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For a hypersurface-orthogonal null congruence under the null energy condition, an initially negative expansion ${\theta }_0$ diverges to $-\infty$ within affine distance at most $2/|{\theta }_0|$.

In the orthonormal frame choose the null vector $k=e_0+e_2$, which has a nonzero angular component. The curvature forms giveFor a null vector, the trace term in the Einstein field equations drops out, so the null energy condition implies $R_{ab}{k}^a{k}^b\ge 0$. Therefore

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Contracting the Einstein field equations with the null tangent eliminates the trace term and givesby the null energy condition. The screen metric is positive definite, so ${\hat{\sigma }}_{ab}{\hat{\sigma }}^{ab}\ge 0$. The Null Raychaudhuri equation consequently impliesWhile $\theta <0$ this is equivalent toIf $heta({\lambda }_0)={\theta }_0<0$, integration givesThe right-hand side reaches zero after affine distance $2/|{\theta }_0|$. A finite negative expansion cannot pass through this value, so $heta\to -\infty$ no later than that point. This is the null focusing theorem.

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Apply the time-reversed Penrose singularity theorem. The two past-directed null congruences orthogonal to the compact anti-trapped surface have negative expansion. The null energy condition, through the Einstein field equations, supplies the null convergence condition, and the null focusing theorem forces each generator to acquire a conjugate point within finite affine length if it can be extended that far.

If every past-directed null generator were complete, the boundary of the causal past of the surface would therefore be generated only for a bounded affine interval. Compactness of the initial surface and continuous dependence of geodesics on initial data would make that achronal boundary compact. A globally hyperbolic spacetime provides a Cauchy hypersurface and a timelike flow projecting the boundary onto it. The standard Penrose argument then makes its image both open and closed, forcing the connected Cauchy hypersurface to be compact. This contradicts its stipulated topology ${\mathbb{R}}^3$.

Hence at least one past-directed null generator ends after finite affine parameter: the universe is null-geodesically incomplete to the past. Global hyperbolicity controls the causal boundary, the noncompact spatial topology supplies the contradiction, and the energy condition supplies focusing.

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The two future null directions are proportional to $k_{+}=(1,1)$ and $k_{-}=(1,-1)$. The null energy condition requiresEquivalently,

Solved by gpt-5.6-sol high.