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We prove a key result regarding the mass and angular momentum content oflinear vacuum perturbations of the Kerr metric obtained through the formalismdeveloped by Chrzarnowski, Cohen, and Kegeles (CCK). More precisely we provethat the Abbott-Deser mass and angular momentum integrals of any suchperturbation vanish, when that perturbation was obtained from a regular Fouriermode of the Hertz potential. As a corollary we obtain a generalization ofprevious results on the completion of the `no string' radiation gauge metricperturbation generated by a point particle. We find that for any bound orbitaround a Kerr black hole, the mass and angular momentum perturbationscompleting the CCK metric are simply the energy and angular momentum of theparticle "outside" the orbit and vanish "inside" the orbit.

The mass and angular momentum of reconstructed metric perturbations