Null Completeness and Directional Rigidity in Bianchi I Spacetimes
Abstract
In Einstein gravity, every null geodesic of a non-static Bianchi I spacetime satisfying the null energy condition is incomplete in at least one direction within the Bianchi I region. We prove a stronger directional rigidity theorem: a single null geodesic complete in both affine directions forces staticity if the null curvature is nonnegative along it and, at each time, in one transverse null direction. The spatial metric need not be diagonal, and the expansion rates may have either sign. The proof uses caustic-free null congruences whose conserved spatial covectors define fixed screen hyperplanes. The two curvature conditions admit averaged replacements along the geodesic and, when cosmic time is complete, along comoving worldlines. In a non-static metric, at most four oriented directions are optically saturated; every other complete null geodesic has strictly negative averaged null curvature. Periodic and finite-pulse examples attain this bound with NEC-violating sources and nonpositive total energy density. For finite pulses, we derive bounds on metric excursions from three directional curvature budgets at fixed cosmic duration, and on the mode mixing of a free, massless, minimally coupled test scalar field from the averaged null-energy deficit at fixed affine duration.
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BibTeX
@misc{zhou2026null,
title = {{Null Completeness and Directional Rigidity in Bianchi I Spacetimes}},
author = {Ye Zhou and Alan Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.29556},
eprint = {2608.29556},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29556},
}
RIS
TY - UNPB
TI - Null Completeness and Directional Rigidity in Bianchi I Spacetimes
AU - Ye Zhou
AU - Alan Zhang
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2608.29556
UR - https://arxiv.org/abs/2608.29556
AB - In Einstein gravity, every null geodesic of a non-static Bianchi I spacetime satisfying the null energy condition is incomplete in at least one direction within the Bianchi I region. We prove a stronger directional rigidity theorem: a single null geodesic complete in both affine directions forces staticity if the null curvature is nonnegative along it and, at each time, in one transverse null direction. The spatial metric need not be diagonal, and the expansion rates may have either sign. The proof uses caustic-free null congruences whose conserved spatial covectors define fixed screen hyperplanes. The two curvature conditions admit averaged replacements along the geodesic and, when cosmic time is complete, along comoving worldlines. In a non-static metric, at most four oriented directions are optically saturated; every other complete null geodesic has strictly negative averaged null curvature. Periodic and finite-pulse examples attain this bound with NEC-violating sources and nonpositive total energy density. For finite pulses, we derive bounds on metric excursions from three directional curvature budgets at fixed cosmic duration, and on the mode mixing of a free, massless, minimally coupled test scalar field from the averaged null-energy deficit at fixed affine duration.
ER -