A Lie-Rinehart 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 ∞ -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.

Generic features of a polymer quantum black hole

“…The existence of such a large ambiguity in the construction of these models should not be surprising on general grounds [17,18]. On the upper side this freedom could prove useful in the investigation of off-shell hypersurface deformation algebra [50][51][52]. Our analysis can also be applied to specific choices of non-sine polymer functions.…”

Emergent modified gravity

“…(Mathematically, the metric dependence can be formulated by using the notion of algebroids, in which (1)-( 3) are realized as brackets on sections of a fiber bundle over a suitable space of metrics. However, as shown recently [8] in an application of general mathematical results to the case of hypersurface deformations, the corresponding bracket, placed in the language of BRST/BFV gauge generators, cannot be Lie but is L ∞ . )…”

“…More generally, we may try to modify the classical expression of T(N) before we apply a linear transformation (8). For instance, as suggested in models of loop quantum gravity [9,20], we could replace the quadratic dependence on curvature components in T(N) by non-classical polynomials or even non-polynomial functions, motivated by quantum-gravity considerations.…”

“…(There are parameter choices in the higher-order modifications that imply additional non-k 2 -dependent terms in T(N), but we will not consider them here.) Moreover, for solutions of the constraints (as opposed to their off-shell gauge behavior that is important for demonstrating general covariance of the modified theories) the linear transformation (8) does not make a difference. We therefore obtain the classical static solutions for the non-zero e 1 and e 2 as well as just as they appear in the Schwarzschild line element.…”

Abelianized Structures in Spherically Symmetric Hypersurface Deformations

“…Only in this case can the theory be considered a geometrical effective theory of some deeper and as yet unknown quantum space-time, just as different dynamical versions of gravity given by higher-curvature effective actions make use of the same Riemannian form of space-time. Because of its importance for covariance and the classification of meaningful effective theories, we will review the structure of hypersurface deformations in the beginning of our first section below, combining classic results from gravitational physics with more recent mathematical developments [5,6].…”