= Affine focusing model for the Penrose theorem
{title2=$\theta'=-\theta^2/2-Q,\quad Q=\widehat\sigma_{ab}\widehat\sigma^{ab}+R_{ab}k^ak^b$}
The <Null Raychaudhuri equation> is the light-ray version of the <Raychaudhuri equation> associated with <Amal Kumar Raychaudhuri>. His original work concerned relativistic <cosmology>.<Amal Kumar Raychaudhuri by Jayant Vishnu Narlikar>[research discussion, PDF pp. 4–6, the 1955 paper Relativistic Cosmology]{ref} For the normal <null geodesic congruence> before its first focal point, <null twist> vanishes. With an <affine parameter> $\lambda$, the four-dimensional equation reduces to the displayed <ordinary differential equation>. The <null shear> contraction is nonnegative on the transverse <screen space>, and the <null convergence condition> makes the other part of $Q$ nonnegative. Consequently $\theta'\leq-\theta^2/2$.
For $\theta(0)=\theta_0<0$, put $w=-1/\theta>0$. Then
$$
w'=\frac{\theta'}{\theta^2}\leq-\frac12,
\qquad w(\lambda)\leq\frac1{|\theta_0|}-\frac\lambda2.
$$
A regular negative <null expansion> therefore cannot persist beyond $\lambda=2/|\theta_0|$ if the <null geodesic> extends that far. The <null focusing theorem> turns this estimate into a focal-point bound. The number refers to an <affine parameter>, not the <proper time> of a light ray, which is zero.
For an original soluble model, prescribe constant $Q$ along one twist-free beam; this does not specify an entire solution of the <Einstein field equations>. Introduce a positive affine scale $L$ and the dimensionless variables
$$
u=\frac\lambda L,\qquad y=L\theta,\qquad q=L^2Q,
\qquad \frac{dy}{du}=-\frac{y^2}{2}-q,\quad y(0)=-2.
$$
For an infinitesimal area $A$ define $a=\sqrt{A/A_0}$ before the first zero. Since $y=2a'/a$, where primes now mean $d/du$, the same model becomes
$$
a''=-\frac q2a,\qquad a(0)=1,\quad a'(0)=-1.
$$
Two independently checkable solutions are
$$
\begin{aligned}
q=0:&\quad a=1-u,\qquad y=-\frac2{1-u},\qquad u_*=1,\\
q=2:&\quad a=\cos u-\sin u,\qquad y=-2\frac{\sin u+\cos u}{\cos u-\sin u},\qquad u_*=\frac\pi4.
\end{aligned}
$$
The nonnegative extra focusing term moves the first zero earlier. Direct substitution into both <ordinary differential equations> verifies the solutions; numerical integration of the nonlinear equation provides a separate comparison with the linear beam equation.
\Image[/penrose-affine-raychaudhuri-focusing.png]
{title=Finite affine focusing in two explicitly defined Raychaudhuri models}
{description=Both beams start with $L\theta_0=-2$. Curves show the exact solutions for $q=0$ and $q=2$; circles independently integrate $dy/du=-y^2/2-q$. The first zeros occur at $\lambda=L$ and $\lambda=\pi L/4$, respectively. These are illustrative <ordinary differential equation> models, not fitted collapse data.}
A focal point makes this beam description degenerate, but individual <null geodesics> may continue smoothly through it. Even converging light in <Minkowski spacetime> can form a caustic. Converting local focusing into <null geodesic incompleteness> requires the global step below.
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