A Lie-Rinehart algebra in general relativity

TL;DR

This paper constructs a Lie-Rinehart algebra related to Einstein's equations using a BV-BFV approach, revealing new algebraic structures in the initial value problem of general relativity.

Contribution

It introduces a generalized Lie-Rinehart algebra for Einstein's equations, extending previous Lie algebroid constructions with a BV-BFV framework.

Findings

Constructed a Lie-Rinehart algebra over initial data for Einstein's equations.

Connected the algebraic structure to the invariance under space-time diffeomorphisms.

Provided a new perspective on the constraint algebra in general relativity.

Abstract

We construct a Lie-Rinehart algebra over an infinitesimal extension of the space of initial value fields for Einstein's equations. The bracket relations in this algebra are precisely those of the constraints for the initial value problem. The Lie-Rinehart algebra comes from a slight generalization of a Lie algebroid in which the algebra consists of sections of a sheaf rather than a vector bundle. (An actual Lie algebroid had been previously constructed by Blohmann, Fernandes, and Weinstein over a much larger extension.) The construction uses the BV-BFV (Batalin-Fradkin-Vilkovisky) approach to boundary value problems, starting with the Einstein equations themselves, to construct an $L_\infty$-algebroid over a graded manifold which extends the initial data. The Lie-Rinehart algebra is then constructed by a change of variables. One of the consequences of the BV-BFV approach is a proof that the coisotropic property of the constraint set follows from the invariance of the Einstein equations under space-time diffeomorphisms.