# The mass and angular momentum of reconstructed metric perturbations

## TL;DR

This paper proves that certain linear vacuum perturbations of the Kerr metric, constructed via the CCK formalism, have zero Abbott-Deser mass and angular momentum, linking these perturbations to the particle's energy and angular momentum.

## Contribution

It demonstrates that the Abbott-Deser integrals vanish for CCK-derived perturbations from regular Fourier modes of the Hertz potential, generalizing previous results on metric completion.

## Key findings

- Abbott-Deser mass and angular momentum integrals vanish for these perturbations.
- The mass and angular momentum perturbations correspond to the particle's energy and angular momentum outside the orbit.
- Perturbations inside the orbit vanish, simplifying the understanding of metric completion.

## Abstract

We prove a key result regarding the mass and angular momentum content of linear vacuum perturbations of the Kerr metric obtained through the formalism developed by Chrzarnowski, Cohen, and Kegeles (CCK). More precisely we prove that the Abbott-Deser mass and angular momentum integrals of any such perturbation vanish, when that perturbation was obtained from a regular Fourier mode of the Hertz potential. As a corollary we obtain a generalization of previous results on the completion of the `no string' radiation gauge metric perturbation generated by a point particle. We find that for any bound orbit around a Kerr black hole, the mass and angular momentum perturbations completing the CCK metric are simply the energy and angular momentum of the particle "outside" the orbit and vanish "inside" the orbit.