A New Family of Gauges in Linearized General Relativity

TL;DR

This paper introduces a new family of gauges in linearized general relativity inspired by conformal invariance properties in Maxwell theory, involving higher-order derivatives and non-local constructions to achieve conformal invariance.

Contribution

It proposes a novel class of gauges based on fifth-order derivatives, proves their admissibility in flat and curved backgrounds, and constructs a non-local tensor to ensure conformal invariance.

Findings

Admissibility of new gauges in flat spacetime established

Construction of a non-local tensor for conformal invariance

Restrictions on the DeWitt supermetric parameter derived

Abstract

For vacuum Maxwell theory in four dimensions, a supplementary condition exists (due to Eastwood and Singer) which is invariant under conformal rescalings of the metric, in agreement with the conformal symmetry of the Maxwell equations. Thus, starting from the de Donder gauge, which is not conformally invariant but is the gravitational counterpart of the Lorenz gauge, one can consider, led by formal analogy, a new family of gauges in general relativity, which involve fifth-order covariant derivatives of metric perturbations. The admissibility of such gauges in the classical theory is first proven in the cases of linearized theory about flat Euclidean space or flat Minkowski space-time. In the former, the general solution of the equation for the fulfillment of the gauge condition after infinitesimal diffeomorphisms involves a 3-harmonic 1-form and an inverse Fourier transform. In the latter, one needs instead the kernel of powers of the wave operator, and a contour integral. The analysis is also used to put restrictions on the dimensionless parameter occurring in the DeWitt supermetric, while the proof of admissibility is generalized to a suitable class of curved Riemannian backgrounds. Eventually, a non-local construction is obtained of the tensor field which makes it possible to achieve conformal invariance of the above gauges.