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On the completeness of gravitational pp-wave spacetimes: A counterexample to the {Ehlers--Kundt} conjecture
Abstract
This paper concerns the geometry of gravitational pp-wave spacetimes. We give a counterexample to the Ehlers--Kundt conjecture. In fact, we show that geodesic completeness is generic, in the sense of Baire category, among gravitational pp-waves. We also define a concept of ``universal pp-wave'', i.e. a pp-wave whose metric approximates that of every other gravitational pp-wave arbitrarily closely on arbitrarily large compact sets, and we show that there are geodesically complete universal pp-waves. Our machinery also suffices to give a new proof of the polynomial case of the Ehlers--Kundt conjecture (previously proved by Flores and Sánchez).
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BibTeX
@misc{cairo2026completeness,
title = {{On the completeness of gravitational pp-wave spacetimes: A counterexample to the Ehlers--Kundt conjecture}},
author = {Hannah Cairo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30419},
eprint = {2609.30419},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30419},
}
RIS
TY - UNPB
TI - On the completeness of gravitational pp-wave spacetimes: A counterexample to the {Ehlers--Kundt} conjecture
AU - Hannah Cairo
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.30419
UR - https://arxiv.org/abs/2609.30419
AB - This paper concerns the geometry of gravitational pp-wave spacetimes. We give a counterexample to the Ehlers--Kundt conjecture. In fact, we show that geodesic completeness is generic, in the sense of Baire category, among gravitational pp-waves. We also define a concept of ``universal pp-wave'', i.e. a pp-wave whose metric approximates that of every other gravitational pp-wave arbitrarily closely on arbitrarily large compact sets, and we show that there are geodesically complete universal pp-waves. Our machinery also suffices to give a new proof of the polynomial case of the Ehlers--Kundt conjecture (previously proved by Flores and Sánchez).
ER -