Gauge Invariant Frames of Bianchi Models in Separated Form
Abstract
Bianchi spacetimes are usually written in separated form: metric coefficients depend only on cosmic time, while spatial dependence is carried by an invariant coframe. In this representation, geometric quantities and the Einstein equations reduce to a time-dependent system. In practice, however, observer- and gravitational-wave-adapted gauges mix time and space (e.g., from integrability-adapted to synchronous comoving coordinates). Homogeneity is preserved, but separability is no longer manifest, which complicates computation. To address this, we give a gauge-invariant reconstruction of separated frames using only the spatial Killing algebra. Given a basis {ξa}, we construct an invariant spatial frame {Xi} from [ξa,Xi]=0, then dualize to {ei}. The metric then takes the canonical form g=γμν(t) eμeν in any gauge. We prove diffeomorphism covariance of the construction, and demonstrate computational gain in a Bianchi II example. The framework recovers canonical adapted-gauge constructions (in the sense of Jantzen) while remaining stable under arbitrary Diff(M) gauge transformations, and is directly suited to gravitational-wave Bianchi models and related exact solutions of recent interest of which we give a example of the Bianchi VII type.
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BibTeX
@misc{verweg2026gauge,
title = {{Gauge Invariant Frames of Bianchi Models in Separated Form}},
author = {Dave B. H. Verweg and Robbert W. Scholtens},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31887},
eprint = {2610.31887},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31887},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Gauge Invariant Frames of Bianchi Models in Separated Form
AU - Dave B. H. Verweg
AU - Robbert W. Scholtens
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31887
UR - https://arxiv.org/abs/2610.31887
AB - Bianchi spacetimes are usually written in separated form: metric coefficients depend only on cosmic time, while spatial dependence is carried by an invariant coframe. In this representation, geometric quantities and the Einstein equations reduce to a time-dependent system. In practice, however, observer- and gravitational-wave-adapted gauges mix time and space (e.g., from integrability-adapted to synchronous comoving coordinates). Homogeneity is preserved, but separability is no longer manifest, which complicates computation. To address this, we give a gauge-invariant reconstruction of separated frames using only the spatial Killing algebra. Given a basis {ξ_a}, we construct an invariant spatial frame {X_i} from [ξ_a,X_i]=0, then dualize to {e_i}. The metric then takes the canonical form g=γ_μν(t) e^μe^ν in any gauge. We prove diffeomorphism covariance of the construction, and demonstrate computational gain in a Bianchi II example. The framework recovers canonical adapted-gauge constructions (in the sense of Jantzen) while remaining stable under arbitrary Diff(M) gauge transformations, and is directly suited to gravitational-wave Bianchi models and related exact solutions of recent interest of which we give a example of the Bianchi VII type.
ER -