Filtered expansions in general relativity II

TL;DR

This paper develops a mathematical framework to construct solutions to Einstein's equations with inhomogeneous perturbations, supporting the BKL conjecture about spacetime behavior near singularities.

Contribution

It introduces a spectral sequence and homological contraction techniques to show that inhomogeneous perturbations are unobstructed in the Einstein equations.

Findings

Spatially inhomogeneous perturbations are unobstructed

Spectral sequence method applied to Einstein equations

Homological contraction based on gauge-fixing used

Abstract

This is the second of two papers in which we construct formal power series solutions in external parameters to the vacuum Einstein equations, implementing one bounce for the Belinskii-Khalatnikov-Lifshitz (BKL) proposal for spatially inhomogeneous spacetimes. Here we show that spatially inhomogeneous perturbations of spatially homogeneous elements are unobstructed. A spectral sequence for a filtered complex, and a homological contraction based on gauge-fixing, are used to do this.