sigma }}_{AB}{\hat{\sigma }}^{AB}/2$. The Null Raychaudhuri equation gives ${\theta }^{\prime }=-{\theta }^2/2-2{\sigma }^2-R_{ab}{k}^a{k}^b$ and $A^{\prime }/A=\theta$. Combining these gives the displayed equation. Positive shear focuses the beam and has a negative sign. For nonuniform expansion this is a local area identity, not the same differential equation for the entire integrated horizon area.