PreprintCC BY 4.0arXiv
Failure of local equi-Lipschitzness for families of Lorentz distances to Cauchy surface foliations
Abstract
The families of Lorentzian distance functions to and from points along a complete timelike line in a spacetime are known to be locally equi-Lipschitz continuous in a neighborhood of the line. This is an essential component in the proof of the classical Lorentzian splitting theorems. We show that this property fails in general for families of Lorentzian distances to and from the level sets of a given Cauchy temporal function. Moreover, we formulate conjectures based on the existence of Cauchy temporal functions in cosmological and timelike geodesically complete spacetimes such that the Lorentz distances to its level sets are equi-Lipschitz, which are equivalent to Bartnik's splitting conjecture.
Related
BibTeX
@misc{galloway2026failure,
title = {{Failure of local equi-Lipschitzness for families of Lorentz distances to Cauchy surface foliations}},
author = {Gregory J. Galloway and Robert J. McCann and Argam Ohanyan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.21997},
eprint = {2606.21997},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.21997},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Failure of local equi-Lipschitzness for families of Lorentz distances to Cauchy surface foliations
AU - Gregory J. Galloway
AU - Robert J. McCann
AU - Argam Ohanyan
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2606.21997
UR - https://arxiv.org/abs/2606.21997
AB - The families of Lorentzian distance functions to and from points along a complete timelike line in a spacetime are known to be locally equi-Lipschitz continuous in a neighborhood of the line. This is an essential component in the proof of the classical Lorentzian splitting theorems. We show that this property fails in general for families of Lorentzian distances to and from the level sets of a given Cauchy temporal function. Moreover, we formulate conjectures based on the existence of Cauchy temporal functions in cosmological and timelike geodesically complete spacetimes such that the Lorentz distances to its level sets are equi-Lipschitz, which are equivalent to Bartnik's splitting conjecture.
ER -