= Area-square-root optical focusing equation
{title2=$\frac{(A^{1/2})''}{A^{1/2}}=-R_{ab}k^ak^b/2-\sigma^2$}
For an affinely parametrized twist-free <null geodesic congruence>, let $A$ be local transported screen area and $\sigma^2=\widehat\sigma_{AB}\widehat\sigma^{AB}/2$. The <Null Raychaudhuri equation> gives $\theta'=-\theta^2/2-2\sigma^2-R_{ab}k^ak^b$ and $A'/A=\theta$. 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.
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