Metric Reconstruction in Kerr Spacetime

TL;DR

This paper improves the metric reconstruction method in Kerr spacetime by providing an efficient iterative scheme for the sourced linearized Einstein equations, enhancing perturbative solutions and clarifying relations to gravitational radiation quantities.

Contribution

It introduces a new iterative integration scheme for the GHZ metric reconstruction method applicable to sourced equations in Kerr spacetime.

Findings

Developed an efficient iterative scheme for metric reconstruction.

Established relations between Hertz potential tails and gravitational radiation.

Provided methods for transforming perturbations to Lorenz gauge.

Abstract

Metric reconstruction is the general problem of parameterizing GR in terms of its two ``true degrees of freedom'', e.g., by a complex scalar ``potential'' -- in practice mostly with the aim of simplifying the Einstein equation (EE) within perturbative approaches. In this paper, we re-analyze the metric reconstruction procedure by Green, Hollands, and Zimmerman (GHZ) [Class. Quant. Grav. \textbf{37}, 075001 (2020)], which is a generalization of the Chrzanowski-Cohen-Kegeles (CCK) approach. Contrary to the CCK method, that by GHZ is applicable not only to the vacuum, but also to the sourced linearized Einstein equation (EE). Our main innovation is a version of the GHZ method giving an efficient integration scheme for the initial value problem of the sourced linear EE. By iteration, our scheme gives the metric to as high an order in perturbation theory around Kerr as one might wish, in principle. At each order, the metric perturbation is a sum of a corrector, obtained by solving a triangular system of transport equations, a reconstructed piece, obtained from a Hertz potential as in the CCK approach, and an algebraically special perturbation, determined by the ADM quantities. As a by-product, we determine the precise relations between the asymptotic tail of the Hertz potential in the GHZ and CCK schemes, and the quantities relevant for gravitational radiation, namely, the energy flux, news- and memory tensors, and their associated BMS-supertranslations. We also discuss ways of transforming the metric perturbation to Lorenz gauge.