Long wavelength iteration of Einstein's equations near a spacetime singularity

TL;DR

<3-5 sentence high-level summary>The paper analyzes the long-wavelength iteration scheme for Einstein's equations in the approach to a spacetime singularity, clarifying its connections to the Belinski–Khalatnikov–Lifshitz (BKL) oscillatory picture and Tomita's antinewtonian scheme. It derives the generic first-order solution for a barytropic perfect fluid in a synchronous frame, and then develops a generic third-order correction, while rigorously addressing the regime of validity and the role of curvature and velocity terms. In the spherical symmetry reduction, the authors show that the first-order solution reproduces Kasner-like behavior and, for dust, connects to the Tolman–Bondi solution, illustrating gauge choices and the impact of the equation of state. The work highlights that curvature and velocity terms generally limit the scheme's applicability near singularities, but under specific conditions (e.g., $\mu^1_{\ 23}=0$ or sufficiently large $\Gamma$) the approach recovers BKL dynamics and provides a concrete, gauge-aware framework for analyzing approach to singularities and gravitational collapse.

Abstract

We clarify the links between a recently developped long wavelength iteration scheme of Einstein's equations, the Belinski Khalatnikov Lifchitz (BKL) general solution near a singularity and the antinewtonian scheme of Tomita's. We determine the regimes when the long wavelength or antinewtonian scheme is directly applicable and show how it can otherwise be implemented to yield the BKL oscillatory approach to a spacetime singularity. When directly applicable we obtain the generic solution of the scheme at first iteration (third order in the gradients) for matter a perfect fluid. Specializing to spherical symmetry for simplicity and to clarify gauge issues, we then show how the metric behaves near a singularity when gradient effects are taken into account.