Local continuity laws on the phase space of Einstein equations with sources

TL;DR

This paper introduces a new covariant method to derive local continuity laws for Einstein equations with sources, using adjoint operators, leading to conserved quantities and gauge-invariant currents expressed via Debye potentials.

Contribution

It presents a novel approach based on adjoint differential operators to obtain covariant conservation laws for Einstein-Maxwell and Einstein-Weyl solutions, enhancing understanding of phase space structures.

Findings

Derived covariant continuity equations for Einstein solutions with sources.

Established gauge-invariant conserved currents expressed through Debye potentials.

Connected the continuity laws to bilinear forms and conserved quantities on phase space.

Abstract

Local continuity equations involving background fields and variantions of the fields, are obtained for a restricted class of solutions of the Einstein-Maxwell and Einstein-Weyl theories using a new approach based on the concept of the adjoint of a differential operator. Such covariant conservation laws are generated by means of decoupled equations and their adjoints in such a way that the corresponding covariantly conserved currents possess some gauge-invariant properties and are expressed in terms of Debye potentials. These continuity laws lead to both a covariant description of bilinear forms on the phase space and the existence of conserved quantities. Differences and similarities with other approaches and extensions of our results are discussed.