New Gauge Conditions in General Relativity: What Can We Learn from Them?

TL;DR

This paper explores new gauge conditions in general relativity derived from conformally invariant constructions, involving covariant derivatives and compensating terms, with existence proofs on various manifolds.

Contribution

It introduces a framework for constructing conformally invariant gauge conditions in Einstein theories, including existence proofs on Lorentzian manifolds.

Findings

Existence of conformally invariant gauges on positive-definite manifolds.

Extension of gauge construction to Lorentzian manifolds with integral equation solutions.

Technical methods for imposing gauges in curved spacetime.

Abstract

The construction of conformally invariant gauge conditions for Maxwell and Einstein theories on a manifold M is found to involve two basic ingredients. First, covariant derivatives of a linear gauge (e.g. Lorenz or de Donder), completely contracted with the tensor field representing the metric on the vector bundle of the theory. Second, the addition of a compensating term, obtained by covariant differentiation of a suitable tensor field built from the geometric data of the problem. If the manifold M is endowed with an m-dimensional positive-definite metric g, the existence theorem for such a gauge in gravitational theory can be proved. If the metric g is Lorentzian, which corresponds to general relativity, some technical steps are harder, but one has again to solve integral equations on curved space-time to be able to impose such gauges.