= Solution
The hypersurface generators have zero <null twist>. Contracting the <Einstein field equations> twice with the null tangent removes the trace term, and the <null energy condition> gives $R_{ab}U^aU^b=8\pi T_{ab}U^aU^b\geq0$. Since the squared <null shear> is nonnegative, the <Null Raychaudhuri equation> implies
$$
\frac{d\theta}{d\lambda}\leq-\frac12\theta^2.
$$
While $\theta<0$ this is equivalent to
$$
\frac{d}{d\lambda}\left(\frac1\theta\right)\geq\frac12.
$$
Starting from $\theta(0)=\theta_0<0$, the right-hand side reaches zero no later than $\lambda=2/|\theta_0|$. The reciprocal expansion must therefore vanish and
$$
\boxed{\theta\longrightarrow-\infty
\quad\text{within affine distance }2/|\theta_0|}.
$$
This is the <null focusing theorem>.
Back to article page
