New Supplementary Conditions for a Non-Linear Field Theory: General Relativity

TL;DR

This paper introduces a new family of gauge conditions for general relativity involving fifth-order derivatives, expanding the tools for analyzing non-linear field equations and their applications in quantum gravity.

Contribution

It develops a novel class of gauge conditions based on geometric structures, extending previous gauges like de Donder and Lorenz, with applications to curved backgrounds and quantum gravity.

Findings

New gauge conditions involving fifth-order derivatives.

Admissibility of these gauges in linearized and curved backgrounds.

Potential applications to Euclidean quantum gravity.

Abstract

The Einstein theory of general relativity provides a peculiar example of classical field theory ruled by non-linear partial differential equations. A number of supplementary conditions (more frequently called gauge conditions) have also been considered in the literature. In the present paper, starting from the de Donder gauge, which is not conformally invariant but is the gravitational counterpart of the Lorenz gauge, we consider, led by geometric structures on vector bundles, a new family of gauges in general relativity, which involve fifth-order covariant derivatives of metric perturbations. A review of recent results by the authors is presented: restrictions on the general form of the metric on the vector bundle of symmetric rank-two tensor fields over space-time; admissibility of such gauges in the case of linearized theory about flat Euclidean space; generalization to a suitable class of curved Riemannian backgrounds, by solving an integral equation. Eventually, the applications to Euclidean quantum gravity are discussed.