Metric reconstruction in a Bondi-like gauge
Abstract
Metric perturbations obtained from a Hertz potential in an ingoing radiation gauge (IRG) generically grow in an outgoing coordinate r at future null infinity, and are therefore not asymptotically flat in the Bondi sense. Hollands and Toomani showed that this growth can be removed, within the Green--Hollands--Zimmerman (GHZ) framework, by an IRG-compatible ``residual'' gauge transformation together with a carefully chosen choice of the undetermined integration ``constants'' of the Hertz potential. In this work we turn that proof into a practical algorithm. We show that this entire completion is generated by a single ``Held Master Scalar'' (HMS), which can be thought of as a function on future null infinity. The HMS is obtained from the leading asymptotic piece of the $0$-Weyl scalar through a Teukolsky--Starobinsky inversion. We give explicit formulas for the four integration constants of the Hertz potential associated with the $0$-Weyl scalar, and for the residual gauge vector in terms of the HMS, thereby bringing the metric perturbation into a Bondi-like gauge. In Schwarzschild the entire construction is diagonal in (ℓ, m)-modes and every completion field is available in closed form; in Kerr the construction is tri-diagonal. We have numerically implemented the full scheme in (ℓ,m) and in m-modes, including the corrector piece of the GHZ framework. We illustrate our scheme numerically for a point-particle on a circular orbit in a puncture scheme in Schwarzschild and in Kerr, by demonstrating the Bondi-like fall-off. We also find agreement with the known Detweiler redshift computations in Schwarzschild and Kerr. Our method directly gives the Bondi news, mass-, and angular momentum aspects, for which we give exemplary results in Kerr.
Related
- Scalaron Quasinormal and Quasi-Bound Modes of Schwarzschild Black Holes in the Bondi-Sachs Formulation of Metric f(R) Gravity
- Kerr metric perturbations in Lorenz gauge via numerical relativity methods
- Polar-metric/axial-gauge perturbations of the Maldacena-Milekhin-Popov wormhole
- Axial-metric/polar-gauge perturbations of the Maldacena-Milekhin-Popov wormhole
- Information-preserving boundary reconstruction for center-flux energetics in a regulated SU(3) gauge ladder
BibTeX
@misc{casals2026metric,
title = {{Metric reconstruction in a Bondi-like gauge}},
author = {Marc Casals and Stefan Hollands and Vahid Toomani and Peter Zimmerman},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06367},
eprint = {2611.06367},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06367},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Metric reconstruction in a Bondi-like gauge
AU - Marc Casals
AU - Stefan Hollands
AU - Vahid Toomani
AU - Peter Zimmerman
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06367
UR - https://arxiv.org/abs/2611.06367
AB - Metric perturbations obtained from a Hertz potential in an ingoing radiation gauge (IRG) generically grow in an outgoing coordinate r at future null infinity, and are therefore not asymptotically flat in the Bondi sense. Hollands and Toomani showed that this growth can be removed, within the Green--Hollands--Zimmerman (GHZ) framework, by an IRG-compatible ``residual'' gauge transformation together with a carefully chosen choice of the undetermined integration ``constants'' of the Hertz potential. In this work we turn that proof into a practical algorithm. We show that this entire completion is generated by a single ``Held Master Scalar'' (HMS), which can be thought of as a function on future null infinity. The HMS is obtained from the leading asymptotic piece of the $0$-Weyl scalar through a Teukolsky--Starobinsky inversion. We give explicit formulas for the four integration constants of the Hertz potential associated with the $0$-Weyl scalar, and for the residual gauge vector in terms of the HMS, thereby bringing the metric perturbation into a Bondi-like gauge. In Schwarzschild the entire construction is diagonal in (ℓ, m)-modes and every completion field is available in closed form; in Kerr the construction is tri-diagonal. We have numerically implemented the full scheme in (ℓ,m) and in m-modes, including the corrector piece of the GHZ framework. We illustrate our scheme numerically for a point-particle on a circular orbit in a puncture scheme in Schwarzschild and in Kerr, by demonstrating the Bondi-like fall-off. We also find agreement with the known Detweiler redshift computations in Schwarzschild and Kerr. Our method directly gives the Bondi news, mass-, and angular momentum aspects, for which we give exemplary results in Kerr.
ER -