Spacetime closedness theorem for homogeneous Lorentzian foliations
Abstract
Motivated by the geometric interpretation of spatially homogeneous cosmological models, we formulate a spacetime closedness theorem directly at the Lorentzian level. A classical space-form classification theorem organizes homogeneous and isotropic spatial geometries, but by itself it does not yield a Lorentzian statement about the ambient spacetime. The main technical step is to derive finite slice-volume from genuinely Lorentzian control hypotheses on the foliation. This is achieved in a strong version, for maximally regular globally hyperbolic (n+1)--spacetimes with a finite-time Big Bang and homogeneous complete spacelike slices, and in a weaker version in which maximal regularity is replaced by time-integrability of the accumulated expansion rate. In both cases, the argument separates an analytic step, deriving finite slice-volume from the Big Bang and temporal control, from a geometric step, upgrading finite volume to compactness by homogeneity and completeness. The theorem is stated in arbitrary spacetime dimension and is accompanied by a Lean~4 formalization of the strong and weak abstract statements.
Versions
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BibTeX
@misc{monjo2026spacetime,
title = {{Spacetime closedness theorem for homogeneous Lorentzian foliations}},
author = {Robert Monjo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09181},
eprint = {2609.09181},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09181},
}
RIS
TY - UNPB
TI - Spacetime closedness theorem for homogeneous Lorentzian foliations
AU - Robert Monjo
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.09181
UR - https://arxiv.org/abs/2609.09181
AB - Motivated by the geometric interpretation of spatially homogeneous cosmological models, we formulate a spacetime closedness theorem directly at the Lorentzian level. A classical space-form classification theorem organizes homogeneous and isotropic spatial geometries, but by itself it does not yield a Lorentzian statement about the ambient spacetime. The main technical step is to derive finite slice-volume from genuinely Lorentzian control hypotheses on the foliation. This is achieved in a strong version, for maximally regular globally hyperbolic (n+1)--spacetimes with a finite-time Big Bang and homogeneous complete spacelike slices, and in a weaker version in which maximal regularity is replaced by time-integrability of the accumulated expansion rate. In both cases, the argument separates an analytic step, deriving finite slice-volume from the Big Bang and temporal control, from a geometric step, upgrading finite volume to compactness by homogeneity and completeness. The theorem is stated in arbitrary spacetime dimension and is accompanied by a Lean~4 formalization of the strong and weak abstract statements.
ER -