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arXiv · 2606.17071

Geometric Phase-Space Structure in Cosmological Solutions of Einstein's Field Equations

Hassan Ugail
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Abstract

Einstein field equations allow cosmological dynamics to depart from the Friedmann-Lemaitre-Robertson-Walker (FLRW) idealisation in several physically different ways. Matter may become spatially inhomogeneous, the local expansion scalar may vary across a hypersurface, the expansion may acquire anisotropic components through shear, and the free gravitational field may be encoded in nonzero Weyl curvature. The key question is not only how far a model is from FLRW, but which geometric mechanism is responsible. A single departure from FLRW number cannot distinguish these mechanisms. This paper introduces a compact geometric diagnostic framework that keeps them separate while using standard quantities in general relativity. The framework is observer-explicit and domain-explicit, intended as a practical tool for comparing analytic and numerical solution families rather than as a new invariant classification of spacetime. Buchert's kinematical backreaction is retained as a derived explanatory quantity rather than a separate axis, since it is already fixed by the expansion-variance and shear contributions. A single curvature normalisation is used for all Weyl diagnostics. The method is tested on six benchmarks, namely FLRW, Bianchi-I, Kasner, Lemaitre-Tolman-Bondi dust, scalar-perturbed FLRW, and tensor-perturbed FLRW. These benchmarks occupy distinct regions of the diagnostic space, and the magnetic Weyl contribution appears only in the tensor case. The classification remains stable under changes of perturbation amplitude, spatial resolution, averaging domain, constraint reliability, and a leading-order observer tilt. The curvature expressions for the exact benchmarks are verified symbolically against metric-derived Weyl invariants, and the supporting computer code, numerical results, tables, and figures are publicly available.

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Hassan Ugail. 2026-06-08. Geometric Phase-Space Structure in Cosmological Solutions of Einstein's Field Equations. https://arxiv.org/abs/2606.17071

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