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 $\{{\xi }_a\}$, we construct an invariant spatial frame $\{X_i\}$ from $[{\xi }_a,X_i]=0$, then dualize to $\{e_i\}$. The metric then takes the canonical form $g={\gamma }_{\mu \nu }(t)e^{\mu }{e}^{\nu }$ 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 $\mathrm{Diff}(M)$ 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.