Pure and Applied Mathematics Quarterly

Volume 19 (2023)

Number 4

Special Issue in honor of Victor Guillemin

Guest Editors: Yael Karshon, Richard Melrose, Gunther Uhlmann, and Alejandro Uribe

A Lie–Rinehart algebra in general relativity

Pages: 1733 – 1777

DOI: https://dx.doi.org/10.4310/PAMQ.2023.v19.n4.a2


Christian Blohmann (Max Planck Institute for Mathematics, Bonn, Germany)

Michele Schiavina (Dipartimento di Matematica, Università di Pavia, Italy)

Alan Weinstein (Department of Mathematics, University of California, Berkeley, Calif., U.S.A.; and Department of Mathematics, Stanford University, Stanford, California, U.S.A.)


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.

2010 Mathematics Subject Classification

37Kxx, 53D20, 83C05

Dedicated to Victor Guillemin, whose work has inspired us for many years

M.S. acknowledges support from the NCCR SwissMAP, funded by the Swiss National Science Foundation. Part of the research has been carried out while he was a visiting scientist at the Max Planck Institute for Mathematics, Bonn.

Received 30 January 2022

Received revised 23 December 2022

Accepted 2 February 2023

Published 20 November 2023