Cutoff-Stable Null Convergence and Directional Rigidity in Bianchi I Spacetimes

Abstract

We derive an endpoint-free optical rigidity theorem and use it to separate two rigidity regimes in Bianchi I spacetimes, without assuming that the spatial metric is diagonal in a fixed basis. For a smooth, regular, twist-free null congruence, the affine Raychaudhuri equation expresses a finite null-Ricci integral as an expansion boundary term minus a nonnegative optical bulk. If the past and future cutoffs are removed independently and their two-end liminf is nonnegative, two-sided completeness forces the optical tensor and the null Ricci contraction to vanish pointwise. In Bianchi I this freezes the spatial metric on the fixed kernel of the conserved covector. We classify the resulting saturation geometry: a non-static metric has zero, one, or two unoriented saturated lines, equivalently zero, two, or four oriented rays, and a third line forces staticity. Under the stronger pointwise null convergence condition, the existence of a single two-sided complete null geodesic already forces the spatial metric to be constant and the cosmic-time interval to be all of $\mathbb R$; on the Cartesian universal cover the spacetime is Minkowski. Periodic models attain the four-ray saturation bound in the weaker cutoff-stable regime, with $I_p^{\mathrm{ind}}=-\infty$ in every nonsaturated direction, while pointwise null convergence fails on open time intervals. Matter and achronal averaged-null-energy consequences are stated under an explicit matching assumption on field equations and cutoff prescriptions.