# The graded Lie algebra of general relativity

## TL;DR

This paper constructs a graded Lie algebra framework for vacuum Einstein equations, linking algebraic structures with the analysis of general relativity and introducing a gauge-fixing algorithm for perturbative solutions.

## Contribution

It introduces a novel graded Lie algebra construction for general relativity and a gauge-fixing method that simplifies the analysis of Einstein equations using homological algebra.

## Key findings

- Maurer-Cartan equation equivalent to vacuum Einstein equations
- Gauge-fixing algorithm produces contractions to smaller complexes
- Framework enables homological algebra application to Einstein equations

## Abstract

We construct a graded Lie algebra $\mathcal{E}$ in which the Maurer-Cartan equation is equivalent to the vacuum Einstein equations. The gauge groupoid is the groupoid of rank 4 real vector bundles with a conformal inner product, over a 4-dimensional base manifold, and the graded Lie algebra construction is a functor out of this groupoid. As usual, each Maurer-Cartan element in $\mathcal{E}^1$ yields a differential on $\mathcal{E}$. Its first homology is linearized gravity about that element. We introduce a gauge-fixing algorithm that generates, for each gauge object $G$, a contraction to a much smaller complex whose modules are the kernels of linear, symmetric hyperbolic partial differential operators. This contraction opens the way to the application of homological algebra to the analysis of the vacuum Einstein equations. We view general relativity, at least at the perturbative level, as an instance of `homological PDE' at the crossroads of algebra and analysis.