# Filtered expansions in general relativity I

## TL;DR

This paper develops a formal power series approach to solving vacuum Einstein equations, focusing on one bounce in inhomogeneous spacetimes, using a graded Lie algebra and homological methods.

## Contribution

It introduces a novel perturbative framework with a filtration that encodes BKL features, enabling systematic solutions to Einstein's equations.

## Key findings

- Constructed formal solutions with one bounce in inhomogeneous spacetimes
- Established an unobstructed perturbative problem using a graded Lie algebra
- Implemented a filtration capturing key BKL features

## Abstract

This is the first 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. We use a graded Lie algebra, homological framework. A dedicated filtration encodes key features of the BKL proposal, and we use it to set up an unobstructed perturbative problem.