We construct a Lie-Rinehart algebra over an infinitesi-mal 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 ex-tension.) The construction uses the BV-BFV (Batalin-Fradkin-Vilkovisky) approach to boundary value problems, starting with the Einstein equations themselves, to construct an L∞-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 invari-ance of the Einstein equations under space-time diffeomorphisms.

### A Lie-Rinehart algebra in general relativity

#### Abstract

We construct a Lie-Rinehart algebra over an infinitesi-mal 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 ex-tension.) The construction uses the BV-BFV (Batalin-Fradkin-Vilkovisky) approach to boundary value problems, starting with the Einstein equations themselves, to construct an L∞-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 invari-ance of the Einstein equations under space-time diffeomorphisms.
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2023
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11571/1486475`
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