Linear Form of Canonical Gravity

TL;DR

This paper presents a formulation of general relativity and complex general relativity where all constraint equations are linear in terms of multimomenta, simplifying the structure of the theory and aiding in quantum gravity quantization.

Contribution

It introduces a linear form of the constraint equations in both real and complex general relativity using jet bundle formalism, extending the framework to complex manifolds.

Findings

Constraint equations are linear in multimomenta in real general relativity.

Extension to complex general relativity yields holomorphic equations linear in multimomenta.

Provides a foundation for quantizing gravity theories using this linear formalism.

Abstract

Recent work in the literature has shown that general relativity can be formulated in terms of a jet bundle which, in local coordinates, has five entries: local coordinates on Lorentzian space-time, tetrads, connection one-forms, multivelocities corresponding to the tetrads and multivelocities corresponding to the connection one-forms. The derivatives of the Lagrangian with respect to the latter class of multivelocities give rise to a set of multimomenta which naturally occur in the constraint equations. Interestingly, all the constraint equations of general relativity are linear in terms of this class of multimomenta. This construction has been then extended to complex general relativity, where Lorentzian space-time is replaced by a four-complex-dimensional complex-Riemannian manifold. One then finds a holomorphic theory where the familiar constraint equations are replaced by a set of equations linear in the holomorphic multimomenta, providing such multimomenta vanish on a family of two-complex-dimensional surfaces. In quantum gravity, the problem arises to quantize a real or a holomorphic theory on the extended space where the multimomenta can be defined.