The graded Lie algebra of general relativity

Michael Reiterer, Eugene Trubowitz

Published 2018-12-30Version 1

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.