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

TL;DR

This paper explores the connections between a long wavelength iteration scheme of Einstein's equations, the BKL solution near singularities, and the antinewtonian scheme, clarifying their regimes of applicability and behavior near singularities.

Contribution

It establishes the links between different approximation schemes of Einstein's equations and derives the generic solution near singularities for perfect fluids.

Findings

Derived the first iteration solution for matter near singularities.

Clarified the applicability regimes of long wavelength and antinewtonian schemes.

Analyzed metric behavior near singularities with gradient effects in spherical symmetry.

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.