Completion of metric reconstruction for a particle orbiting a Kerr black hole

TL;DR

This paper develops a rigorous method to determine the missing stationary, axially-symmetric part of metric perturbations caused by a particle orbiting a Kerr black hole, crucial for accurate self-force calculations.

Contribution

It introduces a new technique to find the completion piece of metric perturbations in Kerr spacetime, extending previous methods to non-circular orbits.

Findings

Method successfully determines the completion piece for eccentric orbits.

Provides a rigorous foundation for previous results on circular orbits.

Extends metric reconstruction techniques to more general orbital configurations.

Abstract

Vacuum perturbations of the Kerr metric can be reconstructed from the corresponding perturbation in either of the two Weyl scalars $\psi_0$ or $\psi_4$, using a procedure described by Chrzanowski and others in the 1970s. More recent work, motivated within the context of self-force physics, extends the procedure to metric perturbations sourced by a particle in a bound geodesic orbit. However, the existing procedure leaves undetermined a certain stationary, axially-symmetric piece of the metric perturbation. In the vacuum region away from the particle, this "completion" piece corresponds simply to mass and angular-momentum perturbations of the Kerr background, with amplitudes that are, however, a priori unknown. Here we present and implement a rigorous method for finding the completion piece. The key idea is to impose continuity, off the particle, of certain gauge-invariant fields constructed from the full (completed) perturbation, in order to determine the unknown amplitude parameters of the completion piece. We implement this method in full for bound (eccentric) geodesic orbits in the equatorial plane of the Kerr black hole. Our results provide a rigorous underpinning of recent results by Friedman {\it et al.}\ for circular orbits, and extend them to non-circular orbits.