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

Abstract

Vacuum perturbations of the Kerr metric can be reconstructed from the corresponding perturbation in
either of the two Weyl scalars ? 0 or ? 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 et al. for circular orbits and
extend them to noncircular orbits.

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Identifiers

ORCID for Amos Ori: orcid.org/0000-0003-4742-9413

ORCID for Adam Pound: orcid.org/0000-0001-9446-0638

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